Beijing Lectures in Harmonic Analysis-Stein-0691084181
Beijing Lectures in Harmonic Analysis-Stein-0691084181
Beijing Lectures in Harmonic Analysis-Stein-0691084181
Number 112
BEIJING LECTURES IN
HARMONIC ANALYSIS
EDITED BY
E. M. STEIN
PREFACE vii
NON-LINEAR HARMONIC ANALYSIS, OPERATOR THEORY
AND P.D.D.
by R.R. Coif man and Yves Meyer 3
INDEX 423
v
PREFACE
E. M. STEIN
vii
Beijing Lectures in
Harmonic Analysis
NON-LINEAR HARMONIC ANALYSIS,
OPERATOR THEORY AND P.D.E.
R. R. Coifman and Yves Meyer
00
F(f) = I Ak(f )
k=0
Ak(fI...fk):B1 xB1...xB1 B2
3
4 R. R. COIFMAN AND YVES MEYER
tial at 0 ).
We will concentrate our attention on very concrete functionals arising
in connection with differential equations or complex analysis, and would
like to prove that they depend analytically on certain functional parameters.
As you know there are two ways to proceed.
1. Expand in a power series and show that one has estimates (1).
2. Extend the functional to the complexification as "formally holo-
morphic" and prove some boundedness estimates.
Let L denote a differential operator like
a(x) dx xfR,
a(z)az zEC
ai j(x) x e Rn
La l+a -dx
with hall < 1 a(x), real valued.
Laf=\_d fh-110h=i
UhdxUhlf
where
Uhf =fh.
sgn d
dx) f = J
elx6sgn 6 (e)de = n P. V.
f
-00
f t dt = H(f
x-t
n sgn(La )f = nUhsgn li dx
a Ulf
h =pv J
r f(t)(l+a(t)
x-t+A(x)-A(t)
dt
-00
F(a) f =
f f(t) (1+ia+i$)
x-t+A(x) + iB(x) - A(t) - iB(t)
dt
f t)[(l+a)/(1+a)](1+a) dt
x+A(x)-t-A(t)+i(B x)-B(t))
= UhCUhif1
where
x-t x-t
-
Observe that the operators are of the form
IA xx_A t I
< M and T(f) = p.v. J (`A(xx-y (Y) f(y) dy
00
00
r 1 f t z' t dt
J z s- Z (t s-t
00 s-t
if we assume
* 0<a<lzs-ztl<1
s-t
2.
antisymmetric i.e.,
K(x,y) = -K(y,x) .
i1
(For example, K(x,y) (A(x) -Y (Y)) --y
-0 c C4, A' c L .) Recently
G. David and J.-L. Journe found a necessary and sufficient condition for
such operators to be bounded on L2 (or LP ). This condition is simply
that T(1) must be of bounded mean oscillation.
We now would like to state certain facts concerning B.M.O. and
prove their theorem.
Recall that b c BMO(R) if
1 /2
111 1 /2
rr *b12 dxtdt
JJ
I 0
1 /2 f 1 /2
lill Ib-m1(b)I 2
1/2
+
r
III f ID2-ml(b2)I 2
1 /2 1 /2
2(-L
II
JI
r1I2dx <c III fI
<cIIbII
<
f
Ix-y1>111
III dylIbll,. < CIIbII .
IX-YI2
Integrating in y we get
which shows that second term is bounded by CIIbII. We have thus shown
the necessity of the condition T(1) c BMO. Before stating the theorem
precisely we would like to reformulate it somewhat.
NON-LINEAR HARMONIC ANALYSIS 11
1+
(xY)2
(where we used the fact that ly-z I < t , Ix-ut < t, Ix-y l > 3t and the
hypothesis layK(x,y)I < ). Ix-yi-2
<T I2 ffK(z,u)Otu)Ot(z)-.At(z)ot(u))dzdul
but lp (u) t (z) -fir (u) (z) I < Iu and the fact that Iu-x I < t ,
t
1
t
Iu-zl < t, Ix-yj < 3t and IK(z,u)l <3 1 imply
lz-u I
,.Ot >I
THEOREM (G. David, J. L. Journe) [7]. Let T:`D - D' such that for some
E>0
<T,`t'St >1 <- t 1 I+E
= pt(u-v)
1+IuyI
t
and
where
and
1 E
k(x1,y)-k(x,y)I < Ix x
I I
for Ix-yI > 2Ix-xI I
E
x-yI
(This last condition followed from k(x,y) = -k(x,y) and 1, 2) then the
conditions * are verified. It can be shown that if * is valid then T
can be represented as a limit of integral operators whose kernels satisfy
the conditions 1, 2, 3 We will refer to 1, 2 as standard estimates,
and to 3 as the weak cancellation (or boundedness) property. This last
condition is independent of 1, 2, and can be proved in a variety of ways,
as we shall see.
To see how the theorem can be applied to reduce the degree of non-
linearity a of a "polynomial" and to obtain estimates consider
C(aIf)
n = ft!J,A(xX_A(Y)ln
! Y J
f(y)
Y
dY
Cn(a,l)(x) = Cn-I(a,a)(x)
Ptf=95t*f Qtf=qit*f.
1 /E
We'll study only the first term (since the second is its adjoint).
We start by observing that
pt 2g
t _2 ( ) = t 2 2(te)g(e)
where ip1(x) = xq5(x) and Q 1,tg = iA1,t *g. This permits us to rewrite
the first term as
00
dt
-2 f Q1 tLt'tf t
0
where
and
00
+ QL,tlLtPtf-Lt(1)Pt(f )Idt
J0
r
0
J <e J Ifl2dx
1 (f
4xtdt) ftxYxPt(f )(y)
< rTQ1,t(g)I2
- Pt(f )(x))dy
1 /2
dxdydt
(j//'Pt(x_Y)IPt(f)(Y)_Ptf )(x)12
1 /2
dudydt
fff Pt(u)IPt(f )(Y)-Pt(f )(Y+u)12 t
- dudedt1 /2
')I2leit1 1I2f(e)2
t
1/2
(fffpt(u) 1u S I$(te)12 Itel 8 2 du ddt
` t I
1^
(6)1 t
1/2 00 1/2
= P(u) us f I0(t)12ts dt (j(2de)1/2
J
o
As for the first term, we proceed as above and are led to estimate
1 /2
(f*bX)I2t*f2 d tdt
This is dominated by llf 112 if and only if (fit * b(x))2 d tdt is a Carleson
measure i.e., b is in B.M.O.
In fact, recall that dv(x,t) on R* is a Carleson measure if for each
interva 1 I
I =1(x,t) cR2:(x-t,x+t)<If.
I
NON-LINEAR HARMONIC ANALYSIS 17
00
J (art*b)(0t*f)dt
0
Since all the problems we will be considering are invariant under such
transformations, it is not surprising that the bilinear operations which
arise look like n.
18 R. R. COIFMAN AND YVES MEYER
F(f)=rr(f;F'(f))+e
where
The proof of this linearization formula is the "same" as for the T(1)
theorem and is achieved as follows.
00
00
is multilinear in the ti
* Ak(th,tz,...,tk)(e) =Ak(tI...tk)(e-h)
where
th(IA ) = ti(V, - h) .
ikie,thi e-ihki
Clearly if we take ti(9) = e ( = ti(O) we find using the
linearity in each argument that
k
A h(l ki)
ik19 ikke ik10 ikkO
e Ak(e , ...,e )(e) = Ak(e ...e )(e-h)
taking h = 0 we get
jk j1
where
such that
(1,..,N)2 -(1,...,N)
where
ffeix(e1+...ek)VeI...ek)(eI)f(e2)...f((k)deidek
Ak(fI...fk)(x) =
Rk
and
NON-LINEAR HARMONIC ANALYSIS 21
(Note that this realization is only valid for multilinear operations verify-
ing some mild continuity conditions.) This realization permits us for
example to show that the study of fo lAt * b (kt * fat can easily be con-
verted to the study of
0
(' t/it*b(kt*f dt _ ('eix(eI 2) (f(tei) 3(te2)dt 2
0
J (ow
5l'1(t(e1
where 0 I(() is any function equal to 1 on 1.9 < lel < 2.1 . In this
case we see that the two expressions are equal. This sort of analysis
comparing frequencies and interaction of various functions can be carried
out permitting one to understand the structure of some multilinear opera-
tions. We are thus led to consider the general question of studying multi-
linear multipliers. We state two known results.
22 R. R. COIFMAN AND YVES MEYER
maps L2 x x L2 L2 continuously.
This is an elementary result which we leave as an exercise. It
would be desirable to obtain more subtle results for L2.
then
A(fI...fk): LPIxLp2...xLpk,L4
k
>pi>1 >Pi
eix(e1+e2)x(41,
A(f, a) (x) = IJ e2)a((1)f(e2)del de2
[fk(x_tx_)a(t)] f(y)dy
where k(x 1,x2) = A(x1x2) satisfies Ik(x 1, x2)I < IVkI <
x12 +Cx22 ,
3
Ixi
IIT(e'xe)IIBMO < C .
III
24 R. R. COIFMAN AND YVES MEYER
In fact
lox-y(y)l
fIT(9t)(x)I <_ dy
-t
<
if lx I > 2 It l thus
I
Since T(4t) is in B.M.O. it follows that
thus we obtain
IT(Ot) ,At 15 t .
J
where
k
f ix(16i)
uk(x,t) = J e 1 ak(e,t)v(e1)... v(ek) del ... d6k
n-1
r3uk ,auk
3
at ax 3 r)x Lr uiuk-l
j=1
k
it e3
we find first that ak(e, t) = e 1 ak(e), and that
/k k
( k
13 k-1
ak(6)1 6i3 l ; J )=_3(e1) Y aj(61 ... ej) 6k)
1 j=1
under permutations of ek, so that the equality should be true only after
symmetrization.)
It can be shown (not so simply) that
e1+...+ek
k
uk(x,t) =13
i J 1 1 ;k-1 k i=l vx,t(ei)de1 ...d,k
If we let p. v. f --L- v,
uk(x,t) = iax J
and
I I- ' f T4,
d
NON-LINEAR HARMONIC ANALYSIS 27
ItbIL,. <E0.
ij 1
a2 -Llu=0
&2 /
or more generally for F(L) where F is a bounded holomorphic function
in a sector jarg zj < S containing the spectrum of L (this can be shown
to be true if e0 is sufficiently small).
We define formally
F(L)
2ni J
d
r
where r is the boundary of the sector.
28 R. R. COWMAN AND YVES MEYER
We let
R1= 92
ax
R =
I
i 1
2
-L "
+A-Lr bil
2
(I-BR) (C +A)
= fix) =
(C-L)-I = (C+A)-1(I-BR)-t
=
, (C+A)-t(BR)k
We thus find
00 00
F(L) = I J (C+A)-t(BRekF(C)dC = IAk(B) .
0 r 0
(A0(B)f) = J Q F(C)dI 1
2ni
1 F(C)dCf(C)
C_ICI2
r r
=F(IKI2)f(C) = m(C)f(C) .
00
AI(B) = f
t B a2 1 F(it) dt
it+A axiaxl itt+ t
J
00
J 1
B s2D;D 1 (s) as (s) = F(1/s2) .
i+s2t i+s2A
0
Now write
00
D.D.
This gives f 1 B 1 j (s) ds
s which we recombine with the other
0 1+s2O
term corresponding to t < 0 to get
D_J
.D
A0(B) B f
If'o 1+s2O
1
B
1+s 2O
1
G) s
ds DiDI
0'
and the spectrum is the whole complex plane. We are thus led to consider
for 0 e Co(C1) expressions of the form
=-2ai fx)1(idxAd.
C C
These formal expressions need to be given sense and one should prove that
(oq,)(L). In the case L0 = 2 such an expression is
easily justified using the Fourier transform. Since
d
1 dX A d f (e) = 21 4(X) 1
dX
21 1(f aX- X-L
C
ax X-6
= O(Of(e)
(here we used the fact that S(z), where S is the Dirac mass
tai az =
at 0 ). z
We see that
then
= L(ei(h(z)C+i )) =
LXC CXC
or more precisely
n
f e'{(h(z)-h(w))C+(z-w #
f(w)dw
and
32 R. R. COIFMAN AND YVES MEYER
)ei1(h(z)-h(w))C+(z-w)Z1dr
c(L)f ( f(w)dw
f1fo
fk(h(z)-h(w),z-w) , f(w)dw
where
k(u,v) = f)eidC
C
Lf=-1 fw dw-!!f.
L (z-w)2 0z
5.
-yy
Ck(aIf) _ J(A(x)_A(y))kf(y)d A(x) = a(x)
sati sfy
f
f(y)
dy
x-y+iA(x) - iA(y)
I A(x)-A(y)
i x-y
f(y)dy < ec lei IIA'Ilo
x-y IIf II2
L2
A(x)-A(y)
x-y
IITe(f)112 = fe (y) dy
x-y < C(IIeajl-+1)NIIf II2
2
P (
A(x)-A(y) f(y) dy
x-y ) x=y fc){fe xydy d
2 2
TA(f)= re dy
The main idea to estimate the BMO norm of T(1) is to replace inside
each interval I , TA by an operator TA where AI = A on a large
I
fraction of I and Ai has a smaller Lipschitz norm, and then compare
TA (1) to TA(1). This is achieved via the following lemma, the first of
I
which is the rising sun lemma (or the one-dimensional version of the
Calderon-Zygmund decomposition).
then for each I there exists a function AI and a constant CI such that
CI-3M<AI(x)< CI+3M.
Proof. We can assume C = M i.e., 0 < A' < M2. There are two cases:
a) mI(A') > M.
36 R. R. COIFMAN AND YVES MEYER
In that case consider the smallest function AI > A with A'(y) > 23 . We
Ik
"_.UIk 'fUlk
=2MIII-3MIUlkI
i.e.,
3M-3M<Ai(y)<43 +3M.
b) m1(A') < M.
NON-LINEAR HARMONIC ANALYSIS 37
4
A(x) = 2M(x-a) - A11 = Al
A'(x)=2M-AI=A1
3M-3M<Ail< 3 +3M
3M-3M<AI<3M+3M.
THEOREM (G. David). Assuming that there exist S > 0, c > 0 such that
for each I there exists KI(x,y) satisfying standard estimates uniformly
in I with
satisfying
IITI11L2(L),L2(I) < Co
Hx rE , Vy eE , K1(x,y) = K(x,y) .
38 R. R. COWMAN AND YVES MEYER
IITIIL,BMO C77 C0 .
<-
iMA(x)-A(y)
x-y
If we let a(M) = sup II f e x-y dylisMo a direct application
IIA-liar 1
of this theorem choosing for each interval I
A1(x)-A1(y)
im x-y
e
K1(x,y) = x-y
shows that
. A(x)-A(y) II
t x-y
e
IITf11 2= x-y f(y)dy L2 < C(1+IIA'II00IIflIL2
f IT(f1)-TI(f1)Idx
E
NON-LINEAR HARMONIC ANALYSIS 39
fIKxy)_Ki(xY)ldY
J
xEE' yd
IK(x,y)-K(x,Yi)+KI(x,Yi)-KI(x,Y)IdY
= J
xEE' I.i
where we have used the fact that K(x,y) = KI(x,y) outside Ii and yi
are endpoints of Ii, (K(x,yi) = KI(x,yi)). This integrand is dominated
by the Marcenkiewitz function of UIi . Consequently the integral is
bounded by c III enabling us to apply Lemma 1.
All of these results can be extended to Rn by various methods. The
easiest is the so-called method of rotation based on a general transfer-
ence principle, valid for multilinear operators commuting with translations,
and some nonlinear operators. We state the result in general although for
the case of Rn this is an easy application of Fubini's theorem.
=J
satisfying
40 R. R. COWMAN AND YVES MEYER
k
I IA(a; f)II, p<_ [I IIaiII,oIlfIIL
1=1
A(A,F) = J
frk(t1,t2. tk,s)[JAi(Ut .x)F(Usx)dt ds 1
satisfies
I}A(A,F)IILP(X)` 11 IIA'IIL(X)IIFIILP(X)
4f - c rf(0-t) 1t dt
J ctg t
If we take
k
Ak(a,f) = x-y f
f(A(x)_A(y)dy
SE--y
k
f(A(x)_A(x_t)) A? t dt
NON-LINEAR HARMONIC ANALYSIS 41
then
k
S2(e)
A(x)-A(x-te) f(x-te) dt
t t
f (A(x)-A(y)lk
x-y I K(x-y)f(y)dy where
is odd for K even
k(Y) _
WYIYI)
IYIn
6.
f(y) (7 A (X
(1-a(y))dy, A'=a
F(a)f=1: J y
0
We wish to find the smallest norm III III (i.e., the largest Banach space)
for which
IIA1(a)IIL2 L2 <_ Ca .
A(x)-A(y ) f(y )
AI(a)f =
f x-y xx-y
dy .
We have already shown that 1IA I(a)11 L2L2 < c Ila II, and it can easily be
(x-x 0)2 A I (a) ()(I) - 2(x -x 0) A I(a) ((y-y 0) XI) + A I (a) ((y-Y o)2 X I)
f(y)dy
1 f(y)dy = fT-_ A (x1X -A x-y
F0(a)f -f x-y (A(x)-A(Y)) _y
1 ,/A(x)-A(y))
x-Y for some q5 c C
1 - A(x) -A(y) 1`
x-y
f
00
aa)f . 217i
p.v.
z(s)-z(t) dt =
Ak(a) f
_ ,p
IIIaIII = IIAI(a)IIL2,L2
and one finds that this norm is equivalent to the BMO norm of a. On the
other hand it is easy to show that if IIaIIBMO < So then 1 -So <
Iz s -z t I<1 and this is precisely the condition permitting us to write
S -t - ,
S-t
Thus BMOis the natural space of holomorphy. As you recall from
S. Krantz's lecture one can express the Szego projection S(a), projecting
L2(1', ds) L2(R, ds) onto H+(I', ds) (H+ is the space of functions in
L2 admitting a holomorphic H2 extension to the "left" of T ) in terms
of the Cauchy operator. This representation yields the result that the
S(a) has BMO as its natural space of holomorphy and is an entire func-
tion on the manifold of chord-arc curves. (A similar result is true for the
Riemann mapping function.)
Another remarkable example leading to an "exotic" space of
holomorphy and to interesting geometry involves the functional calculus
in L = 1L - . As already seen it was necessary to assume that a E L
z
12
and a * E L with small norm. (This was obtained by considering the
z
example T(a) = L .) It turns out that T(a) is analytic in a , relative to
the norm IIIaIII = IIail0 + Ila * II.., and the condition really means that
a
Z2
there exists a bilipschitz map h(z):C -C such that ai h = a(z) + 1 .
NON-LINEAR HARMONIC ANALYSIS 45
REFERENCES
[1] A. P. Calderon, Cauchy Integrals on Lipschitz curves and related
operators. Proc. Nat. Acad. Sci., U.S.A. 75 (1977, 1324-1327.
[2] R.R. Coif man, D. G. Deng and Y. Meyer. Domaine de la racine
caree de certains operateurs differentiels accretifs. Ann. Inst.
Fourier 33, 2 (1983), 123-134.
[3] R.R. Coifman, Y. Meyer, Lavrentiev's curves and conformal map-
pings, Rep 5. 1983, Mittag Leffler Inst., Sweden.
[4] R.R. Coifman, A. McIntosh and Y. Meyer. L'integrale de Cauchy
definit un operateur borne sur L2 pour les courbes lipschitziennes.
Annals of Math. 116 (1982), 361-387.
[5] R. R. Coifman and Y. Meyer, Au deli des operateurs pseudodiffer-
entiels. Asterisque 57. Societe Mathematique de France (1978).
[6] G. David, Operateurs integraux singuliers sur certaines courbes du
plan complexe. Ann. Scient. Ec. Norm. Sup. 4 serie, 17 (1984),
157-189.
[71 G. David, J. L. Journe, "A boundedness Criterion for Calderon
Zygmund operators," Annals of Math. 120(1984), 371-397.
[8] E. Fabes, D. Jerison and C. Kenig, Multilinear Littlewood-Paley
estimates with applications to partial differential equations. Proc.
Nat. Acad. Sci. U.S.A. 79(1982), 5746-5750.
[9] R. Rosales, Exact Solutions of Some Nonlinear Evolution Equations,
Studies in Applied Math. 59, 117-151.
[10] E. M. Stein, Singular Integrals and Differentiability Properties of
Function, Princeton University Press (1970).
MULTIPARAMETER FOURIER ANALYSIS
Robert Fefferman
Introduction
The article which follows is an attempt to give an exposition of some
of the recent progress in that part of Fourier Analysis which deals with
classes of operators commuting with multiparameter families of dilations.
In some sense, this field is not that new, since already in the early 1930's
the properties of the strong maximal function were being investigated by
Saks, Zygmund, and others. However, for many of the problems in this
area which seem quite classical, answers have either not been found at
all, or only quite a short time ago, so that our knowledge of the area is
still fragmentary at this time.
The article is divided into six sections. The first treats some basic
issues in the classical one-parameter theory whose multiparameter theory
is then discussed in the remaining sections. Since the reader is no doubt
quite familiar with the main elements of the classical theory, we have
omitted references to the materials in section one. The book "Singular
Integrals and Differentiability Properties of Functions" by E. M. Stein is
an excellent reference for virtually all of the material there.
Finally, it is a pleasure to thank Professors M. T. Cheng and
E. M. Stein for all of their hard work in organizing the Summer Symposium
in Analysis in China, as well as many others whose generous hospitality
made the visit to China such a very enjoyable one.
47
48 ROBERT FEFFERMAN
a
Rn
into 2n congruent subcubes. Select from these the cubes Q' such that
1 f, Qf > a . For these Q' we stop the bisection process. For the
10
rest, we continue until we first arrive at a cube Q' such that 1 f ,f > a
I Q
at which point we stop.
The cubes Q' at which we stop are then our Qk. By construction
1
f > a. Let Qk be the cube containing Qk which was bisected
IQkI fQk
Qk. Then and since we did not stop at k,
1IkIf1k f < a. It follows that
MULTIPARAMETER FOURIER ANALYSIS 49
1Q k 1
f f < 1Q k
k1 1 f f< 2na ,
Qk
proving (1). Notice that (3) follows from (1) because 1Qk' < a fQk f so
summing on k, we have
('
IUQk1<a Y f f
Qk Rn
Proof. (1) Let x c Qk. Then there exists a ball B(x;r) such that
x c Qk C B(x;r) and B(xx;rl < Cn. Then
k
IQkl 1
Mf(x) >_ 1 Ifl > IfI > 1 a
B(x;r)
,/
B(x;r)
(IX)l) IV Cn
Qk
ff I f+ I f
B(x;r) B(x,r)fld[UQk] Qi
< ajB(x;r)I }
Qi
QiflBj$
That is, Mf(x) < Cna.
MULTIPARAMETER FOURIER ANALYSIS 51
(2) if Tf(x) = fRn K(x,y)f(y)dy for f c LP(X), and suppose for some
g(x) = Qk
f(x) if x / Qk .
Tbk(x) = J K(x,y)bk(y)dy .
Qk
Tbk(x) = IK(x,y)-K(x,yk)Ibk(y)dy
J
Qk
and
diam(Q )s
ITbk(x)IN <_ +3 Ilbkll Lt (x)
Ix-yk1
diam(Qk)s
Ilb k II t dx < CII f II t
f IT(b)(x)IN <k Y. f dist(x,Qk)n , L L (X)
X/Uijk x'!k
Thus
if *
S2(f)(x) t(y)I2 anal
=ff
r(x)
where r(x) _ {(y,01 Iy-xI < ti. Then it is a basic fact that 'IIS(f)IILp <
CPIIfIILP and Ilg(f)IILp <C when 1 < p < oo. If, say, Vi is
Ilf 11
Lpf
P
suitably non-trivial, (radial, non-zero is good enough) then the reverse
inequalities hold:
and K satisfies
+i
L <C InI1
IK(x+h)-K(x)I, if IhI < IxI
IxI 2
and we also have IIf * K11L(L<_ CIIf II since If * ot(x)I < II-kII1 IIf II
L
Then Mf(x) -If* K(x)I , so M is bounded on Lp(Rn) , p'> 1 and
weak 1-1.
is enough to show that T*f(x) = sup If * KE(x)I satisfies the weak type
E>0
estimate IIT*f(x) > all < a f If I . It turns out that by using the Hardy
R
Littlewood maximal operator it is not difficult to prove T*f(x) < CIM(Tf)(x)
+ Mf(x)l which immediately gives the boundedness of T* on Lp(Rn) for
p > 1 . However, it fails to give the weak type inequality for functions on
LI(Rn). This inequality follows easily from the observation that T* is
a singular integral.
Let
1 if IxI < 1
O(x) c C (Rn), O(x) =
0 if lxI >2
and H is
Ix
bounded from L2 - L2(L), so H is weak 1-1 .
1
IQ,
f if x c Q
k
Qjk
f(x) if x U Qk
1/2 1/2
If IIL2(Rn) = (Y IlojfIIL2)= II (Y lojf(x)12) I1L2
1 /2
Iojf(x)12) > cMSf(x)
C i 11
llf II
L(Iog L) k
This yields
00
00
<
J
1
r a
a I
MSf(x)>Cna
If(x)Idx (log a)k-1da
Qk
Stein operator. These operators all had one thing in common. They all
commute in some sense with the one-parameter family of dilations on Rn,
x - fix, 8 > 0. The nature of the real variable theory involved does not
seem to depend at all on the dimension n. In marked contrast, it turns
out that a study of the analogous operators commuting with a multi-
parameter family of dilations reveals that the number of parameters is
enormously important, and changes in the number of parameters drastically
change the results.
Let us begin by giving the most basic example, which dates back to
Jessen, Marcinkiewicz, and Zygmund. We are referring to a maximal opera-
tor on Rn which commutes with the full n-parameter group of dilations
(xl,x2, ,xn) - (Six1,82x2, ,Snxn), where Si > 0 is arbitrary. This
is the "strong maximal operator," M(n), defined by
M(2)(f8)(xl,x2) =M(1)(XIxII<8/2)(xl)M(1)(XIx2I<8/2)(x2) ti
I lI I I
and
I#x EQ0IM(2)(f8)>aI I ? I#xI Ix1I Ix2I < a , and 8< 1x11<1 II ti a log !
if a=1/8.
MULTIPARAMETER FOURIER ANALYSIS 59
R
ffif(xt.x2)idxidx2 = III f(th ff(xi.x2)dx2)dxi
(2.1)
R I J
so (2.1) is
M(2)f(x1,x2)
<- Mx Mx2(f )(xl,x2)
1
We have seen that Mx2 maps L(log L)(Qo) boundedly into L1(Q0) so
that
IIMx2fIIL1(Q0) - CIIfIIL(log L)(Q0)
and finally
Now, this method of iteration in the proof above gives sharp estimates
Now,
for M(n), and it may be suspected that the whole story of the harmonic
analysis of several parameters can be told by applying this iteration
technique. That this is not the case should become clear as this lecture
proceeds. We want to describe some of the multi-parameter theory and to
do this, let us begin with maximal functions. For many years after the
Jessen-Marcinkiewicz-Zygmund theorem, there was no machinery around to
treat problems here, and then, only fairly recently, two such machines were
created. The one we describe here proceeds by means of covering lemmas
while the other, due to Nagel, Stein, and Wainger, which Wainger has
described in detail, uses the Fourier transform [2]. Though the two
methods seem totally different on the surface, they are really quite closely
related and have in common the main theme of reducing higher parameter,
complicated operators to lower parameter simpler ones, which are already
well understood.
The model for our method is the following, where the operator. M(n) is
M(n-1 )
controlled by
Before we prove this theorem, let us show that it implies the Jessen-
Marcinkiewicz-Zygmund result. Let a'> 0, and for each point
x e IM(n)f(x) > at there is a rectangle Rx containing x with
(2.2) LIfI>a.
I
where the above union is taken over all the Rj , j <k for which Rj n R 4 0.
If the answer is no, we move on to consider the next rectangle on the list.
If the answer is yes, we make the rectangle R = Rk+l , and start the
process over again.
Now, we prove (1) as follows: If R is an unselected rectangle, then
Rn U (Rj)d >21R1.
kinR=c
L before R
kj
Rk n U (Rj)d
Wjnkk
j<k
(2.3) <
L)n-1
X k (k C II(bII L(log
so
\1 /(n-1)
f exp X
k/
1
S
0
fI
S0
Xgk0dxl..dxn-1 < I
k J 0 <C I
'k
'
IEkI-L
k
fO
<
<C
f
S0
M(n-1)(4))dxldx2...dxn-1 <C1II0II
L(log L) n-1
Notice that in our argument above the slicing was the most important
idea. If you try the proof without it, you will not wind up with the estimate
you want, on the exp( )1 /(n-1 )norm, but rather on the exp( ) 1 /n norm
instead. Also the slicing is the mechanism by which we control M(n) by
the lower parameter operator M(n-1) and here this enables us to proceed
by induction. Of course, in the end the theory of the boundedness of M(n)
had been known for some 40 years before the covering lemma. But the
lowering of the number of parameters, and the induction procedure will be
used in what follows as the key ingredient to prove new theorems.
64 ROBERT FEFFERMAN
This operator was considered following the ideas of the covering approach
outlined above by Stromberg [4] and Cordoba-Fefferman [5]. Somewhat
later Nagel, Stein, and Wainger [6] used Fourier transform methods to
extend the result we shall discuss below.
What we prove is that
IImf(x)>afl <
a IIfIIL2fR21
The proof consists of showing that, given a sequence of rectangles 3Rkl
belonging to S?j , there exists a subfamily {RkI such that
ti
To prove this we give a rule for selecting Rk, given that we have
k
already selected Rj for j < k. Assume that the Rk all have their
longest side in a direction in the 1st quadrant, and are ordered so thai
their longer side lengths are decreasing. Then consider the rectangle R
ti
following Rk_1- Consider in particular
II y XWkI2 = f X
j,kXk,Xkk =
2j
1
j
j<k
xAk +
k
IRkI
(2.4)
This is (2). To show (1), we let R be some rectangle which was un-
selected. This implies that
IRI
I
before R
IRfRjI > 1
.
In fact,
IRjnSI hH/B' h
IS 1
hL - LO'
and
IRjnRI ti h h O'-9 _ h
IRI h e(e'-0)
I$11 fL
S
& be re
R
. I I
JR before R
2
c
in other words
IUR;I _CIIYX&112<c'IURji
MULTIPARAMETER FOURIER ANALYSIS 67
Then
IIfIIp1P
;mf> all < CC .
a
IIfIIp}uRk)I/P.
a IIflIpiURkII"p <_ C
and the estimate on Ilmf > all follows by a division of both sides by
I U'k I I /P'
Our last topic for this lecture will be the so-called Zygmund Conjec-
ture. I believe it was Zygmund who was the first to realize the difference
between the one-parameter and several parameter harmonic analysis.
Particularly, he remarked that in differentiation theory a "big picture"
was evolving. He considered n functions (A I1 '021 ,'On of the positive
real variable t , with each Oi(t) increasing and the family of rectangles
68 ROBERT PEFFERMAN
f lf(x+y)I dy
M(f )(x) = u JR._
Rt
Then M is of weak type 1-1 , just as in the special case of the Hardy-
Littlewood operator where fi(t) = t. Zygmund noticed that the proof of
this was virtually the same as the Hardy-Littlewood theorem. All one had
to do was to prove a Vitali-type covering lemma for Rt's using the fact
that if is the class of all translates of the Rt and if R,S r 3 and
R R S 10 and if R corresponds to a bigger value of t than does S ,
then S C k, the 5-fold dilation of R. Next, he considered the collection
of rectangles Rs,t , s,t > 0 where
Not long ago, using the methods we have just discussed, Cordoba was
able to prove this [7]. Let us give the proof. Suppose IRkl is a
sequence of rectangles with side lengths s,t, and 0(s,t) in R3. We
must show there exists a subcollection 1Rk4 such that
(1) IURkI > c}URkf
To prove this, order the Rk so that the z side lengths are decreasing.
With no loss of generality, we may assume that IRkf1 ['Uk Rj11 < 2 (Rkl ,
that there are finitely many Rk and that the Rk are dyadic. (In fact, we
may assume this because if IRI fR IfI > a for some R e 91 containing x ,
then there exists a dyadic R1 whose R1 (double) contains x such that
IR' 1R1 +fj> C .) Now let R1 =R and, given R1, ,Rk, select Rk+l
1 fexP(x)dx<c.
(R j<k j
Rk be R1,
ti
We claim that the Rk satisfy
, RN and let
`x
j = Rj
i exp(I X
- U Rk. Then
) < C'. To see this let the
k
k>j
N N (' N-1
I exp X = r exp Xk )dx +J ex
X+...+ Jexxk=1
J 1
Ukj il N 'kN-1 t1
and
j
fexp X <C fex(kk=1
k
k<j }
so we have
1 fex(x)dx>C
1R I
ti
where the sum extends only over those chosen R k which precede R.
Let us slice R with a hyperplane in the xl,x2 direction. Call S,Si the
slices of R and R Then
1 fexP())dxidx2>C.
I S1
S
ti
(Again we sum only over those Sj which appear before S.) Now, each
Rj appearing before R has the property that its x3 (or z ) side length
ti
exceeds that of R. It follows that each corresponding Si . has either its
xl or x2 side length longer than that of S. Call those j havyng
longer x t side length than x 1 length of S of type 1, and the other of
type II. Put S = I x j . Then
Jbexp(,.+jX
X.9i)dxidx2 = I1111J1 f ,,dxldx2
C < II1 f f exp( I J II I
IxJ
so it follows that
III
I
exp
C X dxt -Lfexp
IJI
J
X-,
;}
dx2
MxI[exp(YXWAMx2Cexp(lXk A >C
on URA; hence
so
So far what has been done suggests the following general conjecture
of Zygmund which says: Let 0, i = be func-
tions which are increasing in each of the variables ti > 0 separately.
Define a k-parameter family of rectangles
Rt1 ,t2,...,tk
MJ)(f)(x) =
t1 t 2 supsk >o Rt1,...,tk
1 fR
t1, ,tk
lf(x+y)ldy
Then
following: say 2-k-1 < p1 < 2-k. Choose s e (2-k-1, 2-k) satisfying
'Ys P3 (since 1 < P3 < < 2k+1 we can do this). Then choose t
1(s) P2 - P2 - P1
so that t(Ai(s) = p2. This guarantees t-02(s) = p3, and we are finished.
01(2(ss))
We can make assume every value between 1 and C2k on
[2-k-1 2-k] bt letting 01(s) = e-1 /s and 42(s) = 1(s) on
2-k-1, 2 -k] and
L2 .
p-1
w-1/(p-1)dx <C.
IQI IQI ('
fwdx)
Q Q
r)_1 p-1
< (f W-1/(P-1) < rw-1/(P-1) <C
JE E JQ
and so
w>C.
JE
74 ROBERT FEFFERMAN
So far all the properties of AP weights listed are obvious and follow
straight from the definitions. There are some deeper properties which
though not difficult to prove are not immediate.:
(E) If w c AP then w satisfies a Reverse Holder Inequality:
1/(1+8)
IQI f
Q
w1+s
<
C8
IQI
fw
Q
for all cubes Q with 8 > 0. The constant CS may be taken arbitrarily
close to 1 as 8>0.
(C) From (E) it is immediate (see also (y)) that w c AP implies w c Aq
for some q < p.
(r)) If f is a locally integrable function in some LP space and
0 < a < 1 then (Mf )a c A 1 , i.e., M((Mf )a)(x) < C(Mf )a(x) (for w c A
implies w c AP for all p > 1 ).
To prove this let f e Lp(Rn) be given and a c (0,1). Let Q be a
cube centered at z, and Q its double. Then write f = XQf + XcQf =
f1 + f0. We must show that
and
If
)1/a
II ff1.
M(f1)adx iIdx
J J If
Q Z1
MULTIPARAMETER FOURIER ANALYSIS 75
C C
We have proven that for all x EQ, M(foxx) <A ICI fc Ifoldx so that
I1 G
lf <AaM(f)a(1) .
fM(fo)adx<Aa(_i_
Q C
Let us begin to discuss the weight theory by showing that the Hardy-
Littlewood maximal operator is bounded on LP(w) if and only if w e AP,
1 < p < - [121. In the first place if f = w-1 /(P-1) XQ and if M is
bounded on LP(w) one sees right away that
p
w(Q)
IQI f Q
w-1/(P-1)dx
<C f
Q
w-P/(P-1)wdx
or
76 ROBERT FEFFERMAN
IQ
f1
f- Ck and
k
Qj
P
1
f (Mf)Pwdx<C'Y w(QkCk<C'I w(Qk) (TQkl J
Rn k,j k,j j
P
<C- W(Q) k (Ql) 1
fo odx < by the AP condition on w
1
k, j IQk Qk
1
p
(*) < C "" o(Qk) 1 f (f o 1) odx where o = w-i /(p-1)
kj o(Qk) k
Qi
So far we have used only arithmetic. Now we come to the main point.
If Ek = Qk - U Q then choosing C large enough insures that
Q>k
IEkI > IQkI , and since a c AP u .E A we have o(Ek) > rlo(Qk)
J 2 j J J
and so
MULTIPARAMETER FOURIER ANALYSIS 77
where Mo(f )(x) =scupp g(Q) fQ IfIda. Now the same proof that works to
r
<C fPol-Pdx = j fPw dx .
Note that the operator M11 f(x) = sup 1 f IfId and its boundedness
x(Q (Q) Q
on LP(d) enter in a crucial way the proof of the weight norm inequalities
for the Hardy-Littlewood operator. This operator M is interesting in its
own right, since it is natural to ask what happens if we replace the God-
given Lebesgue measure by another measure d.
In fact, if is any measure finite on compact sets and if, in the
definition of M we insist that the balls be centered at x then M is
bounded on LP(), p > 1 . The proof of this remarkable theorem relies
on a refinement of the usual Vitali covering lemma due to Besocovitch.
We should also remark that the original proof of the weight norm inequali-
ties for M on Rn made use of M as well. In fact, if w c AP it is not
hard to see that
M(f )(x) < CMw(fP)I /p(x)
78 ROBERT FEFFERMAN
(Indeed,
1 /p /p11
(ffPwdx) (JrW_P'IPdXV'
I f f dx = I1 II
ffw'/Pw4/Pdx<_L 1Q1
Q Q Q Q
p (p-1)/p
w 1/p I fw_h/(P_1)dx) < CMw(fP)1/p(x) . )
1Q1 ( JfPw)
Q
I
Q
R a rectangle with sides parallel to the axes. Unlike the case where
n = 1 , restrictions must be placed on whether or not R is centered
at x .
The following result gives conditions on which are rather unre-
strictive, and which guarantee the boundedness on LP(IA) on Mn) [13]:
J wdx>r) f w dx
E It'JxJ
80 ROBERT FEFFERMAN
fw dx
f wdx>71'
I'xJ IxJ
IR n [U(Rk)d]I < 2 IR I
where the union is taken over all those k such that Rk precedes R and
RknR=0.
Now if we slice R , an unselected rectangle, with a hyperplane per-
pendicular to the xn direction we have
Therefore
< C J M(n-1)(O)wdx.
w
U SjBy induction
k - -9k - j<k is bounded on LP(w) so this
last integral is
IIX1j IILP(w)<
LP(w)
pth
This shows that III X3rk1I (w) < Cw(USk)l /P'. Raising this to the
LP
power and integrating in xn we have
REMARK. Given this covering lemma, cover the set IMwn)(f) > al by Rk
such that fw >a. Then we need only estimate w(URk) of
w(k) fRk
the covering lemma. But
IIf !I w(URk) 1 Jp .
2 LP(w)
82 ROBERT FEFFERMAN
The maximal operator is weak type (p,p), p > 1 and we are finished by
interpolation.
One application of this theorem is that with it, one can obtain weighted
norm inequalities for multi-parameter maximal operators which cannot be
handled directly through iteration. We give the following example.
Suppose % denotes the family of rectangles with side lengths of the
form s, t , and s and t > 0 are arbitrary. (Suppose
the sides are also parallel to the axes.) Define the corresponding maximal
operator M by
fMfPw<CJfpw.
The answer is AP(N) where this class is defined in the obvious way [14]:
P-1
w c AP(R) if and only if (_'_. rw <C
lR II
IwJJ
R
1/(1+)
fw(xlfrP)i4dxI) <C III fw(xiP)dxi).
I I
MULTIPARAMETER FOURIER ANALYSIS 83
E I. E I ExI
Since w c A(t), ffExl w < (1- 71) AR w and so fE W(p)dp < C(1-rl) fsW(P)dp ,
and by taking S small enough, C will be so close to 1 that C(1-71) < 1
and W is uniformly A on the collection of all such S. Therefore since
S is just 1-parameter (just a linear change of variable in one of the x2
or x3 variable away from squares) we have that W satisfies a reverse
Holder inequality: (For 8' some value < S )
1/(1+8')
ISI f W14dp <C ISI rW.
w S
and so
84 ROBERT FEFFERMAN
N
N
I I I I
0 1 2 4
Figure 2
Mef(x) = sup 1
XEREB0 IRI ,J
r if I dt
R
that the behavior of T9 on LP(R2) for p > 2 is linked with the be-
havior of Me on L(P/2)((p/2)' is the exponent dual to p/2 ). More pre-
cisely, suppose that To is bounded on LP(R2) and we assume the
weakest possible estimate on M0, namely J{MOXE > 2 I1 < CIEI. Then
Me is of weak type on L(P12) . Conversely, if Me is bounded on
L(P12) (R2) then TB is bounded on Lq(R2), for p'< q < p [16].
To prove this assume first that To is bounded on LP(R2). Then
the first step is to notice that this implies that
I(YITkfkl2) /2 (fkl2)1/2IILp(R2)
1ILP(R2)<CII
and
(lII
li ( Lp ` `Te(eTk xf k)I2/ II
' LP
iT x p \IIP
J
r f i Irk(t)T0(e k fk)(x) dtdx = TT (Irk(t) e
irk *x
fk)I dxdt
J
R2 0 0 R2
1
(' (' P /2:1
<C
fJ R2
J
0
k(x) dtdx<C' (IfkxI2)'
lam ll
LP
MULTIPARAMETER FOURIER ANALYSIS 87
Figure 3
88 ROBERT FEFFERMAN
I1 /2IIq )1/2IIq
IITOfIIq
Y
I{ (skTofI2) (IskTkf 12
fITkskfI20 f IT k(Skf
)12.b
But in R1 we have the classical weight norm inequality for the Hilbert
transform:
(1+00 I /p
IF(x +iy)Ipdx < C for all y c R+ .
00
One of the main reasons for introducing these spaces was the connection
with the Hilbert transform. If F(z) = u(z) + iv(z) is analytic with u and
v real, and if F is sufficiently nice then F will have boundary value
u(x) + iv(x) where v(x) is the Hilbert transform of u(x). It turns out,
since
+00
f
_
IF(x+it)Idx
increases as t 0, we have
+00
So we may view the space HI through its boundary values as the space
of all real valued functions f of LI(RI) whose Hilbert transforms are LI
as well.
90 ROBERT FEFFERMAN
n Cal
i (x, t) 0 (t = x 0)
i=0 1
and
dui dtrj
for all i,j .
J 1
cnt
u(x,t) = f * Pt(x); f(x) = u(x,0) and Pt(x) =
(Ix I 2 +t2)(n+1)12
Let F(x) _ ((y,t)l lx-yl < t(. Then since convolving with Pt at a point
x can be dominated by an appropriate linear combination of averages of
f over balls centered at x of different radii, it follows that
Applying this to G = [F la (which has JG1 'a(x,t)dx < C for all t > 0 )
we see that G* < M(h) for some h c L1 'a. Now M is bounded on
L1 Ia so that M(h) c L1 La and so G* c L1 Ia. It follows that F* a L1 .
Just as for a random f c L1(Rn) we do not necessarily have Rif c L1(Rn)
(singular integrals do not preserve L1 ) it is also not true that for an
arbitrary L1 function f that for u = P[f], u* c L1 . But if f c H1(Rn+1)
then u* c L1(Rn). Thus the nontangential maximal function
S2(u)(x)
= ff JVu12(y,t)dtdy
r(x)
IIu*IILP<00<1IF*IILP<00 FcHP(R+)
It turns out that there is another important idea which is very useful
concerning Hp spaces and their real variable theory. So far, we have
spoken of Hp functions only in connection with certain differential
equations. Thus, if we wanted to know whether or not f e Hp we could
take u = P[f] which of course satisfies Au = 0.
This is not necessary. If f is a function and 0 E C- (Rn) with
IR 4 n=1 , then we may form f* (x) = sup If *Ot(y)I , (kt(x)
(t,y)EF(x)
C n O(x/t) and if lr a C0 (Rn) is suitably non-trivial (say radial, non-zero)
and fci = 0 we may form
Rn
These are the basic facts of Hp spaces that will concern us here and
which we shall later generalize to product spaces.
Let us now prove that for a harmonic function u(x,t) in R++1
12
ltu*>Call < c S2(u)(x)dx +
a J
S(uYa
From this our claim follows. This is because for )Lg(a) = If Igi > all we
have
00 00 a
<
f
0
#xS(u)(,6) f aP3dadfl +
00 f aP_1As(aa
00
0 00 0
f0
OP-1's(u)(R)df3
NS(u)IIP .
To prove the estimate on Itu* > Calf , we set the notation that
M(XE) > 2 , and then claim
Then
f
S(u)<a
S2(u)(x)dx = J
xJS(u)aI
J ivul2(y,t)tl-ndydt
T(x)
dx
(4.1)
Ioul2(y,t)tl-nIIxl(y,t)tr(i),x/IS(u)>a#ildydt
= ff
R n+l
as claimed.
II. The next step is to write IVu12 = A(u2), and apply Green's theorem
to R :
ffA(U2)(y,t)t dydt = f !) t - u2 Alt da
R aR
Since, for purposes of all estimates we may assume that u is rather nice,
we may assume u(Vu)t vanishes at t = 0, so
MULTIPARAMETER FOURIER ANALYSIS 97
where OR is the part of aR above IS(u) > al. It is not hard to see that
lpult < a on aR so that
1 /2
(+
J Jul lpult da< all' /2 (f u2da
aR aR
aR S(u)<a
ti ti
III. Next we wish to define a function f by f( x) = u(x, r(x)) where
(x,r(x)) c aR defines the function r. We claim that in the region R
(4.2) lul<P[fl+Ca.
This is done by harmonic majorization. It is enough to show this on OR
and this in turn is just saying that for any point p c aR , JU(p)l is
dominated by the average over a relative ball on aR of u + Ca. This
follows from the estimate JVult < Ca on aR . Anyway, from (4.2) we
have, for x I IS(u) > al, u*(x) < CP[f ]*(x) + Ca , so that finally
The proof that IIu*IIp < CPIIS(u)IIp which we just gave has been lifted
from Charles Fefferman and E. M. Stein's Acta paper [18]. To prove the
reverse inequality we want to go via a different route, and we shall follow
Merryfield here [20]. We prove the following lemma. In the next lecture we
show how this lemma proves IIu*IIp ? CpIIS(u)IIp
LEMMA. Let f(x) and g(x) c L2(Rn), and suppose 0 c C (Rn) radial
and u = P[f ] . Then
ff IouI2(x,t)Ig*ot(x)12t dtdx
R n+1
+
< fn'
R Rn+1
Proof.
ff A(u2)(x,t)Ig*ot(x)I2t dtdx = f
Rn+1 Rn+1
f
-2 ,
R n+1
u(x,t)
n+t n+1
R+ R2
MULTIPARAMETER FOURIER ANALYSIS 99
where
I
ff
Rn+l
+
u(tV(g*.kt))t-1/2. Vu(g*0t)t1/2dtdx
1/2 1/2
< ff U2 1 g*otl 2 dt dx I rIV,I21g*ot(x)I2 t dtdx
Rn++l RJn++I
and
u !AL (g*ot)2dtdx
ff
Rn+l
-JJ u22(g*(kt)
A fn
R
f2g2dx .
We see that
but
But
so
(ffIvu2(g*ct)2 t dtdx)\
1)/2 1 /2
I fu2(g*(xjc)t)2 d tdt
+(fg/ 2
Ydx
1/2,
*(xio))2 dtdxi)t
t
1/2
fflvuI2(g*t)2tdxdt
/
f
2
C [Y 1 (
1u2 g* 14) dt dx )rru2(g * ( X d tdx
+f f2g2
(ai4) 2
dt dx
5. More on Hp spaces
At this point we wish to discuss the theory of multi-parameter Hp
spaces and BMO. We saw, in the last lecture, that HP(Rn) could be
defined either by maximal functions or by Littlewood-Paley-Stein theory.
All of these spaces, Hp and BMO were invariant under the usual dila-
tions on Rn, x -Sx, and this is hardly a surprise, since they can be
defined by the maximal functions and singular integrals which are
MULTIPARAMETER FOURIER ANALYSIS 101
S2(u)(x) = ff IV1V2u(y,t)I2dy1dy2dtidt2 .
F(x)
x x'
I 0 = 1, otI,t2(xi,x2) = tl1t21 tz
then
S ,(f )(x) =
ffIf*&tt(yi,y2)I2 dyl d Y2 dt dt 2
I 't t 2
t 2
F(x)
f =f+++f+_+f_++f__
If a<p,
u*(x)a < M(Ifla) e LP/a(R2), if f f LP
-Qtf(x1,x2) _ f(xl-y,x2)Qt(y)dy
dydt
F(x1,x2) a L2 (0);
t
by
Now we know that in the one-parameter case iiS1fIIp > cpIlflip and a
glance at the proof of this fact reveals that it remains valid for Hilbert
space valued functions. Fix x2. Then
00 00
R2 R2
But fixing x1, since IF(x1,x 2)1 L2(iis the value of the one-parameter
)
integral of f(x 1, - ) at x2 , we have
MULTIPARAMETER FOURIER ANALYSIS 105
J
R
IF(x1,x2)Ip 2
L J,)
dx2 > cp f
R 1
If(xl,x2)IPdx2
cp ff Iflpdxldx2
R2
On the other hand when the SI operator acts on the first variable we have
jJu2(x,t)(g*cfrt(x))2 dt dx
+
R2
(5.2)
jf
R2xR2
[Qt1Qt2f(xl,x2)]2[tt Pt2g(xt,x2)]2
dt1dt2
t it2
dxldx2
x2,t2 xl,t2
Now
ffPt2Pt1f(x))2(?g(x))2 dtdx
I= d
X1t1 x2 t2
t
Then
+ (Pt2f(x))2(Qt2)2(x)dx2dt2/t2
J J
x1ER1 (x2,t2)ER+
f J J f2(x)g2(x)dx
Rz
x2ER1\(x1,t1)ER+
where
I
M()( u*>a )<1 /200
S2(u)(x)dx when u - P[f]
< ff 1V1V2ul2(y,t)t1t2dydt
R*
f
M(Xl *>a #)<1 /2 00
u
S2(uXx)dx < ff IV,V2u12(Y,t)Pt(g)2(Y)dyt1t2dt
(R+)2
f If
f Pt1)f(Y)2.
t
t
(g)2(Y)dydt
l+ ff2g2dy
+ J 1 1 1 J
Y2 (Yi,tl)fR+2 R2
= i + ii + iii + iv.
fft(g)2(y)dy tdt = a2
fft(1_g)2(y1y dt < a2II1-gIIL2 < a2lju*>a}I
a2 12
(R+)2 (R+)2
Ix2 -Y2I < t2 and u * (yl,x2) < a. This implies that IPt2)f(y1,y2)1 <a so
2
ii is less than or equal to
ff(2)()2(Y)th2dY)dYl
a2 J 2 t2
Yl Y2't2)
Ag ain
ff Qt2)(g)2(Y)
dt 2
dY2 dyl g)2(Y)dY2 t2 dYl
2 J Jv 2
Yl Y2.t2 Yl 2,t2
u*(x)<a u(x)<a
So we have
satisfies (5.5) if and only if p(S(R)) < CIRI for all rectangles R C R2
with sides parallel to the axes, where the Carleson region S(R) is
defined by S(I x J) = S(I) x S(J) for R = I x j . In terms of these Carleson
measures, it is not hard to show that 95 satisfies (5.4) if and only if its
bi-Poisson integral u satisfies
And finally, all of this in some sense is equivalent to asserting that every
f e HI(R2 xR2) can be written as I Akak where Y' IAkj < CIIf1I 1 and
H
ak(x1,x2) are "atoms," i.e., ak is supported in a rectangle Rk = IkxJk
such that
and
1
IIak112<_
1 /2
IRkI
In 1974 [22], L. Carleson showed that p(S(R)) < CIRI was not suffi-
cient to guarantee the inequality
1J
I
Ic(xi,X2)-C1(x1)-C2(x2)12dxldx2 < C
R
In fact, this follows immediately from the inequality (5.3). To see this,
notice that IV1V2uI is invariant under the Hilbert transform HXi(i = 1,2)
so that if we prove this when f e L(R2), we will have proven it also
when f is of the form
MULTIPARAMETER FOURIER ANALYSIS 113
gl + Hx 92 + Hx2 93 + Hx Hx 2g4 gi c L .
1 1
and F__ c L1(R2) such that a = c F++ and reflections of the F++ are
boundary values of bi-analytic functions. A bianalytic function F with
(distinguished) boundary values in L1(R2) has F* c L1 by a subhar-
monicity argument applied to IF I', a < 1 . So a* a L1 and a e H1. Let
$ e H1(R+xR+)*. Define a map from H1(R+xR+) L1(R2)i by
i=1
Then Ij6f jI
l lif 11
HI .
0 is obviously one to one, so 0-1 = s exists
and is bounded on Im(6). The map $ extends, by Hahn-Banach to
an element of the dual L1 = L. Then
= rfgl + Hx f g2 + Hx f g3 + Hx Hx2f g4
rJ 1 2 1
X1 \(x2t2) / X2 \(X1,t1)
11f 112
< CIIf11211gII2 < C ICI
00
ft 1
1
2
(x1,x2) _
tl
X
1)0(x2)
t2
t11t21,
f
0
=1
we have
dtldt2
f(x1,x2) = f *tPtl,t2(Yl,y2)otl,t2(xl -y1,x2-y2)dyldy2 tit t2
R+x R+
00 00
R2xR2
+ +
= U W(R), if we define
Rc91d
fR(x1,x2) _ ff
f(y,t) f= 1 fR, and each fR is supported in
Rc9? d
where 14 =TxT.
It will be convenient to define a norm I I R on functions supported on
a rectangle R , as follows.
N
_ oaf
IfIR IIIa111ia2
IaI=O axa1ax22
g4 in L(R2).
(3) If u = P[] in R+ x R+, then
S(Q)
S(fl)
MULTIPARAMETER FOURIER ANALYSIS 117
Proof. To begin with, we proved in section 5 that (1) --> (2). It is also
trivial that (2) - (1), since if f e H(R2xR+
f
J f(x) Hx Hx (g)(x)dx =
1 2
fHx1 Hx (f Xx) g(x) dx
2
ak(x l ,x2) = I
Re91d
f R(x)
IRflUk1 >1/21RI
IRf1Slk+1I < 1 /2I RI
ak(x I x )
Then, as we shall elaborate later on ak(xlPx2) = 2 is an
H1(R+xR+) atom where
J ak(xl,x2)'O(xlx2)dx <C
R2
1
t i
tit
R9k Sff(y,ty,t'y__ 2
(R)
2k1 RU
ff If
(y,t)I2dY
ate
1 /2
ff I56(Y.t)12dy d t2c2
/2
kI c%k RU
-1 91k
MR) 91(R)
MULTIPARAMETER FOURIER ANALYSIS 119
(6.2)
J
IS,,(f)2(x)dx >_ U
RE9
ff If(Y,t))2dy td
1t2
fl k/Qk+1 k 91(R)
To show this
S, (f Xx)dx =
J
ff If
(Y,t)12dy
t2
I)k/uk+l xE?k/SZk+1 F(x)
But then
il kAlk+1
As for
R(R.
91(R)
U
ff I0(y,t)I2dy
tat2 < if
195(y,t)I2dy tat- < C ?1
RE9lk
91(R) S(? k)
by (4). This shows that I f a k Odx I < C and completes the proof that (4)
implies (1).
We shall show next that (2) implies (4). Let g e L(R2). We claim
that if 1 C R2, then
dt CIIgII2lgl
f Ig(y,t)12dy tit-00
00
ff
S((1)
Ig(y,t)12dy tit 2 ff (g)(y,t)dy dt
(R+)2
12
CIO
IIgX 1112 f
0
MULTIPARAMETER FOURIER ANALYSIS 121
Since, f +1 1 q, = 01
i(0) = 0 and fr E Co
() =
0(161 -N) as I CI - 00, for each N > 0.
so
00
f
0
de
<
It follows that
f
S(U)
Ig(Y,t)I2dy
dt < C IIgII,2oIjjl
tlt2
as claimed.
If we wish to prove that the same Carleson condition holds for g
replaced by Hx1 Hx2 (g), then we proceed as follows. Observe that
S(SZ)
t12
1/2
(d)
(.9)aq,j = 0(2-kN-jN) as k,j o if Ial <2
If
1 /2 1 /2
12
tlt2 k,j
S(fl) s(1)
Now, to estimate
I9 *(Tk,j)t12dy tat
12
kj
we use the same argument as that given above, except that now
supp(q1k j) < 4R(0; 2k,2)) and not the unit square. If (y,t) a S(cl), then
R(y;t) C B and the support of (Wk,j )t(. -y) will be contained in
Thus
I 5 Ig*('Fk,j)t(y)I2dY
dt2 = ff 1(9,,,,
kj) *'Ykjl2dy tat2
S(B) S(Q)
I'kj(6,,C2)12
2 <1i Ill 1 2
11gx
112 ff
kj 2
0
f 0
1
1 2
kit
JJ
0
0
00
2
MULTIPARAMETER FOURIER ANALYSIS 123
ak(x) = fR(xl,x2)
2kls kl RERk
minimumr (sill ,
R il <
IS
Slsi q11 /2 =P
1` 2
A * ctlt2x) = IA I A * q5t1t2(x)
2i/Is2I<p
2)/Is1I<p
Y fR * Otlt2(x)
RE8
iR2II
where '23 C 91k consists of rectangles R so that iRi < p, I
< p and
SII S2
kf1S 0. Thus R C''`9 for all R E 18, and the reason this subtracted
term occurs is that we have double counted these fR whose R sides are
both very small.
In order to estimate A' * q5t1t2x) we use the following trivial lemma.
f a(x)-O(x)dx
aq
We estimate Al * (btt(x) using the fact that for each fixed x2,
A1(- ,x2) has N vanishing moments over disjoint x1 intervals over
l
length 2 .2) . (Actually, we sould have to break up A into 3 pieces to
insure this, but we spare the reader this trivial complication.) It follows
from the lemma that
2i )N+I(IA' I
IA *1 0t 1 (x)I < * 1 X[-t 1 t 1 ] (x)) '
t1 1 I
I dx'
IAA *
IS -gI
126 ROBERT FEFFERMAN
2'/IS1 I<p
Aj <CpN/2 1
IiI J
f (IA2)
-9
II
1/2
< CpN /2 1 Y 1 /2
r(IAhl2)
By symmetry
1 /2
2 .0tlt2(x) < CpN/2 1 121
L. Aj * IAi
.
2'/IS2I<p
IR R 2I
Now let RCS , with 1I < IS2I . Then
S11
and also
<C\1SI/N/2 IS I 1
IfR * .0t1,t2(x)I IfRI
Thus
r
L IfR *1t t (X)I
2
<- C --L J
fI fR(x)
1 /2
(
(Is
R
IlN /21 /2 RN/4
?
ReAap 1 IS I RERk / RE%k
I
1 /2
CpN/4 1 Y IfRI2
I I R E991k
I)N/4
(6.5) JA * otlt2(x), C fJ(Y)dY
Snl SI
where
1/2 1/2 `1/2
+((A2)2)
-++ (I fR)
RE%k
_ R2 I fffthYt)dYTTdt
W(R)
rf f &'t)12dy dt2)
/2 /2
< (Pz JJ
9I(R)
1 f
R+)2
Ih(y,t)12dy dt
tIt2)
1/2 1/2
c IlS .(h)II2 < (Y't)Izdy dt
(51If(Y,t)12dY tat
tit 2
(ffif tIt2
128 ROBERT FEFFERMAN
R E(R )
k kk+1
/0
1 /2
A* < 111 /2 (fA*2 < c1,011/2 IIA112 < c2k 1.1 .
ti J
so
1 /2
fA*dx < (fM(2)(x2o(xdx)'(fM2a)2(x)dx
R /
ROBERT FEFFERMAN
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF CHICAGO
CHICAGO, ILLINOIS 60637
BIBLIOGR APHY
Ill B. Jessen, J. Marcinkiewicz and A. Zygmund, Notes on the Differ-
entiability of Multiple Integrals, Fund. Math. 24, 1935.
[2] E. M. Stein and S. Wainger, Problems in Harmonic Analysis Related
to Curvature, Bull. AMS. 84, 1978.
MULTIPARAMETER FOURIER ANALYSIS 129
Carlos E. Kenig*
PREFACE
have had throughout the years, which resulted in the work explained in
this paper.
Introduction
A harmonic function u is a twice continuously differentiable function
on an open subset of Rn , n > 2, satisfying the Laplace equation
132u
Au = = 0. Harmonic function arise in many problems in mathe-
i=I 9X?
7
The theorem asserts that fr(Q) = u(rQ) converges to f(Q) not only in
LP norm, but also in the sense of Lebesgue's dominated convergence.
In the analogous estimates to (*) in the Neumann problem, u is re-
placed by the gradient of u. In that case the estimate fails for p = 00,
even if aN is continuous.
In both the case of the Dirichlet problem and the Neumann problem,
the radial limit can be replaced by a non-tangential limit: if X tends to
Q with IX-QI < (l+a) dist(X, dB), for some fixed a > 0, then u(X)
has the limit f(Q) for almost every Q .
The theorem is most easily proved by writing down a formula for the
2
solution, u(X) = fas P(X,Q) f(Q) do(Q) , where P(X,Q) = n Iin
IX-Q
The estimate now follows as an easy consequence of the Hardy-Littlewood
maximal theorem. An analogous formula holds for the Neumann problem.
ELLIPTIC BOUNDARY VALUE PROBLEMS 135
This time, it is more difficult to obtain the estimates. One needs to use
the Calderon-Zygmund theory of singular integrals, and the Hardy-
Littlewood maximal theorem.
The case of the Laplacian in the ball is relatively easy because of
the existence of explicit formulas for the solution. What should we do in
the case of a general domain, where explicit formulas are not available?
What should we do to study systems of equations? What happens to our
solutions as the domains become less smooth? We hope to give a system-
atic answer to these questions in the rest of this paper.
a(E) = f (1 + IO95(x')12)I
/2 dx
On the other hand, if we know G(X,Y), we can formally write down the
solution to the Dirichlet problem.
In fact,
= Ju(Q) aN G(X,Q)da(Q) ,
4
aD
where the fourth equality follows from Green's formula. Thus, we have
derived the formula
for the harmonic function u with boundary values f. The problem with
138 CARLOS E. KENIG
1
n IX--Q InI<X_Q,NQ> da(Q)
aD
<Q -P,Np C
IP-Qin IP-QIn-2
<X-Q,N >
u(X) = wn g(Q) IX-Ql do(Q),
dD
IP--P,QNp>
Cl,a
Because of the estimate I <Q I on
n
domains, it is easy to see that K is compact as a mapping on LP(dD).
Also, standard arguments show that
<Q-P,Np C
lP-Qln < IP-QIn-I
and so, even the LP boundedness, much less the compactness of K, is
far from obvious.
Let us now turn to the Neumann problem. Let D be a smooth domain.
We seek to solve Au = 0 in D, !AL I = f. By Green's formula, we
aD
must have f3D fda = 0. When D and cD are connected this is the only
compatibility condition needed. We will only consider that case for
simplicity. A good first guess at the solution u is the so-called single
layer potential of f given by v(x) = faD f(Q)F(X,Q)da(Q) =
u(X) = f(T1f)(Q)F(X,Q)da(Q),
aD
the subspace of Lp((3D) of functions with mean value, 1 < p < w. Hence:
u(X) = ff(Q)da(Q)
dD
142 CARLOS E. KENIG
statics. The results obtained had not been previously available for
general Lipschitz domains, although a lost of work had been devoted to
the case of piecewise linear domains. (See [24], [25] and their bibli-
ographies.) For the case of CI domains, these results for the systems
of elastostatics had been previously obtained by A. Gutierrez ([15]),
using compactness and the Fredholm theory. This is of course, not
available for the case of Lipschitz domains. The authors use once more
the method of layer potentials. Invertibility is shown again by means of
Rellich type formulas. This works very well in the Dirichlet problem for
the Stokes system (see part (b) of Section 3), but serious difficulties
occur for the systems of elastostatics (see part (a) of Section 3). These
difficulties are overcome by proving a Korn type inequality at the
boundary. The proof of this inequality proceeds in three steps. One first
establishes it for the case of small Lipschitz constant. One then proves
an analogous inequality for non-tangential maximal functions on any
Lipschitz domain, by using the ideas of G. David ([10]), on increasing the
Lipschitz constant. Finally, one can remove the non-tangential maximal
function, using the results on the Dirichlet problem for the Stokes system,
which are established in part (b) of Section 3.
As a final comment, I would like to point out that even though through-
out this paper we have emphasized non-tangential maximal function esti-
mates, also optimal Sobolev space estimates hold. All the Sobolev esti-
mates can be proved in a unified fashion, using square functions and a
variant of some of the real variable arguments used in part (b) of Section 3.
The details will appear in a forthcoming paper of B. Dahlberg and
C. Kenig, [7].
both the interior and exterior cone condition. For such a domain D, the
non-tangential region of opening J8 at a point Q c aD is I'18(Q) _
IX cD : IX-Qj < (1+,8)dist (X, 3))J. All the results in this paper are valid,
when suitably interpreted for all bounded Lipschitz domains in Rn,
n > 2, with the non-tangential approach regions defined above. For
simplicity, in this exposition we will restrict ourselves to the case n > 3
(and sometimes even to the case n = 3 ), and to domains D C Rn ,
D = I(x,y): y>4i(x)), where : Rn-I R is a Lipschitz function with
Lipschitz constant M, i.e. j0(x)-4(x')I < Mix-x'I . D- _ I(x,y):y<cb(x)1
For fixed M'< M, I'e(x) _ I(z,y): (y-(k(x)) < -M'Iz-xI4 C D and I'i(x) _
I(z,y): (y-0(x)) > M'Iz-xII C D. Points in D will usually be denoted by
X, while points on aD by Q = (x, (k(x)) or simply by x, Nx or NQ
will denote the unit normal to aD = A at Q = (x, 4(x)). If u is a func-
tion defined on Rn'A and Q c aD , u t(Q) will denote lim u(X) or
X-3Q
XEIi(Q)
sup Iu(X)I.
XcF1(Q)
Au = 0 in D Au = 0 in D
(D) , (N)
u' aD = f c L2(aD,do) f c L2(3D,do)
a-N
aD
THEOREM 2.1.1. There exists a unique u such that N(u) c L2((9D, do),
solving (D), where the boundary values are taken non-tangentially a.e. .
Moreover, the solution u has the form
- 1 <X-Q'NQ>
u(X) (On g(Q) do(Q)
J
IQ_XIn
aD
for some g c L2(3D, da).
ELLIPTIC BOUNDARY VALUE PROBLEMS 147
<X-QNQ>
g(x) = (an J IX-QIn g(Q)do(Q)
aD
and
Sg(X) _ - wn( f
aD
I n 2 g(Q)do(Q)
IX-QI
g(x) dx
g(z.Y)=w f [Ix-z I2 + [fi(x)-0(z)121n/2
Rn-I
Sg(z ,Y) _ -
ca
1
n-2)
'J r 1+IV _(X) I2
-n 2- g(x) dx
2
Rn-I [Ix-zI2+I4(x)-y]21
(b) lim 1
r z)-q(x)-(z-x) VS6(x) g(x) dx = Kg(z) exists
Wn [IX-zI2 +[S6(x)-0(z)]2]n-2
Ix zI>E
Lp(aD, do), and its LP norm is bounded by CIIgII , 1 < p < oo.
LP(aD,do)
(c) (xg) t(Q) _ 2 g(Q) + Kg(Q)
(VSg)t(z)=1
2
g(z)Nz+- lim J (z-x,0(z)-0(x)) (1 + Q(k(x) g(x)dx
[Iz-xI2+[(k(z)-(k(x)]2]n/2
n E -+o
Iz xI>E
operator
M*g(z) = sup
E-0
I
f
It XI>E
K(z,x)g(x)dx(
The proof of (b), (c) follow from the theorem above, together with the
following simple lemmas.
n--1
_-
1 zk-xk
X
[(z)_(x)1 (x)dx
J n-I x-z axk
1
1
c.n
I a-O(a)- (a-x) V (x) f(x)dx =
[Ia-xI2+[0(x)-a]2]n/2
n-1
fix) a
=
2
sign (a --O(a)) f(a) - cI
on n fI xk ak
Ix-amn-I x--a)
c3f (x)dx
c7xk
It is easy to see that (at least the existence part) of Theorems 2.1.1
and 2.1.2 will follow immediately if we can show that (2 I+K) and
150 CARLOS E. KENIG
cit
Lp(Rn-1) , 1 < p < - with bound independent of t , by the theorem of
Coifman-Mclntosh-Meyer. Moreover, for each t, IITtf II > C Ilf II L2 , C
L2
independent of t. The invertibility of T now follows from the continuity
method:
In order to prove (2.1.7), we will use the following formula, which goes
back to Rellich [30] (also see [28], [29], [27]).
f<Nen>iVui2da = 2 ay ado.
aD aD
J( )2da<C fVtuI2da.
aD` t3D
2 faD
<Nx,en>( )2do+2
M faD <a,Vu> TN
('u) do. Hence faD<Nx,en>(o)2do=
flVtul2da< c f( )2 do.
aD aD
In order to prove 2.1.7, let u = Sg. Because `of 2.1.3c, ptu` is con
C0N`
tinuous across the boundary, while by 2.1.4, = (T. z I-K*) g . We
This
now apply 2.1.11 in D and 5, to obtain 1.1.7. finishes the proof
of 2.1.1 and 2.1.2.
We now turn our attention to L2 regularity in the Dirichlet problem.
THEOREM 2.1.13. The single layer potential S maps L2(A) into L2(A)
boundedly, and has a bounded inverse.
EL(3113).
I I - ICI I
where the first inequality follows from the fact that both G and v are
positive, and harmonic on B, and 0 on ()iZP fl B (this is Lemma 5.10
in [19]). Assume now that 2 I+K were invertible on Lq(ds). Let
g ? 0 e C(dl3 ), and h(X) be the solution of the Dirichlet problem with
data g. Then,
h(X*) = gda, = J gkds
J
aiiP 0-n
1 /q 1 /q
h(X*) < JJ fhc(fNhds) < C J ggds
because of the second formula for h(X*), and the assumed Lq bounded-
1
ness of (2 1+K . But this implies that k ( LP(ds), a contradiction.
<X-Q,NQ>
u(X) -
X Ql g(Q)da(Q) ,
n I
aD
u(X) = g(Q)da(Q) ,
wn(n-2) J IX-Q In-2
aD
THEOREM 2.2.3. There exists E = E(M) > 0 such that given f c LP(A),
1 < p < 2+E, there exists a harmonic function u, with IIN(Vu)lt <
L'(A) -
CIIVtfIILp(A), and such that Vtu = Vtf (a.e.) non-tangentially on
is unique (modulo constants). Moreover, u has the form
1 1
u(X) _ - g(Q)da(Q) ,
wn(n-2) fIX-QIn-2
aD
The case p - 2 of the above theorems was discussed in part (a). The
first part of 2.2.1 (i.e. without the representation formula), is due to
B. Dahlberg (1977) ([51). Theorem 2.2.3 was first proved by G. Verchota
(1982) ([331). The representation formula in 2.2.1, Theorem 2.2.2, and
the proof that we are going to present of 2.2.3 are due to B. Dahlberg and
C. Kenig (1984) ([61). Just like in part (a), 2.2.1, 2.2.2, and 2.2.3 follow
from.
We first turn our attention to the case I < p < 2 of Theorem 2.2.5. In
order to do so, we introduce some definitions. A surface ball B in A
is a set of the form (x,4(x)), where x belongs to a ball in Rn-I .
L2(A) /2
I
larity theory for divergence form elliptic equations. Consider the bi-
Lipschitzian mapping (D: D - D- given by 4D(x,y) = (x, O(x) - [y-(x)1).
Define u* on D- by the formula u* = u at-1 , u* verifies (in the weak
sense) the equation d iv (A(x,y) Vu*) = 0, where A(x,y) _ J X) (X) ,
where X = 1-1(x,y) . It is easy to see that A c L(D-), and
< A(x,y)e, t; > > CI; I2 . Notice also that supp dN C B 1 < B* fl aD. Define now
J I2 < C. But, since ' solves Lu = 0, max [u1 < C(f rul21 /2 < C
2B B 2B
([261). Therefore, u E L(Rn\B*), IrII LOO(Rn\B*) < C. Hence, since
a = 0, Vu=Vv, and Iv(x,y)I VC/(IxI +IyI)n-2+v'
(a) supp A C B, a surface ball, (b) IIAII < 1/a(B), (c) f Ada= 0.
We will use the following interpolation result:
160 CARLOS E. KENIG
u(x,y) (x,y) c D
u(x,y) _
-u*(x,y) (x,y) E D_
We turn now to the LP theory, 2 < p < 2 +E. In this case, the results
are obtained as automatic real variable consequences of the fact that the
L2 results hold for all Lipschitz domains. We will now show that
IIN(?u)IILp(n) <C`IIILp(n) for 2 <p <2+E.
The geometry will be clearer if we do it in Rn , and then we transfer
it to D by the bi-Lipschitzian mapping (D: R+ -, D , 4D(x,y) = (x,y+q5(x)).
We will systematically ignore the distinction between sets in R+ and
their images under It.
Let y=t(x,y)ER+:Ix(<y1, y*=((x,y)ER+:alxl<yt, where a
is a small constant to be chosen. Let m(x) = sup IVu(z,y)I , and
(Z' Y) Ex+y
*(x)
m = sup IVu(z,y)I . Our aim is to show that there is a small
(z, y) E x+y*
E f0 AE-1 fEA m2dA < C E fO Al+E IIm* >AIIdA+CE fQ c'o AE 1(fh>A m2)dA.
162 CARLOS E. KENIG
2+E , we see that fm2+E < C(f m2+E)2+E(fM(f2) 2 )2+E, and the
E
Bkr = aQkrflR+\Akr
so that aQk,r = Qk,r U Ak,r U Bk,r. Note that the height of Bk,r is
dominated by Ca length (Qk), and that IVul <,k on Ak,r. Let ml
be the maximal function of Vu, correspond ing to the domain Qk,r (i.e.
where the cones are truncated at height ti P(Qk) ) Then, for x c Qk,
m(x) < mI(x) + X. Also,
0u+(a+)Vdivu=0 in D
(3.1.1)
ulaD =f eLZ(aD,da)
0u + (k +) p div u = 0 in D
(3.1.2)
A(div u)N+l pu+(pu)tlNlaD =f EL2(aD,da).
164 CARLOS E. KENIG
Our main results here parallel those of Section 2, part a). They are
The proof of Theorem 3.1.3 starts out following the pattern we used
to prove 2.1.1, 2.1.2 and 2.1.14. We first show, as in Theorem 2.1.3. that
the following lemma holds:
ELLIPTIC BOUNDARY VALUE PROBLEMS 165
LEMMA 3.1.4. Let Kg', Sj be defined as above, so that they both solve
0u + (A+)p div u = 0 in R3\oD. Then:
+ (.v. f
aD
aPi
F(P-Q)g(Q)da(Q)
j
,
useful to explain the stress operator T (and thus the boundary value
problem 3.1.2), from the point of view of the theory of constant coefficient
second order elliptic systems. We go back to working on Rn, and use
the summation convention.
Let ars, 1 < r, s < m , 1 < i, j < n be constants satisfying the
ellipticity condition
and the symmetry condition ars = asr. Consider vector valued functions
u = (u1, ,um) on Rn satisfying the divergence from system
ars us = 0 in D. From variational considerations, the most
TT ij TX- T
natural boundary conditions are to Dirichlet condition (43D =f ) or the
s
Neumann type conditions, = n ars = f r . The interpretation of
av 1 ij
problem (2) in this context is that we can find constants ail , 1 < i, j < 3 ,
1 < r, s < 3, which satisfy the ellipticity condition and the symmetry con-
dition, and such that p0u + (A+p) V div u = 0 in D if and only if
I/2 I/2
('
2C fh, n
E
ars
Lj
s do, < 2C
i
J pur12 da J da
aD 030 D
(9D
Thus,
fa D IVur12du < C I3 D j
2 do.
For the opposite inequality, observe that, for each r,s,j fixed, the
vector hinLaes - hLnfars is perpendicular to N. Because of Lemma 3.1.5,
J j
REMARK 3. In the case in which we are interested, i.e. the case of the
systems of elastostatics,
168 CARLOS E. KENIG
s
ars au au r j i
au-
2
(div u)2 + ,
i , JX-i ' Fxj z ax; dx.
i.i
(heeij
n ars - h' n ee,
ars) is a tangential vector. Thus, f
aD
h n ars our aus da <
N
2 I 12 -2 12
C (j IVtuI do) (f3D IVu1 do)' Consider now the matrix drs =
(ark n;nj)-1 . This is a strictly positive matrix, since ark e; ej77 t77 s >
C 1e121'i12 ta"
Moreover, d rs UV (a'A - ars
``av i, Ri Ni
r
`us = d rs n. art aut , n sm
i, Ni ek 3Xk 1
0,U m
ars our aus d rs nk art aut nmast our - atr at our = d rs nk art `out
ij JX; aXj kgaXQ my aXv vec)Xv aXe kv3Xv
nmameN, ave7v a Q
= 1drsnkarkv n ma ml - avf v X Now, note
n= atr
art
kvaknvdrs ast
me nm atr
mem vlv nkdrs asr
mem n -- atr
memn--Sis ast
mem n -atrmenm
A
arsn I aus __ (mars - arsn au. But, for i, r,s fixed, ars -
ki knin J aX i] ik k I J
I
3Xi i]
We now choose h = en , so that hene > C , and recall that (drs) and
(artnkn) are strictly positive definite matrices. We then see that
I
aD (ID
Iotul2da J
OD
+ J jVtul2da
OD
.
-. 2 ,12
Now, as IVul = Iptul + aN, 2 , the remark follows.
2
REMARK 5. In order to show that IaD Ivtul do < C fan TuJ2da, it
suffices to show that faD Ipul2da< C faD IA(div u)I + pipu+Qut}I2da.
In fact, if this inequality holds, we would clearly have that f3D Ipul2da<
C faD Ipu+putl2 da (Korn type inequality at the boundary). The Rellich-
Payne-Weinberger-Necas identity is, in this case (with h = en ),
The proof of the above theorem proceeds in two steps. They are:
Lemma 3.1.7 is proved by first doing so in the case when the Lipschitz
constant is small , and then passing to the general case by using the
ideas of G. David ([91 ). Lemma 3.1.8 is proved by observing that if v
is any row of the matrix A(div u)I+l4pu+putj, then v is a solution of
the Stokes system
Av = pp in D
(S) div v = 0 in D
vl3D = f f L2(dD,da)
LEMMA 3.1.11. Given M > 0 and 0 with IIIVOIII < M, there exists a
constant C = C(M) such that for all functions u in Do, which are
Lipschitz in Do, which satisfy pAu +(A+iz)V div u = 0 in Do and
pu(Ao) = pu(Ao)t , we have IINo(V)IIL2(aD,do) < CIINo(A(div U _)I +
where C = C (M, e) .
Note that if Proposition holds, then the corresponding estimates
automatically hold for all translates, rotates or dilates of the domains
Do, when 95 satisfies the conditions in Proposition (M,E). In the rest
of this section, a coordinate chart will be a translate, rotate or dilate of a
domain Do. The bottom Bo of aDo will be T95(dDOU (x,0): x ERn-I).
ELLIPTIC BOUNDARY VALUE PROBLEMS 173
We will not give the proof of Proposition 3.1.12 here. We will just
make a few remarks about its proof. First, in this case the stronger
estimate IINA(Vu)IIL2(aD,do)<CIIA(div u)N+1A Vu+VutINIIL2(dD,do)
holds. This is because in this case, the domain DS6 is a small perturba-
tion of the smooth domain D. . For the smooth domain D_ , we can
ax aX
solve problem 3.1.2 by the method of layer potentials (see [24], for
example). If a is small, a perturbation analysis based on the theorem of
Coifman-Mclntosh-Meyer ([2]) shows that this is still the case. This
easily gives the estimate claimed above.
Proof of Lemma 3.1.11. We will show that Proposition (M,a) holds for any
M,e. Fix M,a, and choose R so large that if e(10M) is as in Proposition
R
3.1.12, then (1.1)Re(10M) > E. Pick now aj > 0 so that II (1-aj)=1/10
j=1
for example). Pick now 0 with IIIVO-a III < 1-1e, 111-a 1115 (1-a)M. We
will choose No as follows: Since a1) 0\Bo is smooth, it is easy to
see that we can find a finite number of coordinate charts (i.e. rotates,
translates and dilates of Do ), which are entirely contained in Do ,
such that their bottoms By are contained in c3Do, such that
Tq((x,0): I IIx III < 1/2) cover dD(k, and such that the pi's involved
satisfy IIIVIII < (1 - 2) M, and there exist u such that III ao III
1.11E . The non-tangential region defining N0l, on TV (x,0) : 11141 < 2
then a(E3) < (1-rJM)a(Uj) where 77M > 0. Assume the claim for the time
being. Then
f
00
00 00 00
00
2
tof m>tIdt = (32 (1-77M) f m2da+ 5 m2da. Thus, if
S
0 aDo 3D0
we choose /3 > 1 , but so that /32 . (1- riM) < 1 , the desired result follows.
It remains to establish the claim. We argue by contradiction. Suppose not,
then a(EJ) > (1- r)M) a(Uj). Let E) = T,I(E)). If 71M is chosen suffi-
ciently small, we can guarantee that IEi -9911-1. Let now Fi _
Ei n Ij , and construct now the Lipschitz function Vi corresponding to it,
as in the definition of NS6 . Thus, > , III < (i_)vi
IIo - aIII < 1.111E . We now apply Lemma 3.1.10 to Vi, one variable
at a time, to find a Lipschitz function f, with f > 0 on Ij , such that if
F = Ix (Ii : f =01, then IFS nFjI > C a(Uwith IIIVfIII < (1 - 10) M,
and such that there exists af, with III afIII < (1 - 10} M so that
IIIVf-afIII < 5 1.111E < e . We can also arrange the truncation of our non-
tangential regions in such a way that on the appropriate rotate, translate
and dilate of Df (which of course is contained in the corresponding
coordinate chart associated to Dv,, which is contained in Do ),
IA(div u)I + jA Vu+ Vut I I <5 t . To lighten the exposition, we will still
denote by Df the translate, rotate and dilate of Df. Note that Proposi-
tion (M,e) applies to it. We divide the sets Ul into two types. Type I
176 CARLOS E. KENIG
are those with diam Ui > 710, and type 11 those for which diam Ui < ri0.
We first deal with the Uj of type I. In this case, Df has diameter of
the order of Because of the solvability of problem 3.1.2 for balls,
1 .
C o(Uj)162t2 <
I
(Fif1Fi)
m2do<C f N2(pu)do,<
aDf
a Df - 1)
Dut(Af)
Nf( Q) > m(Q). Hence, Nf Vu -r2u(Af) 2 (Q) > (R-1-Ca)t >
L
ELLIPTIC BOUNDARY VALUE PROBLEMS 177
1
2
)t if S is small and Q E TIA (F f1Fj). Thus, applying (M,e) to Df,
[Vu(Af)
we see that CJ((,3-1)2t2o(U.) < J Nf pu
T (FJ.(1F.) C
2 Vut(Af
1)
2da<
J
1 aij
(T'ij(X)), where rij(X) = 8n IXI + 1 XiXj , and its corresponding pressure
8n X 3
X1
vector q(X) = q'(X)), where q'(X) = Our solution of (3.2.2) will
4nIXI3
be given in the form of a double layer potential, u(X) = Kg(X) _
faD (H'(Q)F(X-Q){g(Q)do(Q), where (H'(Q) r(X-Q))if = Sijge(X-Q) nj(Q) +
178 CARLOS E. KENIG
arig
(X-Q)nj(Q). We will also use the single layer potential u(X) _
(b) (Kg)(P) =
2 g(P) - p.v. fdD IH (Q)F(P-Q)Ij(Q)da(Q)
+ nl(P)g3(P) nl(P)n)(P)
(d) aXl (Sg )j (P) == +-.) <N(P), g(P)>
2 2
For the proof of this lemma in the case of smooth domains, see [25].
The proof of Theorem 3.2.2 (at least the existence part of it), reduces
to the invertibility in L20D, da) of the operator 2 I+K, where Kg(P)
- P.V. faD {H'(Q)T'(P-Q)jg(Q)da(Q). As in previous cases, it is enough
to show
(3.2.4)
I2
I-K*)2L (aD,da) +I(2
I+K*) 9
L2OD,da)
1
ELLIPTIC BOUNDARY VALUE PROBLEMS 179
f
aD
hFnPaS 1
a 1
Sdo=2 J
aD
hFaX
F
da-2 f
3D
PnS
h
fa F
da.
aD 3D 3D
+2 fhns_.da.
ass our
3D
The proofs of 3.2.5 and 3.2.6 are simple applications of the properties
of u,p, and the divergence theorem.
Choosing h = e3, we see that, from 3.2.6 we obtain
fP2do<C fIvn2da
aD aD
av aN
- then we have
180 CARLOS E. KENIG
fVt2da I fnS2da,
IN, 2
dam
l
aD aD i aD
Proof. 3.2.5 clearly implies, by Schwartz's inequality, that f D 1Vu 12dor <
2
C f,9D ICI do. Moreover, arguing as in the second part of the Remark 2
after 3.1.5, we see that 3.2.5 shows that
1 /2 1 /2
(' j2da)
CJ Vu-1 2do
fins
aD ' (9D
continuous across 3D. Using this fact, 3.2.3 e) and Corollary 3.2.8,
3.2.4 follows.
(Au = Vp in D
(3.2.9) divu=0 in D
((pu+Vut)N-p-N)1j aD = f e L2(3D,do) .
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF CHICAGO
CHICAGO, ILLINOIS 60637
REFERENCES
[1] A. P. Calderon, Cauchy integrals on Lipschitz curves and related
operators, Proc. Nat. Acad. Sc. U.S.A. 74(1977), 1324-1327.
[2] R. R. Coifman, A. McIntosh and Y. Meyer, L'integrale de Cauchy
definit un operateur borne sur L2 pour les courbes lipschitziennes,
Annals of Math. 116(1982), 361-387.
[3] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their
use in analysis, Bull. AMS 83 (1977), 569-645.
[4] B.E.J. Dahlberg, On estimates of harmonic measure, Arch. Rational
Mech. and Anal. 65 (1977), 272-288.
[5] , On the Poisson integral for Lipschitz and CI domains,
Studia Math. 66(1979), 13-24.
[6] B.E.J. Dahlberg and C. E. Kenig, Hardy spaces and the LP Neumann
problem for Laplace's equation in a Lipschitz domain, to appear,
Annals of Math.
[7] , Area integral estimates for higher order boundary value
problems on Lipschitz domains, to appear.
[8] D.E.J. Dahlberg, C.E. Kenig and G.C. Verchota, Boundary value
problems for the systems of elastostatics on a Lipschitz domain, in
preparation.
182 CARLOS E. KENIG
Steven G. Krantz*
185
186 STEVEN G. KRANTZ
2n
fC(C0
f (O)
Tir-i f eie- 0 (iei0d= d (1.4)
o Y
0(0_oz(0=i +zC ,
f( z) = 'ri f f(O
1 1d C
Y
171 _(o
INTEGRAL FORMULAS IN COMPLEX ANALYSIS 187
27i f 00 +
z dC (1.5)
1
f(z) = -z 1=z
y
= 1 fCQ
-Z
dC 27ri
+
fr
Now the numerator of the second integrand in (1.6) is a holomorphic
function of C which vanishes at 0. By (1.4), the second integral
vanishes. Formula (1.6) now becomes
f( z) _
2ni
fy
Cf(C)
-z dC
2n
y,
f(z) = Zn f(ei9) 110_ z2 2 d
le -z10
<dx>=<$,dy>=1,
<dy>=<1,dx> =0.
In complex analysis, it is convenient to define differential operators
a -1 (ax
a a a _1 (a a .
, aj_2c+iay)
az 2 lay
The motivation for this notation is twofold. First,
a
3j7z z=1, z= -z=0
Secondly, if f( z) = u(z) + iv(z) is a C 1 function, with u and v real
valued, then
dz = dx + idy, dz = dx - idy .
It is immediate that
<,dz> = 1 ,
au = ab dz A dz , X = . dz A dz . (1.9)
fu = fdu.
an sZ
fu= fau+u
ale 11
(1.10)
(t as j)dzA.
a
f(z)
2ni
f.!2dC, all zci .
= -z
aci
190 STEVEN G. KRANTZ
Proof. Fix z e it. Let E < distance (z, aft). Define D(z,E) = 1C E C : IC-zI < El
and it = it\D(z,E). We apply Stokes' Theorem to the 1-form u(C) _
f(t
dC on the domain it (note that u has smooth coefficients on a
-z ti
neighborhood of the closure of it, but not on all of it ). Thus, by (1.10),
fu(i= fdu=_j(2)dAd.
ti ti ti
ti
This last is 0 by (1.7). Thus, since aft = dl2 U 3D(z,e) (with suitable
orientations), we have
fu() = J u(O
aft 3D(z,E)
21r
f-!d= J ('
f(z+Eei0)id9 2aif(z)
o
This formula, valid for all f c C'(KI), will be valuable later on.
INTEGRAL FORMULAS IN COMPLEX ANALYSIS 191
0
do E
sup
0
f If(()I2da( )I/2 I II f
L2 (f)
E
(f I
au
E) (17 E)-I' f in L2 (do)
an
192 STEVEN G. KRANTZ
1f(z)1 1
I
2m z dC
3D(z,el) I
(Stokes)
1
dC
2771
C -z
a1Z
el
(Schwartz)
1/+
Now (1.13) and (1.14) imply that H2(fl) is a Hilbert space. Fix
z c fl and define the functional
Oz :H2(11), C
fF-f(z).
INTEGRAL FORMULAS IN COMPLEX ANALYSIS 193
ail
Formula (1.12) is called the Szego formula and kz(c) = S(z, C) the Szego
kernel (see [31, Ch. 11 for further details).
a 1 !a
=2 '
aa 1
2
(1)"ij
with smooth coefficients a a/. (If 0 < p,q e Z and the sum in (2.1)
ranges over jai = p, 1131 = q only, then u is called a form of type
(p,q) .) We then define
Likewise
We define a constant
W(n)= J W(C)AW(C)
B(0,1)
n
I wj(z,4)'(Cj-zj)-1 on iZxSZ\A. (2.2)
jj=1
f(z)
_W (n)
f 1(i) i w) A W (C) . (2 . 3)
do
rl(w)_---, 1 _2ni
1
C-z nW(n)
f(z)
m
f f(C) dC,
z
do
(wi(z,0,...,wn(z,0)
1-z1 .,
w(z,0 _ _ Cn-n .
I.
Let us calculate what the theorem says for this w in case n = 2. Now
w1
aa1
2dc1 + w I - dC2
aC2
1
- w2 w2 1 1A dC, A dC2
a 1 (X2
4=z2)
f(z) = z(2) 5 f(O IC-zI
A dC1 Ad 2
2
a=(a1,a2) cC00(SlxSZ\O): 1 aj(z,c) (cj-zj) =1 .
j=1
INTEGRAL FORMULAS IN COMPLEX ANALYSIS 197
(C2-z2)/K zI2
ag do
=0+0=0.
198 STEVEN G. KRANTZ
Hence, letting 0 < e < dist (z, SO), we have by Stokes' Theorem that (2.5)
= f(C)B(a1,a2) A 0)(C)
J
aB(z,E)
+ B(j6 1,62)AW(C).
JaB(z,E)
= J f(()dA A W(C)
aB(z,e)
= f d(f(() A A A W(C))
3B(z,e) (2.7)
=0,
f f f(C)B(j61,,62) A W(C)
Al 3B(z,E)
which, by (2.4),
INTEGRAL FORMULAS IN COMPLEX ANALYSIS 199
= A W(C)
J I C-z l4
3B(z,E)
(Stokes)
= 4f(z) 2((-z)Aw(C)+O(E)
64 J
B(z,E)
= 4f(z)E4 2(a(J)Au(C)+0(E)
J
B(0,1)
1 det
(C1-z1)(C2-z2)
200 STEVEN G. KRANTZ
1 det
(C1-zl)(C2-z2)
((C22)a2 aC ((C2-z2)a2)
_
(Cl z1) ? a2
Now an easy calculation, as in (1), shows that this last equals aA where
A = det
a2 a2 2
Hence au = f necessitates of = 0.
A simple calculation shows that, for n > 2 , this compatibility condi-
aft af k
all j, k. Notice, however, that when
tion is equivalent to
n=1
vkj = ,
11 = (B(0,4)\B(0,2)) U B ((2,0), 2/
202 STEVEN G. KRANTZ
d(F+v) = 0
or
=0.
n)
zn (3.2)
Our two examples show that solving the 3 equation is (i) subtle and
(ii) useful. Thus we have ample motivation to prove our next result.
u(z) =
2i dC A dC
J J ; -z
C
Proof. We have
INTEGRAL FORMULAS IN COMPLEX ANALYSIS 205
0 "U
az
a (-L
)g-2ni
ff
C
f(a/)(+z) do A dC
2ni fC
w d6 (o d A dC
II
1
2ni
D (0, R)
(C)dC
2ni J z
aD(0,R)
0= J udC= J
aD(0,R) D(O,R)
THEOREM. Let n > 1 and let D(z) _ q5 1dz 1 +-.. + (kndz 1 be Xclosed
on Cn and suppose each (kj CC(Cn). Then for any 1 < j < n the
of
function
(kl(z dC A dC
uj(z) _ - 217i
1
ff
C
C_zj
Proof. Fix 1 < m < n. We need to check that auj = (kj 1 < m < n. If
fim
m = j then the result follows from the lemma. If m ' j then use the
compatibility condition
d0j
=
-m to write
Sam azj
j (zl'...,zj-iX'zj+l,...,zn)
uj(z)
'3z_rn dC A dC
M- C-zj
C
4m
dCAdC.
2ni
ff C
By the lemma, this last equals <bm. Thus 9uj = 0. Notice that, if
f V j , then uj = 0 for zf large (since then (kj = 0 ). Also uj is
holomorphic for zQ large (since du = 40 is then 0 ). So, by analytic
continuation, u = 0 off a compact set. Next, uj - up - 0 since it is
compactly supported and holomorphic. Finally, uj e CCn) by differen-
tiation under the integral sign. 0
a(F +v) = 0
or
49 ((k f)+av=0
or
Notice how the hypothesis n > 1 was used in the proof to control
supp v. The Hartogs extension phenomenon has several interesting
consequences:
(i) A holomorphie function f in Cn , n > 2, cannot have an isolated
singularity. If it did, say at P, then f would be holomorphic on
B(P,2E)\B(P,E) for a small hence, by the Hartogs phenomenon, on
B(P,2E), and hence at P. That is a contradiction
(ii) A holomorphic function f in Cn, n > 2 , cannot have an isolated
zero. If it did, say at P, then apply (i) to 1/f to obtain a
contradiction.
(iii) If U C Cn is open, E C U, f is holomorphic on U\E, and E is
a complex manifold of complex codimension at least 2, then f
continues analytically to all of U. To see this, notice that for
n = 2 the set E is discrete and the result follows from (i). For
n > 2 , the result follows from the case n = 2 by considering
f j(il\E)ne ranging over all two dimensional complex affine spaces
QC Cn.
Now we return to discussion of the d operator. There are essentially
four aspects to this matter:
(1) Existence of solutions;
(2) Support of d data and d solutions;
(3) Choosing a good solution, where "good" means smooth or
bounded;
(4) Estimates and regularity.
Regarding (4), we have noted that ellipticity considerations imply
that when av = g then v is smooth wherever g is. As an exercise, use
the theorem of Section 3 to give another proof of this assertion. (Hint: if
g is smooth on B(P,r), then let 0 E C (B,(P,r)) satisfy = 1 on
canonical solution has this property. See [11], [17], [22], [16], [35].
Sibony [39] has shown that there are smooth pseudoconvex domains
on which uniform estimates for d do not hold. It is not known on
which parameters the uniform estimates depend (however see [13]).
Range [38], Henkin [17], and others have proved uniform estimates on
certain weakly pseudoconvex domains.
i.i '
U X A2(9). Assume the claim for now. Take F e A2(ct)\ U X...
This is the F we seek. The claim now follows from-
i.i Il
0 =}x(RN:p(x)<01
In
j, k=1
a2
dx dx
J J
(P) a] ak > 0 Aa E TP(dul) . (5.1)
n
<z,w>H = I zjwj
j=1
and if we identify
(t 1,...,t2n)
and likewise
2n
<z,w >Re = I tjsj .
j=1
(3) With Tl, P as in (2), let 5' p(9) = {a a Cn : <a, ap(P)>H = 01. We
call `.11(atZ) the complex tangent space to c%) at P. If
a e J'p(9) then is e `.'p(atZ). Also .`I p(9S2) C Tp( ) and it is
the largest subspace of Tp(atl) which is closed under multiplica-
tion by i .
a2p
0 Vw e 3'p(on) . (5.2)
!mar aZ.aZk
i, k=l 7
This theorem means that pseudoconvex domains are the natural arena for
complex function theory. Also (exercise, or see [31]) any pseudoconvex
domain is the increasing union of smooth strongly pseudoconvex domains.
So strongly pseudoconvex domains, in a certain sense, are generic.
In order to unify and illustrate the ideas introduced so far in this
section we prove
n
d2
(P)aj ak
1,k=1 j k j,k=1 l
n 2
ap (P)aa
+12 , k=1
1
dzjdzk jk >0
But a similar inequality also obtains for is e fp(dil). Adding the two
inequalities yields the result. o
o o =T L20 o
L2(fl) 1)(1) L(o 1)(ft)
11
H1 H2 H3
The operators T,S are of course unbounded, but they are densely defined
216 STEVEN G. KRANTZ
See [201 for details. Rather than prove (6.1), it is more convenient to
study the symmetric inequality
f(z) = 1 P077
A W(C)
nW(n)
an
- ff()A7J
Q
-z
(0 = f(i)n A W(C)
on the domain fl\B(z,e). Imitate the proof of the Cauchy Integral Formula
(or see [311).
Now fix 1 a bounded, C2, convex domain. Choose e > 0 so small
that 1 == l z c Cn : dist (z, 11) < e I is convex (hence pseudoconvex). Let f
be a smooth, 0-closed (0,1) form on fl. By Hormander's theorem, there
is a smooth u on fl such that du = f. We apply the Bochner-Martinelli
formula to u (which is certainly in C(fl)
I ). Thus
218 STEVEN G. KRANTZ
Z2
u(z) = fu(017
WW(n)
au
nW(n) f An
-z{
z2
A)(t) (6.3)
The first term on the right is not useful, since it involves u , so we will
remedy matters by subtracting an appropriate holomorphic function from
the right side of (6.3) (see the discussion in Section 4 on choosing a good
solution). The Cauchy-Fantappie formalism now comes into play:
If 0 =(p<01, let
ap -ap
Xn
(D(z, 0 4)(z,
with 4(z, I d
(C) (zj -Cj) and observe that
j=1 Oxj
H(z) = -1 fu(xw(z)) A
v(z) = nW(n)
1 {fu n
/ A - J uw-z
12
asp ale
nW(n) f
Q
A ri
z_z
=z2
Aw(C)-I-II.
z-z
Let G = SZ x [0,1] and define g(z, C, A) = (1-A) + Xw(z,0 to be a
K z I2
form on G. Then
I= n
W(u) full(s) A 0)(C)
aG
1
nW n)
f d(u(C)rl(b) A
G
But
+ u(C)d(n(g)) A W(C)
= f A-J(g)AW(C)
v(z) - n . W(n)
5 f(c) A n(g) A W(C)
-J A 17
-z
a
(D(z,
and
(P(z ( )(z- ) .
The standard reference for Henkin's work is [181; see also [311 Similar
formulas were derived by Grauert-Lieb [111, Kerzman [221, and (6vrelid [35].
Now we assume that i1 is strongly convex, and show how to use
Henkin's formula to obtain uniform estimates for solutions to the
problem. For simplicity we work in C2 only. So
a
f ,(g) Aa )f ^ [g1 dg2 dA-g2 dA AdC1
1
a 1 (C)(r,2-z2) +
2
(C)(1-z1)
v(z) = 2W(2)J
4'(z, S)I4_zI2
all
A (f +f2( ) n
1 f1(b) (_I= 1) +
2W(2)
I K z I4
d1S n d Cz n dCl ^ d s2
1
2W(2)
f ndC1 AdC 2
as
ag
190
and
Sl B(z,R)
<C f r3dr=CR<- .
r3
0
2
ap
(C) a j= 0 a c `.`P(R)
and
2
dp
Re 5 (C)aj = 0 a ETp(SZ).
j=1
It follows that
2 2
>C 1t1+t2+t3+Ip(z)I1
fAi(z)da(i)= J + J .
an B(nz,r0)lao as\B(rrz,r0)
224 STEVEN G. KRANTZ
The second integral is trivially bounded since when iz-Cl > r0 then Al
is bounded. The first is majorized by
=C
tl+t2+t3<CrO ti+t2+t3<Cr0
Y1+Y2
Now
1
IY11<C dttdt2dt3
I (t2 +t3)
IPI+I t1 I<t2+t3
t1 +t2+t3<C r0
< C J 1 dt2dt3
t22 + t23
t2+t3<Cr0
r0
<C J
Likewise
INTEGRAL FORMULAS IN COMPLEX ANALYSIS 225
dtldt2dt3
IY21 < C
t22 +t23 (it,1 I + IPI)
IPI+It1I>t2+t3
ti+t2+t3<Cr0
<C
I
t2+t3<C r 0
t2+t
1
2
3
(Ilog(t2+t2)I+llog(Crdl)dt2dt3
2 3
C r0
<C
f
0
(Ilog rl + log CI)rdr
<C-<00.
Thus
f JAIIda<C <-o .
an
IV II < C Y-IIfjIl L, .
L(tt)
(i) F(12) C U
(ii) F(6\5) C Cu
(iii) F(4) C au
(iv) image F is transversal to X.
The upshot of this theorem is that the Henkin singular function (D which
we know how to construct on U can be pulled back to Q. The construc-
tion of the Henkin solution to the a equation and the uniform estimates
follow just as before.
(C)wjwk>CIwl2 dw (Cn,
J k
2, y
L(z, ) _Cj) + aP
J J k k
i=t J J k-t
INTEGRAL FORMULAS IN COMPLEX ANALYSIS 227
The function L, called the Levi polynomial, has the property that there
a neighborhood Uc of C such that sz n UC n #z : L(z, t) = 0} _;c } . One
can modify L, by solving a a problem (see [181 or [311) to obtain 4i
n
such that 5 n {z : d>(z, c) = 01 = I I and d>(z, C) =I Pi (z, 0 (z)-Cj)
j=1
with Pi holomorphic in z . Henkin's program may be carried out using
this and w(z, C) = (-PI(z, 011b(z, 0, ..., -Pn(z, 0/(D(z, 0)
Notice that, by the discussion in the preceding paragraph, 1/L( , C)
is a local singular function at C. This, together with the uniform esti-
mates for the a equation which we have obtained, completes the program
outlined in Section 4 to show that a strongly pseudoconvex domain is a
domain of holomorphy.
S(z,4) = fS(w)S(z.w)da(w)
aQ
= f
ag
= S(C,z )
=S(C,z)
228 STEVEN G. KRANTZ
S:f i+
(c) S is idempotent.
Therefore S is the Hilbert space projection of L2(8l1) onto H2(1l).
It turns out that the Henkin operator on a strongly pseudoconvex
domain very nearly has properties (a)-(c). First, by a theory of non-
isotropic singular integrals developed especially for boundaries of
strongly pseudoconvex domains (see [81, [361), the Henkin operator
H:fi.nWn) f(4)r!(w)A
J
N(z, C) N((,z)
n(z, 0
is a kernel which is less singular than the original Henkin kernel. Thus
H - H* , rather than being a non-isotropic singular integral operator (as is
H ), is a smoothing operator. This observation of Kerzman and Stein is
now exploited as follows. Denote H* -H = A .
The reproducing properties of S and H guarantee that
(1) S = HS
and
(2) H=SH.
Thus
S -H - S(H* - H) = SA
S = H + SA
= H + (H+SA)A
=H+HA+SA2
(7.2)
= H + HA + (H+SA)A2
-H+HA +HA2 + SA3
.. = H + HA + HAk + SAk+1
Now we know that each of the operators HA, HA2, -- are smoothing. If
we apply both sides of (7.2) to a sequence Oj E Cc '(4) such that ) -, S
230 STEVEN G. KRANTZ
<f,g>= ffdv
ft
[fil = f If 12 dV' /2
Q
for K CC [Z, is easily derived from the mean value property for holomor-
phic functions. As in Section 1, the abstract Hilbert space theory yields
a reproducing kernel for A2 which we call the Bergman kernel.
Just like the Szego kernel, the Berman kernel (denoted by the letter K)
satisfies K(z, C) = K(C,z). Thus the associated operator
INTEGRAL FORMULAS IN COMPLEX ANALYSIS 231
B:f H ff()K(zs)dV()
12
K(z,z) = J K(z,w)K(w,z)dV(w)
12
dV(w) > 0 .
I K(z,w)
2
gij(z) _ log K(z,z)
l 0-Til
integral while the Bergman integral is a solid integral. How can we com-
pare functions with different domains? What we would like to do is apply
Stokes' theorem to the Henkin integral and turn it into an integral over Q.
However, for z c fI fixed, Henkin's kernel has a singularity at = Z. So
Stokes' theorem does not apply.
The remedy to this situation is to use an idea developed in [19], [30],
[331: for each fixed z c fl, let
)
N(z,i
(Dn(z,
S) Easl
(i) 0tiB
or
On the disc and the ball this conjecture is correct; for the ball in Cn one
can calculate (see [31]) that
K(z, ) = nn! 1
nn (1-z. )n+1
Some results about the modified conjecture may now be formulated. Let
us agree to topologize the collection of all smoothly bounded strictly
pseudoconvex domains by equipping their defining functions with the C
topology. Then we have (see [12], [13]):
Sketch of proof. We may as well suppose that no ball, else the result
is straightforward. Then normal families arguments show that, for it
sufficiently near S10, [1 ball. Thus Aut 0 is compact and, by
averaging the Euclidean metric, one can construct a new metric y, smooth
236 STEVEN G. KRANTZ
DEPARTMENT OF MATHEMATICS
THE PENNSYLVANIA STATE UNIVERSITY
UNIVERSITY PARK, PA. 16802
BIBLIOGRAPHY
[11 L. Boutet de Monvel and J. Sjostrand, Sur la Singularite des noyaux
de Bergman et Szego, Soc. Mat. de France Asterisque 34-35 (1976),
123-164.
[2] R. Burckel, An Introduction to Classical Complex Analysis,
Birkhhuser, Basel, 1979.
[3] D. Catlin, Necessary conditions for subellipticity of the a-Neumann
problem, Ann. of Math. (2)117(1983), 147-172.
[4] , Boundary invariants of pseudoconvex domains, to appear.
[7] G. Folland and J. J. Kohn, The Neumann Problem for the Cauchy-
Riemann Complex, Princeton Univ. Press, Princeton, 1972.
[8] G. Folland and E. M. Stein, Estimates for the ab complex and
analysis of the Heisenberg group, Comm. Pure Appl. Math. 27(1974),
429-522.
[9] J. Fornaess, Strictly pseudoconvex domains in convex domains, Am.
Jour. Math. 98 (1976), 529-569.
[10] B.A. Fuks, Introduction to the Theory of Analytic Functions of
Several Complex Variables, Translations of Mathematical Monographs,
American Mathematical Society, Providence, 1963.
[11] H. Grauert and I. Lieb, Das Ramirezche Integral and die Gleichung
of = a im Bereich der Beschrankten Formen, Rice Univ. Studies
56 (1970), 29-50.
[12] R. E. Greene and S.G. Krantz, Stability of the Bergman kernel and
curvature properties of bounded domains, in Recent Developments in
Several Complex Variables, J. E. Fornaess, ed., Princeton Univ.
Press, Princeton, 1981.
[13] . , Deformations of complex structures, estimates for the
d equation, and stability of the Bergman kernel, Adv. Math.
43 (1982), 1-86.
[14] , The automorphism groups of strongly pseudoconvex
domains, Math. Ann., 261 (1982), 425-446.
[15] -, Characterization of complex manifolds by the isotropy
subgroups of their automorphism groups, preprint.
[16] P. Greiner and E. M. Stein, Estimates for the a-Neumann Problem,
Princeton Univ. Press, 1977.
[17] G.M. Henkin, A uniform estimate for the solution of the 3-problem
on a Weil region, Uspekhi Math. Nauk. 26 (1971), 221-212 (Russ.).
[18] , Integral representation of functions holomorphic in
strictly pseudoconvex domains and some applications, Mat. Sb.
78 (120) (1969), 611-632; Math. U.S.S.R. Sbornik 7 (1969), 597-616.
[19] G. M. Henkin and A. Romanov, Exact Holder estimates of solutions
of the d equation, Izvestija Akad. SSSR; Ser. Mat. (1971), 1171-1183,
Math. U.S.S.R. Sb. 5(1971),1180-1192.
[20] L. Hormander, L2 estimates and existence theorems for the a
operator, Acta Math. 113 (1965), 89-152.
[21] T. Iwinski and M. Skwarczynski, The convergence of Bergman func-
tions for a decreasing sequence of domains, in Approximation Theory,
Reidel, Boston, 1972.
INTEGRAL FORMULAS IN COMPLEX ANALYSIS 239
241
242 ALEXANDER NAGEL
Much of this paper is an exposition of joint work with Eli Stein and
Steve Wainger, and I am particularly grateful to them for many years of
stimulation, encouragement, and collaboration.
n 1/2
Ix-yI = Ixj-yj12
j=1
In this example, the important first order operators are just the partial
derivatives with respect to the n variables The Laplace
O-A 1 n
VECTOR FIELDS AND NONISOTROPIC METRICS 243
operator
where
n+1
cn=r (n21`/n 2
Then:
(a) Pf is harmonic on R++1
(b) Pf extends continuously to the boundary and takes on the
boundary values f.
(c) fRn IPf(x,y)ipdx < Iliii for 1 < p < 00.
Proofs of these assertions, along with many other of the results dis-
cussed here, can be found in Stein [16], Chapter III.
Assertions (c) and (d) above suggest a generalization of the Dirichlet
problem to certain classes of discontinuous boundary functions. For
1 < p < -, let hp denote the space of functions u(x,y) harmonic on
R++1 which satisfy
Note that if B(x0, S) = lx r RnI Ix-xoI < 8l are the balls defined by the
standard Euclidean metric, then
where the supremum is taken over all Euclidean balls B which contain
x0. The basic estimates for the maximal operator are given in:
THEOREM 2 (Hardy and Littlewood). For I < p < 00, there are constants
Ap < 00 so that
(i) IIMfIIp <ApllfIIp if 1 < p < 00
These are the two quantitative estimates which underlay the qualita-
tive statement of Fatou's theorem. Complete proofs of these results can
be found for example in Stein [16), Chapters I and III. However, since we
shall appeal to this kind of argument again, we now recall how Fatou's
theorem follows from these two theorems.
Let p < oo and let u c hp. If s > 0 and if we let fs(x) = u(x,s),
then
sup Iu(x,y+s)I = Na[P(fs)I(x0)
(x,y)Era(xp)
< CaIIM(fs)](x0) .
Therefore if A > 0
< [CaA-IIIM(fs)IIp]P
where the limits are taken as (x,y) approaches (x0,0) and (x,y) E Fa(xO).
Then the following facts are easy to verify:
(a) fau(x) < 2Nau(x)
(b) Sla(u+v)(x) < Slau(x) + Slav(x)
(c) Slau(xo) = 0 if and only if u has a limit within Fa(xo)
(d) Slau(x) - 0 if u = Pf and f is continuous.
Now let un(x,y) = u (,y + n) . Then
< [2CaApk-Illu-un11hp]p
balls are also involved in studying the fundamental solution for the
Laplace operator.
An important fundamental solution for A is given by the Newtonian
potential:
N(x) =
c(x) = (2)
Rn
and if
Rn
f N * f(x) = r N(x-y)f(y)dy .
Rn
IN(x,y) < C
(4)
IV N(x,y)I + IVyN(x,y)I < C sIB(x, s)I-1
where S = Ix-yI. Written in this way, these inequalities again make clear
the important role played by the Euclidean metric and the Euclidean balls.
This importance can also be seen when we consider certain singular
integral operators. We claimed earlier that N * f is two orders smoother
2
than f . Using formula (2), this means that for all i, j , should be
9256
as smooth as A S6. Thus we are led to the study of the operator which
32
carries A-0 to . There are two ways in which we can think of
i j
this.
Rn
where kij(y) = cn 2
ylyl
IyI
IyI-n
for an appropriate constant cn 0. We note
that the kernel kij is not locally integrable at 0 , so we must study the
integral in equation (5) in the principal value sense. Now the kernel
kij(x,y) = kij(x-y) satisfies the following estimates in terms of the
Euclidean metric:
250 ALEXANDER NAGEL
Ikij(x.y)I <CIB(x,6)1-1
S)I-t
IVxkij(x,y)I + Ioykij(x,y)I <
CS-1IB(x,
f kij(x)dx = 0 ,
a<Ixl<b
2
and can also be established by studying the operator AO in
terms of Fourier transforms. Define:
(e) = r
Rn
so that
-O(x) = f e
Rn
so
a295 ei el
lei I
(2) The measure has the "doubling property" relative to the family of
falls B(x,5) = IyeXJp(x,y)<5I: There is a constant A so that for all
x eX, 5>0
g(B(x,25)) < AW(B(x, 5)) .
where the supremum is taken over all "balls" B which contain x0. We
prove:
Proof. For X > 0, let E = IMf >X1, and let F C E be any compact sub-
set. If x e E, since Mf(x) > A there is a ball Bx containing x so that
(Bx 1-1 I
If(y)Id(y) > A
Bx
Suppose B is a ball from our original finite sequence which does not
appear in the subsequence. Then there is a first k so that B; fl B 0.
k
By property (2), > &.. Let xik be the center of and xj the
Sik k
center of B1. Then if z < Bik fl Bl ,
< 2KSi
k
< K(2K+1)S. .
k
N 1m`
L.t C U Bj C U Bi k
j=1 k=1
m
Hence (F) < Y p(Bi ). But by property (2) of spaces of homogeneous
k=1 k
type
* 1+1092K(2K+1)
P(Bih) < A (Bik)
=A1p.(B'k).
Thus
since the balls (BikI are disjoint. Since I was an arbitrary compact
subset of E , we obtain the same estimate for the measure of E.
3t
In
j=J
a2
'= a
x
if t>0
E(x,t) =
0 if t<0.
Then E is C on Rn+1\1(0,0)L, and LE = S in the sense of distribu-
tions. (See Folland [5], Chapter 4.) Thus if 0 c C(Rn+1)
0 +
r00 n
(k(x,t) = J (4rrs) 2 e-IYI2 /4s I(x-Y,t-s) dy ds
J
0 Rn
= f S (4(t-s)
2e--2/4(t-(Y,s)dyds
J
with a= (Ix12+t2)1 /2
is false. To see this, let t =x12 with lxi
small, so E(x,t) =CIxt-n, while S =(Ix12+Ixl4)1/2 ti IxJ, so
S-n+l ti xj-n+1 Thus we cannot obtain the appropriate kind of estimate
so that
are now ellipsoids of size S in the directions of x1, ,xn, and of size
S2 in the direction of t. Thus
8n+2
IBP((x,t), S)1 ti
256 ALEXANDER NAGEL
< Ap IIL0IIp
Ilp
a20
A p IIL0li p
p <-
H= {(z1,...,zn'zn+1)
n
= (z,zn+l) c Cn+1 I1mzn+l
> I izjI2 = IzI2
j=1
n+1
Recall that U is the image of the unit ball B = (w1,-',wn+1)I jwj12<1
wk
zk = 1<k <n
l+wn+1
l-wn+1
zn+1 = I l+wn+l
= I(z,t+ilz12)Iz c Cn,t c RI
where
= h2 h1 .
= Imzn+l - Iz I2
= p(z,zn+l) '
have identified (w,s) with the point (w,s+iiw)2) f d2, and this is the
same as Th(0,O). Thus under the identification, Hn acts on (90 as
follows: if (z,t) fan and if h = (w,s) f Hn , then
VECTOR FIELDS AND NONISOTROPIC METRICS 259
Th(z,t) = ThT(z.t)(0,0)
= Th.(z,t)(0,0)
_ (w,s) (z,t)
then
Hence
and
+ 12Im(<z,u>-<z,w>-<w,u>)I
Theref ore
Iz-wI < S
= IRe(wn+1-zn+l)+i(lw12+Iz12-2Cw,z>)I
= IRe (wn+l-zn+1-2i<z,w >)+i lw-z 121
ti lw-zI2+(Re(wn+1-zn+1-2i<z,w>)l
ti< S2
VECTOR FIELDS AND NONISOTROPIC METRICS 261
B(- Z,
ti 82n+2
Thus the doubling property of the balls is verified, and di1 (or Hn )
equipped with the pseudometric d is indeed a space of homogeneous type.
We now want to discuss the analogue of Fatou's theorem for boundary
behavior of holomorphic functions in Q. This problem was first studied
by Koranyi [9] for domains like 1, and was later generalized by Stein [17]
to general smoothly bounded domains in Cn. Here we want to emphasize
the role of the nonisotropic balls on the boundary, in analogy with the
role of Euclidean balls on Rn in Fatou's theorem.
We begin by defining appropriate nonisotropic approach regions in it.
Let n : SZ K1 be the projection
n(z,zn+1) = (z,zn+l-ip(z,zn+1)) -
Aa(w) = l(z,zn+l)cg!n(z,zn+l)EB(w,ap(z,zn+l)
It-s+21m<z,w>I < ay .
If f c Lloc(3t1) , define
COROLLARY. For 1 < p < oo and a > 0. There are constants Ap a < 00
so that if F f Hp(11)
(i) INaFNLP < Ap,aIIFIIH for '<P< oo
P
(ii) IlCE3f INaF(C)>AII < A1,aa-1IIFIIHI if p=1.
and
INaF>Al = U INaFE>AI
E<0
We can now apply exactly the same argument we used for Fatou's
theorem to prove
2
Iu(O,iy)1 < -I
Ay If 2
IS12+It-yI2< Y)
f211
< 2
Ir2y2 If 2 0
u(N/tei0,s+it)Id9dsdt
y 3y
2 rr 2 2
r
0
f Iu(V/teie,s+it)Idtdsd9.
< 4
Try2 f
B((0,0),2\)
Z IB((0,O),2,Vy-)I-1
J Iu(C)Ida(C) .
B((0,0),2 J )
But now using the doubling property of the balls, it follows that there is
a constant A so that if (z,zn+1) E Aa(0,0)
S((z,zn+1) =
asp
where
=cn[i(wn+t-zn+1)-2<z,w>]-(n+1)
S((z,zn+1),(w,wn+i))
with cn = 2n-1n!/nn+1 (see Nagel and Stein [11], page 23). In particular,
for z = (z,t) and w = (w,s) on dQ we have,
.
S(z,w) = cn(i)-(n+1)[(s-t-2Im<z,w>)-ilz-wl2]-(n+1)
Xj =
+2YjYj = -2xj. , T
where we write zj = xj + iyj. These vector fields form a basis for the
left invariant vector fields on Hn. Put
Zj = (Xj-iYj), Zj = (Xj+iYj)
2 2
and consider for a c C.
VECTOR FIELDS AND NONISOTROPIC METRICS 267
n
'a=-2 1 (ZjZj+ZjZj)+iaT
j=1
n
abf = Zjdzj on functions,
j=1
and we extend this in the usual way to (0,q) forms on Al. In L2(Hn)
we can define a formal adjoint (ab)*, and the Kohn Laplacian is then
b=ab+ab .
_jn+a a
(1212+jt) -(n 2
`` 2
Y'a(Z,t) = (IZ I2-it)
I'ka(z,t)I <
In the case of the Heisenberg group, the basic vector fields are
X1,"',Xn'Y1,"',Yn, and the vector field T which is given "weight"
two. We shall later see how the general construction applied to these
vector fields gives the nonisotropic pseudometric d .
cek c C(Q). Here [X,Y] --- XY -YX is the commutator of the two
J
vector fields.
There are several basic examples to keep in mind.
(A) Let N = q = n , let Yi = and let dj = 1 for 1 < j < n . In this
n+l I
I c3xj
012
Yj +2yj 1<j<n
with dj = 1
Yn+j = - 2xj 1<j<n
Yj
In this case of course we are dealing with the Heisenberg group, and
we shall recover the invariant metric on Hn defined earlier.
(D) (An example of Grushin type). Let N = 2, q = 3, and let
X Y3 = [Y 1,Y 2] = a
2
0x2
X(1) =
iX1,...,xp1
5. Global definitions
DEFINITION 1. Let xp,xl c f2, and say p(xp,xl) < S if and only if
there is an absolutely continuous map 0: [0,1] -+ fl with c(j) = xj ,
j = 0, 1 , so that for almost all t c (0,1)
VECTOR FIELDS AND NONISOTROPIC METRICS 271
q
(t) _ I aj(t)Yj(cS(t))
j=1
d(Y.)
where laj(t)I < S
q q
4'(t) = a j(t) Y -(O(t)) , &'(t) = F b j(t) Y j('(t)) , with
1
d(y,z).+E)d(Yi)
Define 0: [0,1] - 9 by
0(at) 0<t<1/a
0(t) =
at-1 1<t<1
as -1 a
d(y,z) + E
where a = 1 + Then 0(0) = x, 0(1) = z and 0'(t)
d(x,y) } E
q
E cj(t)Yj(0(t)) where
j=1
C[Ix-yId(Yj)]d(Yj)
Icj(t)I < CIx-Y1 =
< C[Ix_yIm]d(Yj)
O'(t)dt
J0
1
q
< I laj(t)I IYj(O(t))Idt
0
j=1
< C8
[X1, [x1,x2] ]] = k! vy
Case 1. If we let
and
then this shows that the points (1 +5,0) and (1, S) belong to the p3
ball centered at (1,0) of radius S. In fact, this ball is essentially the
Euclidean ball of radius S centered at (1,0).
This shows that the p3 distance from (0,0) to (0,8k+1) is at most 38.
In fact the p3 ball with center (0,0) is an "ellipsoid" of size S in
the x direction and size Sk+1 in the y direction.
Thus we have given three possible definitions of a metric or pseudo-
metric in terms of the family of vector fields It is also clear
from the definitions that we have the following inequalities:
In fact, these three quantities are locally equivalent. One can prove that
for every x0 E (I there is a neighborhood U of x0 and constants C1,C2
so that for all x,y c Q, x 4 y , and j =2,3.
VECTOR FIELDS AND NONISOTROPIC METRICS 275
P3(x,y)
0<cI<P(x,y)<c2<oo.
6. Local definitions
In order to begin to see why these pseudometrics are locally equivalent,
we need to consider a local definition of metric, and this in turn relies on
the notion of an exponential map.
Given a point p c (1, we let TpiZ denote the tangent space to SZ
at p. Suppose we are given C vector fields defined near
p which form a basis for RN at p. Then we can construct a map from
a neighborhood of 0 c T 1 to a neighborhood of p in [ as follows:
N
every tangent vector v at p can be uniquely written as I
j_1 J j
(p) = v
N
with E RN, and soj=1
I J J
a smooth vector field defined
near p. We can flow along the integral curve of this vector field for unit
time if IlajI is sufficiently small, and the result is by definition
N
exp jS (p) , the exponential map of V.
J
If V_ E Tf) we say
Nx(v)<a
if and only if
q
v = ajYj(x)
j=1
d (Y ) -.
where J <5 i . Of course this representation of v need not be
Ia j
IV ETxSZINx(v)<6
d (Y .)
QS = 1(a1,...,aq)(RgI IajI<S I?
For any N-tuple I = let d(I) = d(Yi )+ ... + d(Y. ), and let
1 N
AT = det (Yi1,.. ,YiN).
VECTOR FIELDS AND NONISOTROPIC METRICS 277
Proof. For each N-tuple 1, the image 0(Q3) contains all vectors of
N
d(Yi)
the form ajY1 where Iaj I < and this is just the image under
i=1
a linear map`from RN to RN with determinant AI. Thus
5d(I)
IXII < (;l)
N
Yj=4 bjkYk 1<j<q,
k=1
A,
Ijk
bjk X1
0
d(Yk)-d(Y j) .
Ib jkI _ 5
278 ALEXANDER NAGEL
q
Now let v c O(Q8), so v -_ ajYj with Iajl < 8 d(Y I) . Then
I
j=1
N 9
= " ajbjk Y k
V k- j=1
and
q
1: ad(Y
< J)Sd(Yk)_d(Yj) = q Sd(Yk)
j=1
(gS)d(Yk)
Hence
d(I) I d(1 )
!)L1
IB(Q8)I < q
0
N d(Yi
BI(x,$) = YcHIY=exp(I ajYi) (x) with jajI <8
1 )
IA1
(x)I ad(10) > IAJ(x)I
8d(J) for all J
o
q
0,(t) _ bj(t)Yj(o(t))
j=1
Sd(Y'
with lbj(t)I < almost everywhere. Now assume without loss that
I0 = and let denote canonical coordinates near x
relative to the exponential map using Then the curve c(t)
is given in canonical coordinates by (uI(t),...,uN(t)) and our object is
d(Y.)
to show that there is a uniform constant C with Iuj(1)I < (CS) . But
f Y, bk(t)Yk(c(t))(uj)(t)dt
k=1
0
N
Yk =1 akYP
=1
S
X =X+ ajf(x,t) .
f=1
where S = p((x,t), (y,s)), and p is the metric constructed from the vector
fields They then define a restriction operator
Rk(x,y) = fk((xiO)(Yis))(s)ds
RS
and it N>2,
<CS2-]IB(x,5)I-1
J
where S = p(x,y).
it = Iz cCn+llp(z)<01
Lj (9P a (9P a
Nn+I dzj - Nj
TaP
[fin+i
a
Wzn+l
_ dp a
()n+l Wn+1
Ak(0 =I IAil,...'if(S )I
and
m
A(C.8) = Aj(t)8j
j=2 2
where the first sum is over all the generators of 9k. Note that Ak(O is
a function whose size measures how much T component the commutators
of XI,---'X2, of length < k can have. In particular, since c3il is of
type m, then Am(0 > 0 > 0.
VECTOR FIELDS AND NONISOTROPIC METRICS 285
2n
77co-QIrl=exp ajXj+yT (c), where
1
THEOREM 12. The "balls" B((, S) are equivalent to the balls B(C, S)
defined in part fl in terms of the vector fields i.e., there
is a constant C so that B((, S) C B(C, CS) and B((, S) C B(C,C&).
n=exp tajXj+yT (0
M
It 1
y=IbjAI
2
m 2n
yT = Ibi X1. + I .6I.,EXE
2 Q=1
then g has a formal Taylor series at the origin, and it is given by:
00
k
g(at,...,a2n,Y)
^' I
k!IajXj+YT\\)
flo
k=0
SZ =(z ECn+llp(z)<01
P(z) = I as zap
then
R(z,w) = S` as za W13
n
R(z,w)=I zjwj-Zi(zn+l-wn+l).
j=1
288 ALEXANDER NAGEL
Also, on the Heisenberg group [Xj, Xn+jI _ -4T so A(w, S) 32 for all
w E M. Thus in this case
1 n
B#(w,S) = z -F ul Iz-wI <6,
L
14
j=1
zjwj 2i (zn+1 -wn+1) <32
and it is easy to check that this defines essentially the same balls as we
did earlier in part I.
We now want to show that the balls B'a(w,S) are equivalent to the
balls B(w,S) defined in terms of the exponential mapping. For simplicity
we will do this only in the special case
L _ a - 2i a", (zl) a
(9f1 azl az2
O(z)_Cz)-MO)-2Re( (0)zl}
VECTOR FIELDS AND NONISOTROPIC METRICS 289
zl=zl
m j
z2 = z2 - i c (0)+2 1 i (0)zi
j=1 dzi
1
j
a-o
00 1<j<m
chi 00
L= +i'(z)N, 5j (97
(Z) Jt
T=?.
We can calculate the various functions XiI", ik using the vector fields L
and L (instead of Re L, Im L ). Thus:
[L,L]=-2i'!
2
so the ideal 92 is generated by O or Ac . Thus
aza
A2(z,t) = IoO(z)I
290 ALEXANDER NAGEL
alo
Ak(z,t) = I
a+,B<k aza&-#
a,13>1
[0,1] CxR
with
q (t)=a
G
12 (t) = y + is iff (01 (t))
f (aas)
0
_a (as) ds I2A2(0) +...+ JalmAm(0)
= A(O, lab
VECTOR FIELDS AND NONISOTROPIC METRICS 291
In particular, we see that the image under the exponential map of the box
is essentially
ItI<A(o,a), IzII<8
lu(zl,z2)I <CIBI-'
f lu(C)Ida(()
B
4 u(0,s+it)I ds dt .
lu(0,iy)I < (8)
7ry2
Now for each s + it we want to imbed an analytic disc into fl, whose
boundary is mapped to A) and whose center is mapped to (0,s+it). To
do this, we try to find S > 0 and a function G(4), continuous for
141 < 1 , holomorphic on 1 , so that the holomorphic map
(84,G(4))
and
G(0)=s+it.
Let 0&(ei0) _ O(Sei0). Let P[0S] be the Poisson integral of (kS , and
let Q[958] be the conjugate Poisson integral, with Q[0,51 (0) = 0. Then
VECTOR FIELDS AND NONISOTROPIC METRICS 293
t
2n f .0(8 ei0)d9 . (9)
2n
dS 2 f 0(S ei0)de =
2 775
r AO(x,y) dx dy > 0
0 X2+Y2<5 2
fo21r
so the function t(S) = Zn j(S ei0)d0 is a monotone increasing func-
tion of S . Thus given t, there is a unique S = S(t) so that equation (9)
holds. We thus obtain
3Y /2
Iu(0),iy)1 < na 22
J Iu(s(t)e
Y
0
-Y /2 Y12
s+ dt dOds
A(S) = 1
nS2
ff Ad!(x,y)dxdy
x2+y2<52
Y/2 2s 32
u(O,iy)< Lff
77Y
2
J
-Y/2 0
S(I)
2
We defer the proof for a moment, and return to the estimate for u(0,iy).
In our last integral, when r is between S 2 and S , t(r) is
between and 2 . 2
It follows from the lemma that in this range,
2
A(r) ti y/6(y)2 , and it follows from part (iii) of the lemma that the range
of s integration in the integral is contained in { Is I < Cy I for some
constant C. Thus:
VT
CS(y)
Iu(0,iy)I < C
yS(y)2 ff
IsI<Cy 0
f Iu(rei0,s+ic(reie)Irdrdeds
0
t(S)=2n
('
J0
2n
L
0(Sei0)d9>AmL r as+(30
azar}i
(0) I Sa+R = AmA(0, a)
2n S
f
I
0
f 0
Ab(rei0)rdrde> Am
llazaaza
(0) Sa+1i = AmA(0, S) .
J
j (0)=0, 0<j<m; 66(b(z)>0 if ,zj<1 ;
and
I as
+(3
o (0)
1(gzaaZo
1
296 ALEXANDER NAGEL
0 fe10o
0
Sf(z) = ff()S(zs)da() , z e iZ .
0 il
VECTOR FIELDS AND NONISOTROPIC METRICS 297
where the supremum is taken over all F e H2(1) with IIFII 2 < 1 . But
H
by our theorem, if F E H2(SZ),
1 /2
< CIBI-1 /2 r IF( )I2 da(C)
aJiZ
= CIBI-1 /2IIFIIH2
COROLLARY. If z E SZ
e95(z) = AO(x+iy)
is actually independent of y .
Our approach is the following: if
L= -2ic3z
1 1
(z )a 2
L
07 Tt
2[aao 49
+2 (I-
We now make a change of variables on C x R
D(x,y,t) = (x,y,t-A(x,y))
where
A(x,y) _ - f (t,y)dt .
0
Then if we put
x
b'(x) = f A95(t)dt
0
VECTOR FIELDS AND NONISOTROPIC METRICS 299
and
L = +i r + b'(x) a
where b"(x) > 0. Our finite type hypothesis is now E IbW)(x)l > go > 0,
i=2
and we want to obtain estimates in terms of balls defined by the vector
fields and 22- + b'(x) in R3.
dly -t d
`.fu = u(x,rl,r) = f e
R2
so that
u(x,y,t) = ff e21ri(Yri+tr)u(x,i7,r)di7dr
R2
Luf-1ifu
where
Lu . e2"(nx+b(x)r) dx (e 2rr(rix+b(xp)u)
Let
q,(x,rr,r) = e 2rr(nx+b(x)r)
300 ALEXANDER NAGEL
and let
M.g(x,rl,r) = O(x,r1,r)g(x,71,r) .
Then
Lu=f 0- 1-iA,fu.
dx
Now
and
f g(x,r1,r)12e4r7(t7x+rb(x))dxdr7dr
< +- .
Let
P: L2(e4n(rlx+rb(x))dxdrldr) -, L2(e4n(rlx+rb(x))dxdrldr)
be defined by
Pg(x,rl,r) =
P .` -IM PMtA
<g,1>1
Pr1,r g = < 1,1 >
00
r g(r)e4n(rlr+rb(r))dr
00
00
r e41r(Tlr+rb(r))dr
Thus
00
where
Kll,r(x,Y) =
where
S((x,Y,t); (r,s,u)) = ff
K,?,r(x,r) dry dr
00 00
e21rq((x+r) + i(y-s))
fe 27rrt(b(x)+b(r))+i(t-u)]
foe 00
di dr.
0
f
-00
e41r(nr-rb (r)) dr
-00 r e4n[r7r-rb(r)]dr
-00
VECTOR FIELDS AND NONISOTROPIC METRICS 303
F(A+it, r) _
I_
2rrr 2b +itb' (`]
/1
00
f+ e2ni77td77
e (2)
-00
J
00
2(r+)+rbe
47rl77r+rrb ' (-rb
LLL
2(2)1J
-00
G"(r)= -rb"(r+2)
m
G(r) r 1 b()) 1
j=2
( -) r)
Hence
00
[2b(3) + itb'(2)1 e2rrjr7td,7
F(a+it,r) = e
277r
L JJ f
J
-00
00
4n nr-r
m f
1 b (j) ( ) r7
I 1 b(j)(a>rj12
In
2 =1
j=2 1!
and in the last integral, make the change of variables r - gr, 77 77.
304 ALEXANDER NAGEL
Then
where ai = 1 j 2
and hence Ym
2
Ia.2 = 1 .
7
m
(X, r) 1 ti Y Ib(j) ( l /i
z
\2/ I rl
f4a r - a i
6a (n) e m
2 dr
m
where a = am), Y-IajI2 = 1 , and r 'I' air) is convex, is a com-
2
From this, one can make estimates on the size of F(A+it,r) and its
derivatives.
Finally we have
00
e-2ar[b(x)+b(r)+i(t-u)]F(x+r+i(y-s),r)dr
S((x,y,t); (r,s,u)) 5
0
VECTOR FIELDS AND NONISOTROPIC METRICS 305
ALEXANDER NAGEL
DEPARTI4IENT OF MATHEMATICS
UNIVERSITY OF WISCONSIN
MADISON, WISCONSIN
REFERENCES
[1] Bets, L., John, F., and Schechter, M., Partial Differential Equations,
Interscience Publishers, John Wiley and Sons, Inc., New York 1964.
[2] Caratheodory, C., "Untersuchungen fiber die Grundlagen der
Thermodynamik," Math. Ann. 67 (1909), 355-386.
[3] Coifman, R. R., and Weiss, G., Analyse harmonique non-commutative
sur certain espaces homojenes, Lecture Notes in Math. #242,
Springer-Verlag, 1971.
[4] Fatou, P., "Series trigonometriques et series de Taylor," Acta Math.
30 (1906), 335-400.
[4a] Fefferman, C., and Phong, D. H., "Subelliptic eigenvalue problems"
in Proceedings of the Conference on Harmonic Analysis in Honor of
Antoni Zygmund, 590-606, Wadsworth Math. Series, 1981.
[5] Folland, G. B., Introduction to Partial Differential Equations,
Mathematical Notes Series, #17, Princeton University Press,
Princeton, N. J. 1976.
[5a] Folland, G., and Hung, H. T., "Non-isotropic Lipschitz spaces" in
Harmonic Analysis in Euclidean Spaces, Part 2, 391-394; Amer. Math.
Soc., Providence, 1979.
[6] Folland, G. B., and Stein, E. M., "Estimates for the complex and
analysis on the Heisenberg group," Comm. Pure Appl.Math. 27 (1974),
429-522.
[6a] Grushin, V. V., "On a class of hypoelliptic pseudo-differential opera-
tors degenerate on a sub-manifold," Math. USSR Sbornik 13(1971),
155-185.
[7] H6rmander, L., "Hypoelliptic second order differential equations,"
Acta Math. 119 (1967), 147-171.
[8] Kohn, J. J., "Boundary behavior of d on weakly pseudoconvex
manifolds of dimension two," J. Diff. Geom. 6 (1972), 523-542.
306 ALEXANDER NAGEL
E. M. Stein
Introduction
Oscillatory integrals in one form or another have been an essential
part of harmonic analysis from the very beginnings of that subject.
Besides the obvious fact that the Fourier transform is itself an oscillatory
integral par excellence, one needs only bear in mind the occurrence of
Bessel functions in the original work of Fourier (1822), the study of
asymptotics related to such functions in the early works of Airy (1838),
Stokes (1850), and Lipschitz (1859), Riemann's use in 1854* of the
method of "stationary phase" in finding the asymptotics of certain
Fourier transforms, and the application of all these ideas to number theory,
initiated in the first quarter of our century by Voronoi (1904), Hardy (1915),
van der Corput (1922) and others.
Given this long history it is an interesting fact that only relatively
recently (1967) did one realize the possibility of restriction theorems for
the Fourier transform, and that the relation of the above asymptotics to
differentiation theory had to wait another ten years to come to light!
The purpose of these lectures is to survey part of this theory and at
the same time to describe some new results. We have found it convenient
to divide our discussion into oscillatory integrals of the "first kind," and
those of the "second kind." The main difference between the two is that
for the first kind we are studying the behavior of only one function as the
parameter increases to infinity, while for the second kind we are dealing
307
308 E. M. STEIN
b
1(X) = r e"(x),A(x)dx
a
*The reader will note that there are several related topics not touched on in
this survey. Chief among them is the subject of oscillatory integrals arising in
the solution of hyperbolic equations and their generalizations - the class of
"Fourier integral operators." For an elegant introduction to that subject see [1],
Chapter 4.
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS 309
(b) Scaling : Suppose we only know that dk'(x) > 1 for some fixed k,
dxk
and we wish to obtain an estimate for f b which is independent
a
of a and b. Then a simple scaling argument shows that the only possi-
ble estimate for the integral is 0(A-Ihk). That this is indeed the case
goes back to van der Corput.
fa
ea(bi"ldx <ck0/k
holds when
310 E. M. STEIN
(i) k>2
(ii) or k = 1 , if in addition it is assumed that 0'(x) is monotonic.
b b b b
e'4dx =f D(eiAO)dx = - r e1X4t-D(1)dx + i1 ix 4v
e
a a a a
b b
r ei t_D(1)dx
r eat i-j1 x (-1) dx '0,
aJ
a a
b
<1 I d 1
_ W
dx
a
a
b
=
f
a
c-S
}f
J
c+S
r
c& +d
b
f
c+S
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS 311
< ck/(XS)l /k
0
Similarly
e'4 dx /k
L
Clearly however
c+s
fC-S
e'kO dx <2S.
Thus
f a
e'40 dx < 2ck
(xs)l /k
+25.
(1.2)
f
0
e"W(x)0(x)dx < Ck,\-1 /k
J
Jv(x)Idx .
312 E. M. STEIN
f a
eaO(x)dx <ck1-1/k, for a<x<b.
r b e'AOV
(c) Asymptotics : We already know that the behavior of Ja dx is
00
j=o
N
(1.3') (1E (A) - ajA-jO(a-(N+1)/k-)
as A, 00
j=0
(1.4) x xPe x z
eiXX2 dx - 1 /2-Q/2 ,
cj(Q) -j
J
-,o
j=0
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS 313
(1.5)
f CIO
eax2xe77(x)dx < AX-1/2 /2
1
with tDf = A simple computation then shows that this term
dx f .
is majorized by
CN
,\N
f lxl>E
lxle-2N dx = C' \-NEe-2N-1
N
N
eX2c (x) _ Y` bjxj + xN+IRN(x) = P(x) + XN+1RN(x)
jj=0+
00
(2)
f 00
eiAX2 e_ C2
xi dx
00
(b)
J00
(C)
fe2P(x)e2(1_(x))dx.
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS 315
For (a) we use (1.4); for (b) we use (1.5); and for (c) we use (1.6). It is
then easy to see that their combination gives the desired asymptotic ex-
eiAx2
REMARKS:
(1) The proof for higher k is similar and is based on the fact that
f 0
eixxke xkxedx = ck,Q(1- i'\)-R+I)/k .
References : The reader may consult Erde1yi [8], Chapter II, where further
citations of the classical literature may be found.
Rn
Proof. For each xo in the support of ci, , there is a unit vector 6 and a
small ball B(x0), centered at x0, so that (e,vx)o(x)> c > 0, for
x c B(xp). Decompose the integral fek'0(x)t4(x)dx as a finite sum
I fe(')clFk(x)dx
n
where each c/ik is C and has compact support in one of these balls. It
then suffices to prove the corresponding estimate for each of these
integrals. Now choose a coordinate system xt,x2, ,xn so that xI
lies along 6. Then
=f(fe"(x
'...,xn)Vk(xt,...,xn)dxI
fe('c)ci/k(x)dx I dx2,...,dxn .
We can only state a weak analogue for the scaling principle, Proposi-
tion 2; it, however, will be useful in what follows.
with k = ial , and the constant ck() is independent of A and t/i and
remains bounded as long as the Ck+1 norm of 0 remains bounded.
0(x0I
axa >
For the inner integral we invoke (1.2) giving us an estimate of the form
-1 /kA-1 / a f' f
ckak (x1,...,xn) dx1
L
J I1 (x1,...,xn) dx1
I
has a critical point at x0, and q' does not vanish of infinite order at
x0, then after a smooth change of variables (b can be transformed to a
simple canonical form , with '(x) = xk (for x near 0 ). There is
no analogue of this in higher dimensions, except for k = 1 and in a
special case corresponding to k = 2. To the asymptotics of the latter
situation we now turn.
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS 319
(2.2)
e"(x)0(x)dx A-n/2 ai X-j , as A
j=o
Rn
each of the bounds occurring in the error terms depend only on upper
bounds for finitely many derivatives of 0 and 0 in the support of Vi ,
a2(b(xo)
the size of the support of and a lower bound for det
xk
The proof of the proposition follows closely the same pattern as that
of Proposition 3. First, let Q(x) denote the unit quadratic form given by
Q(x) xn, where 0 < m < n , with m fixed.
The analogue of (1.4) is
00
n
eax2x_x2
J Pdx J
(' e 2
P
x J dx (1
(f j=1
-oo
n -lh-P./2
and expand the function 11 (1/X+i) (for large A ) in a power
j=1
series in 1/X .
The analogue of (1.5) is the statement that
(2.4)
f
Rn
ei,\Q(x)xPrl(x) dx < AX-n/2-IP!/2;
if r, a C0(Rn)
tsince 2n n
Then
j=1i 1
J
= Rn we can find functions lZ ,
J
i = 1, , n, each
homogeneous of degree 0, and Co away from the origin, so that
n
1= fI.(x), x = 0, with ft. supported in F.. Then we can write
j=1
fe V Q(x)xPn(x)dx = fe'AQ(x)xP,r(x)SZ(x)dx
i
This, together with the fact jxii > 1 Ixl in F., and I (tD.)Nj .(x)1
J2n
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS 321
CN Ixl-2N allows one to conclude the proof of (2.4) in analogy with that
of (1.5).
A similar argument also show that whenever e 8 and e vanishes
near the origin, then
/
pal curvatures of S at x0, and their product (= det 2 r) is the
Gaussian curvature at x . `
THEOREM 1. Suppose S is a smooth hypersurface in Rn, with non-
vanishing Gaussian curvature at each point, and let d = tfida as above.
Then
AIfI-(n-1)/2 .
(3.1) J(d)^(()I <_
(3.2)
fRn
< AA-n/2
Let us consider 1 first. The function ((x, r,) has the property that
(Vx(k)(0, 71N) = 0. We want to see that for each q sufficiently close to
)IN there is a (unique) x _ x(,1) , so that (Vx) c(x(n), rl) = 0. The latter
is a series of n equations and one can find the desired solution by the
implicit function theorem, which requires that we check that the Jacobian
determinant det ((VxpxqS) (0, RN)) 4 0; but this of course is our assump-
tion of non-vanishing curvature. Notice that if the -q-neighborhood of 'IN
general assumption that at each point S has at most a finite order contact
with any hyperplane. We shall call such sub-manifolds of finite type.
(These have some analogy with the finite-type domains in several complex
variables, which are also discussed in Nagel's lectures [21].) The pre-
cise definitions required for our considerations are as follows. We shall
assume that we are considering S in a sufficiently small neighborhood of
'
a given point, and then write S as the image of mapping <b: Rm -> Rn,
defined in a neighborhood U of the origin in R n. (To get a smoothly
embedded S we should also suppose that-the vectors , ,
axI ax2
, NM
are linearly independent for each x, but we shall not need that assump-
tion.) Now fix any point x0 c U C Rm , and any unit vector in Rn . 71
each unit vector q then 3a , jal < k, with (O(x) 77)1 0 0 will
UJL a x=x
and in fact we can take e = 1/k, where k is the type of S inside the
support of t/i .
0(IeI-I /k)
ei (x)* (x)dx =
J
R`"
ti
with (A as described above, and the support of 0- sufficiently small.
Now we can write l; = AY), with Jill = , and X > 0 . Then we know that
11
1 /q
(4.1) If(e)lgda(e) < Ap,q(So) IIf tIp
JSo
(4.2) f 1 /2
o
AIIfII p
Rn
for PO = 2n + 32, and f e 5; the case 1 < p < p0 will then follow by
interpolation.* By covering the support of V1 by sufficiently many small
open sets, it will be enough to prove (4.2) when (after a suitable rotation
and translation of coordinates) the surface S can be represented (in the
support o f Vi) as a graph: en Now with d = V'da we
have that
K(x) = fe
*In fact the interpolation argument shows that we can take q so that (4.1)
holds with q = (n+1} p'' which is the optimal relation between p and q.
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS 327
e s2 7
(4.4) Ks(x) =
1'(s,'2) fe 2nix
)I
1+s
n( W(e')de
Rn
(' e 2ni(x'-e1+xn0(e'))Yf(e')de'
s(xn) = Cs(xn)K(x)
Rn-I
with
00
2
2nix nenlfnl-l+s n(fn)den
e
Cs(xn) 1'(s/2) ./
-00
In fact (c) is immediate from our initial definition (4.4), and (b) follows
from Theorem 1.
Now consider the analytic family Ts of operators defined by T5(f) _
f * Ks' From (b) one has
REMARKS:
(i) For hypersurfaces with non-zero Gaussian curvature this theorem
is the best possible, only insofar as it is of the form (4.1) with
q > 2 . If q is not required to be 2 or greater, then it may be
conjectured that a restriction theorem holds for such hypersurfaces
in the wider range 1 < p < 2n/(n+1). This is known to be true
when n = 2 (see also 7 below).
(ii) For hypersurfaces for which only k principal curvatures are non-
vanishing, Greenleaf [121 has shown that then the corresponding
results hold with 1 < p < 2k 2 , giving an extension of
+
Theorem 3.
(iii) In the case of dim(S) = 1 (i.e. in the case of a curve) there are
a series of results extending our knowledge of the case n = 2
alluded to above. For further details one should consult the
references cited below.
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS 329
COROLLARY. Suppose S is real analytic and does not lie in any affine
hyperplane. Then S has the LP restriction property for 1 < p < po,
for some po > 1 .
0 0
relation among exponents is the same as PO = 2nknk1 Q.E.D.
'
00
d
where Pa(t) is a real polynomial in t of degree d, Pa(t) = F a.O. It
j=o
was proved by Wainger and the author in [27], that the integral is bounded
with a bound depending only on the degree d and independent of the
coefficients ao,al,...,ad. The relevance of such integrals can be better
understood by consulting Wainger's lectures [32]. We shall be interested
here in giving an n=dimensional generalization of this result. We formulate
it as follows. Let K(x) be a homogeneous function of degree -n;
suppose also that IK(x)I < AIxI-n (i.e. K is bounded on the unit sphere);
moreover, we assume the usual cancellation property: f x,l-1 K(x') d a (x') = 0.
We let P(x) _ I aaxa be any real polynomial of degree d.
Ial<d
THEOREM 5:
(5.2) P.V.
f
Rn
eiP(x)K(x)dx < Ad
with the bound Ad that depends only on K and d, and not on the
coefficients as .
Nagel and Wainger observed that if K were odd, one could prove (5.2)
from the one-dimensional form (5.1) by the method of rotations (passage to
polar coordinates). To deal with the general case we need two lemmas.
d
Let Pa(t) = F a tJ denote a real polynomial on R1 , and write also
j=1 j
d
Pb(t) = F
j=1
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS 331
LEMMA 1:
d
Jipa(t) 1Pb(t) dt
(5.3) (e -e )
t
<Ad 1+ log`-II,
j=1
e
mp =f IP(x')I da(x') .
Ix k=1
Then,
(5.4)
f
Ix'I=1
Ilog (I P 1.)) da(x') < B ,
00
(eiat_eibt) dt <A(1+Ilog .
5 E
t )
00 00
r (e1Pa(t) - e iPb(t)) dt = 5+ r
J t J
E E 1
and we treat these two integrals separately. (If e'> 1 , we have only
f00
f"0 and that integral is estimate like .) Let us consider the second
integral. It equals
00 00
f eiPa(t) dt _ (' eiPb(t) dt
J t
f t
00
t
1
becomes
00 1
e iVa(t) dt =
f
+f
ladll/d ladll/d 1
e"Va(t) dt <cd,
t
while
r dt_1
<
J t d log l(Tl=a to 1
ladl1/d
since bd = 1 . Next
f(e Pa(t)
i
a
IPb(t)
)t-J
d_ 1
(e
1Qa(t) e 'Qb(t) )dtt+ 0('dt
J t,
E E E
334 E. M. STEIN
with
d-1 d-1
Qa(t) = I ajtj ,
J=1
Qb(t) = 4 bjti
j=1
since IPa(t)-Qa(t)I < Iti and IPb(t)-Qb(t)I < Iti . However, by induction
hypothesis (using (5.3) for E' = E , and E =1, and d - 1 ),
el1(x)K(x)dx <Ad,
Ix l=1 E1
f f
2
Ix'I=1 \ E1
d
+
Aa 1 m)
fE2
Now on the Heisenberg group one can consider two types of dilations and
their corresponding quasi-distances. The first are the usual dilations
(z,t) (pz, pt), p > 0, and the metric could be defined in terms of the
usual distance. The second are the dilations (z,t) -+ (pz,p2t), and the
appropriate quasi-distance (from the origin) is then (Izl4+t2)1 /4. The
latter dilations and metric are closely tied with the realization of the
Heisenberg group as the boundary of the generalized upper half-space
holomorphically equivalent with the unit ball in Cn+1 . This point of view,
as well as related generalizations, is elaborated in Nagel's lectures [211.
336 E. M. STEIN
(6.4) (Tf)(x) = I
K(x-y)eiB(x,Y)f(y)dy
Rn
We shall give only the highlights of the proof, leaving the details,
further variants, and applications to the papers cited below. Let us con-
sider first the L2 part of assertion (a) when n/2 < g < n. Suppose ri
*For further details see Mauceri, Picardello and Ricci [19] and Geller and
Stein [10].
338 E. M. STEIN
is a Co function, with 77(x)=l for IxI < 1/2, and ii(x) = 0, for
1x1 > 1 . We write T = To is defined as in (6.4), but
with K replaced by Ko = riK, and T with K replaced by K = (1-77)K.
Observe first that since Ko(x-y) is supported where Ix-yl < 1 ,
estimating T0(f)(x) in the ball IxI < 1 involves only f(y) in the ball
IYI < 2. We claim
(6.5)
5Ixl<I ITo(f)(x)12dx < A f If(Y)I2dy .
IYI<_2
fKo(x_y)ei'cf(y)dy -J Ko(x-Y)eis(y,Y)f(y)dy
<cJ Ix-YI-'`+'If(Y)IdY ,
Ix-YI<I
J
ITo(f)(x)I2dx < A r If(Y)I2dY
Iy-hI<2
Ix-hi<l
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS 339
Rn Rn
This will be done by proving the corresponding result for the operator
T,*,T. The kernel L of this operator is given by
L(x,y) = J e-iB(z,x-Y)1c(z-x)K.(z-y)dz
IL(x,y)I < A .
where
fQ IQI f f U_Q
Q
(6.9) OilLp Lp
To prove this we need only observe that (if I)# < 2f0 , and use the result
(see [10]) for the standard # function.
The point of all of this is that for operators of the form (6.4), there is
a naturally associated HE and BMOE theory, and it is given by
choosing
-iB(x,cQ)
(6.10) eQ(x) = e ,
(6.11) 1
Q,
f JTf -yQ dx < A
Q
Q28' fl = 0 elsewhere,
CQ25 n QS-1 ,
f2 = 0 elsewhere,
IF2(x)-yQI <A 8
f Q2S
dy
IYIn+1
+S
Qs-1
dy
IYI-1
l<A'.
Finally
K(x-Y)eiB(x,Y)f3(Y)dY
F3(x) = I
= J (K(x-Y)-K(-Y))eiB(x,Y)f3(Y)dY + J K(-Y)f3(Y)eiB(x,Y)dY
5 IF(x)I2dx
3
< J IF3(x)I2dx = A J IK(-Y)f3(Y)I2dy < Al J den = A5"-
IYI
Q Rn Rn cQ
S-1
Combining these estimates proves (6.11), and hence the fact that T takes
L to BMOE .
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS 343
Rn
where K is homogeneous of degree -n, smooth away from the origin and
with vanishing mean-value.
(i) One may also show that the operators (6.12) are bounded on LP,
1<p<00.
(ii) Given p, with 1 < p < o, then there is an e =E(p,d), so that if K
is homogeneous of degree - g, n - E < < n , T is still bounded on
LP. However now the bounds may depend on P, and in addition one
must assume that P(x,y) is not of the term P(x,y) = P0(x) + Pl(y).
(iii) One can replace K(x-y) in (6.12) by a more general "Calder6n-
Zygmund kernel" K(x,y), a distribution for which the operator when
P e 0 is bounded in L2 , and which in addition is a function (when
x y ) which satisfies IK(x,y)l < Alx-yl-n, lpxK(x,y)l + IoyK(x,y)I
< Alx-yl-n-I
fei(x)du(x) = 0(I6I-(n-I)/2)
as e 00
r(7.2)
Proof. Write
R2
However
IIfIILr(R2) 5f12t12
= f If(t)Ir If(t)I`IJI I-r ds dt
So we take g(t) = If(t)Ir, then Ilgllu = IIfIIp when p = ur. Then if we fix
a so that -1 +a =1-r, then 2 rr = u , and 3r The limitation
0 < a becomes r < 2, and with q = 2r' we obtain from (7.4) that
1 X2/P
IIFiIILr (R2) < c'
0
with a similar estimate for F2(e) which is the analogue of (7.4), but
taken over R2 . Since F = FI +F2 and F = (Tf )2 we obtain
TA(f)(e) = J ei1'(x,4)&(x,e)f(x)dx
Rn
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS 347
axij
(7.6) det a2L(x, e) 0
PROPOSITION:
A)C-n/2IIfJIL2(R')
(7.7) [ITA(f)IIL2(Rn) < .
COROLLARY:
(7.9) J
Rn
348 E. M. STEIN
Now since
L
(a,Vx)[c(x,n)-F(x,e)] =l _ _ a,17-e'\ + OIn-ej2
OXOV
It follows from (7.10) with N = n+1 , that the operator TX*TA which has
kernel KA has a norm bounded by AKn and the proposition is proved.
We shall now formulate some theorems for oscillatory integrals of the
form
Rn-I
(7.12a) B is of rank n - 1 .
THEOREM 10. Under the assumptions above the operator (7.11) satisfies
(2) The proof of part (a) follows the same lines as the proof given for
Theorem 9, once we use (7.8) as the substitute for the Hausdorff-Young
theorem; further details as well as relations with Bochner-Riesz summa-
bility may be found in the papers of Carleson and Sjolin [3] and Hormander
[15]. Since part (b) has not appeared before, we will outline its proof.
This will also serve as a good review of many of the notions we have dis-
cussed here.
Proof of part (b). It suffices to prove the case p = 2 , since the case
p = 1 is trivial and the rest follows by interpolation. Now the case p = 2
is equivalent by duality to the statement
with r = 2(n+ ,
where
Rn
We can calculate
J TX*(F)Tj*(F)dt
Rn-I
and write as
KA(6,r!)F(e)F(n)d6drl
J
RnxRn
with
es2
f eia( (x,n
I'(s/2) J
Rn
(7.16)
since
Next
This follows by applying the estimate (7.7) of the proposition above and
using the non-degeneracy of the Hessian of $(x,e). Finally we claim
that
00
2
vs(u) = F(ss/2) r eixnuv(xn)+xnl-1+sdxn .
-Do
352 E. M. STEIN
8. Appendix
Here we shall prove Lemma 2 and Theorem 6 which were stated in 5.
First let `f'd denote the linear space of polynomials in Rn of degree
< d . We claim that there is a constant Ad , so that
1 /2
(f IP(x)I2 dx)1
2 and f IP(x)ldx are two (equivalent) norms on the
Q0 Q0
finite-dimensional space 5d , so (8.1) holds for Q = Q0, and then for
general Q.
Now it is well known (see e.g. [6]) that a function which satisfies a
"reverse Holder" inequality belongs to the weight space A.. Examining
the proof of this fact one obtains an r = r(d), 0 < r < co, and a constant
Cd , so that
1Jr
(8.2)
(7Q,1 f IP(x)I dx
Q (fQ_o
I f IP(x)I"dx << Cd
(8.3)
fIXI=1
I P(x)I-r da(x) < ca .
E. M. STEIN
DEPARTMENT OF MATHEMATICS
PRINCETON UNIVERSITY
PRINCETON, NEW JERSEY 08544
REFERENCES
[1] M. Beals, C. Fefferman, and R. Grossman, "Strictly pseudo-convex
domains," Bull. A.M.S. 8(1983), 125-322.
[2] J. E. Bjorck, "On Fourier transforms of smooth measures carried by
real-analytic submanifolds of Rn," preprint 1973.
3S4 E. M. STEIN
I. Introduction
These lectures deal with work primarily due to Alex Nagel, Nestor
Riviere, Eli Stein, and myself dealing with certain averages of and singu-
lar integral operators on functions, f , of n variables, n > 2 . These
averages and singular integrals differ in character from the classical
theory in that the integration is over a manifold of dimension less than n.
Let us begin with an example of the type of problem we have in mind.
The classical differentiation theorem of Lebesgue asserts for any locally
integrable function f
a.e.
f(x) = limo IQ
r
I f f(x-y)dy
f(x) = lim 1
r-+0 JBr) J
r f(x-y)dt a.e.,
Br
357
358 STEPHEN WAINGER
Problem IA:
Does
Problem IB:
Does
3) MQ f(x) =
r
r
,J
f(x-y)dar(y)
aQr
and
4) MB f(x) = J f(x-y)dkr(y)
r
aBr
We are asking if
and
and
The argument showing that 9) and 10) imply 1) and 2) is the same as the
argument showing that Lebesgue's differentiation theorem follows from
the weak type inequality for the Hardy-Littlewood maximal function given
in chapter 1 of [S]. While it is not quite as well known, there are appropri-
ate estimates on maximal functions that guarantee 1) and 2) hold for all
L functions. In our case this means the following;
Let E be a measurable set and XE its characteristic function.
Then if
where C(a) may depend on A but not on E, then 1) holds for every f
in L. If
for all x.
We could still ask if 1) and 2) hold in some interesting class even
though 9), 19), 11), and 12) fail. However an important idea of Stein
shows that the failure [SI] of 9), 10), 11), and 12) implies that 1) and 2)
fail even in the class of locally bounded functions. The statement of the
main theorem of [SI] requires that the underlying space be compact. But
if 1) or 2) were true for an LP class on Rn, it would also hold for the
corresponding LP class on the torus. Furthermore, the theorem of Stein
requires the hypothesis that 1 < p < 2. However due to the positive
nature of the averages under consideration, his ideas can be modified to
show that 1) fails for at least some L functions. See [SW]. Thus we
obtain negative results in one dimension. Similar reasoning gives the
same negative conclusion for question IA in any number of dimensions.
AVERAGES AND SINGULAR INTEGRALS 361
h
1
lim 1 f(x-tv(x))dt = f(x) a.e.
h-'0 h _
0
for f in L or L2 or LI ?
Corresponding to problems II and III there are interesting singular
integrals. We let y(t) be a curve and v(x) be a smooth vector field as
in problems II and III. We set
Tf = K*f
Hf(x) = ff(x-t) dt
(f a function on R1 ).
We will explain how the method of rotations can lead to problem II'.
Let K(x,y) be a function of two variables x and y which is odd,
K(-x,-y) = -K(x,y)
00 00
x = rcos 0
y = r2sinO
and find
00 277
27r o0
2rr 00
-J N(O)K(cos(O+n),sin(O+n))dd f
=
0 0
i f(u-rcos 0, v-rsin(O)) dr
since K is odd. Thus
217
since
N(6+n) = N(6) .
Finally
277 00
2n ao
f
Tf(u,v) =
2 0
N(O)K(cos O,sin 0)dd f
-00
i (u-rcos6,v-r2sin6)dr
f
where
AVERAGES AND SINGULAR INTEGRALS 365
f
00
Hef = dt
f(x-yg(r)) ,
-00
with
Now we prove
IITfIILp<cI{fIILP
by showing
IIHefIILP <cplif11Lp
MEf(x,Y) = E2 ff f(x-r,Y-s)
(is)(i+
1
drd s
E2
(")
If K(x,y) were dominated by a decreasing, radial, LI function, the
classical theory would imply
1
K(x,Y) =
(1+x2)(1+y2)
366 STEPHEN WAINGER
1 ff
0 t2
f(x-r,y-s) dr ds
We now consider the intersection of two of these balls of the same size
Tf(x,y) = J f(x-t,y-t2)dt
f(x-t,y-v) . S (1- -1 dv dt
= JJ 0)
Or
22) Tf=K*f,
where
22A) K(x,Y)=XS(1--X ),
368 STEPHEN WAINGER
c) Covering lemmas
d) Calderon-Zygmund decomposition.
Perhaps the natural attack on our problems would be to find appropriate
covering lemmas and suitable variants of the Calder6n-Zygmund decom-
position. Some progress in finding covering lemmas for related problems
was made by Stromberg [Str] and [STRO] and Cordoba [COR1], [COR2],
Cordoba and Fefferman [CF1], [CF2], [CF3], and Fefferman [FEf].
Our approach will however be different. We shall try to use the Fourier
transform or other orthogonality methods and interpolation to reduce our
problems on averages and singular integrals to the more standard averages
and singular integrals. In retrospect we see that some of these ideas
occurred in [SPL], [CS], and in [KS].
We have said earlier that curvature and Fourier Transform would be
important for us. Actually they go together. If one has a nice measure on
a curved surface, the Fourier transform of that measure decays at infinity
even though the measure is singular. Let us consider some examples.
Define, for a test function ,,
r O(t,O)dt .
0
AVERAGES AND SINGULAR INTEGRALS 369
g(e'exel)?y) = r eietdt
0
00
_
Then
v(e exei7y)
00
fe_t2eietet2dt.
Ce
1+1771
= CnJ(lei) Ii (v).
2
370 STEPHEN WAINGER
^ _ (n 1
VAN DER CORPUT'S LEMMA. Let h(t) be a real function. For some j ,
assume Jh(1)(t)I > A in an interval a < t < b. If j = 1, assume also that
h'(t) is monotone, then
b
r exp (ih(t))dt
a
For the proof of Van Der Corput's lemma for j = 1 and 2 see [Z]. The
proof for higher j is similar.
Let us consider the measure
24) du(O) f1
2
(k(t,t2)dt.
Then
2
du(e,i) = el4telnt dt
1
This integral cannot be evaluated explicitly, but we wish to see that Van
Der Corput's lemma may be applied. We take h(t) = e t4 rt2 . First we
use the fact that h"(t) = rt . Thus by Van Der Corput's lemma with j = 2 ,
we see
AVERAGES AND SINGULAR INTEGRALS 371
25)
n
ldu(, , C
1
(1 + Inl)I12
if
27) C
Tf=du*f
maps LP into L2 continuously for some p < 2. For
372 STEPHEN WAINGER
=f +e 2 +1771)2,5
If( ,ii)I 2
(1+e 2+I17I
If(e,,7)12
(1+e2+1,,127
2q ) /q 1 /q
If(e,rl)I
J (Ll+e2+'2')
The second integral is bounded if q' is sufficiently large which means
for some q > 1 . But then the first integral is bounded for f e LP where
P +2q=1
where
29) my(e) = f
_,p
Lt
where D is a distribution
So
m((,q) = r exp(ite+iIt1a(sgnt)rt) Lt
-00
f exp (iah(t))dt
374 STEPHEN WAINGER
00
r dt
J
The proof was by way of the Van Der Corput lemma but was unnecessarily
complicated because at that time we only knew the lemma for j = 1,2 .
Let us see how Van Der Corputs'lemma works in the case y(t) =
(t,t2'..., tn). We then have to show that
30)
I
E <Itl<R
dt < C(n) .
t
exp (iels + ... + ien- sn-1 + < C(n) .
31) iSn)
32)
1
1<JtJ<R
Now
Hence we have reduced the proof 30) for n to proving 30) for n - 1.
Also the case of n = 1 is easy. So we are done by induction.
We now turn to the LP theory. We wish to emphasize how curvature
and Fourier Transform are joining together to help us. So we shall com-
pare the case of the parabola (t,t2) to the straight line (t,t). In the
case of the parabola we are studying
34) Hpf = DP * f
35) Dpi dt
36) HLf=DL*f,
where
00
eieteirlt2 dt
38) Dp(e,rl) = Dp(e1eXei"ly) = F
-00
and
f00ete'it.
-00
t
We can calculate DL explicitly, and we find
41) lim
,q-to J
r ei 'tei77t2 dt _
t
-00
ei6t dt
t
Assume for simplicity that 71 > 0.
Of course
1
1/3
77 00
and
I
,71 /3
43)
1
eiet(ei??t2_1) dt
t
.771 /3
711r/ 3
44)
I
f
1/3 <I fl
e'eteigt2 dt o.
378 STEPHEN WAINGER
f -2/3
eiese"gs
2
ds < c
XFII
45) eieteir7t2 d t
00
2/3
77-
So
-2/3
77
e tese l'7s 2 ds
1/3
77
46)
f eieteil7t2 dt
t <C f 1
dt +77 1/3<C711/3
t2
1/3
77
AVERAGES AND SINGULAR INTEGRALS 379
45) and 46) together with similar estimates for negative t prove 44)
and hence the continuity of Dp(e,-9) away from the origin. If one is a
little more careful in the above argument, one can prove that Dp(C,.9)
satisfies a Lipschitz condition away from the origin. One may then use
Riviere's [R] version of Hermander's multiplier theorem [H] to obtain
some LP results for p 2 . In fact if y(t) = (t,t2) one obtains
J=
f
R<Ixl<2R
IK(x+h)-K(x)Idx
J=
f
R<IXI<2R
I la
IxIaIK(x+h)-K(x)ldx
112
<(
f
R<I x l< 2 R
IXI2a
1
d)'
x (J
IxI2aI(x+h)-K(x)I2dx)
/
Now there was the feeling that a use of Schwartz inequality like that
above lost too much, and that more careful estimates for j for particular
kernels might lead to better results.
To get an idea of what to do we calculated D(e,n) very precisely by
the method of steepest descents. We found
2
48) Dp(6.77) = sgne + C I77I e1e2/n
`- 2/
+ Better terms ,
where
50) m(,n) =
Ir111/2
1
r)2 1
e
where
2)
(171
52) mz(e,rl) = m(e,v)
Then by a duality argument T would be bounded in all LP, 1 < p < oo.
It turns out that one can show by a messy calculation that Tz is of
Calderon-Zygmund type if Re z > 0.
Let us try to understand why the kernel for Tz , Re z > 0, might be
a little better than the kernel for To. T is essentially HY y = (t,t2),
and so the kernel K0 of To is essentially
K0(x,Y)=X8(1-z )
K0(x,y) = a S(9-90)
r3
where
sin0 0
cos200
S3) K0(x,Y) _ O
r
1
54) KE(x.y) =
r 21011-E
(The factor for ordinary polar coordinates plays the same role as
r
r
for parabolic polar coordinates.) We are trying to see whether
)Id9rdr
J J r2
I-D- r20
r>C nearo
y
r
0 y-1
r
K0(r, 0) - Ko(r , 0)
n/4
f
J0
JK0(r, 6)- K0(r', O')j dO = 1 .
00 1 /r
KE(r, 0)d0rdr
5
1 50
00 1/r
Irfderdr
2
el-F
5 0
f1
00
< dr .
,J rl+e
5
57) Hy,f = DP * f
w here
58) Dz(e,rl)
p
e a irlt2 dt
(1+t2)z/2
It turns out that 58) is not a good idea for a very important reason. By
changing variables in formula 56) we see that Dp(Ac, A271) = Dp(c,rl) , for
any A > 0. Note that also the function mz(e,r/) defined in 52) also has
this type of homogeneity, namely mz(Ac,A2r7) = mz((,77), for A > 0.
Now experience has shown that homobeneity is a powerful friend not
to be tossed away lightly. However Dp does not have this homogeneity.
This situation can be remedied by defining
59) Hyf=Dzxf
where
00
+r72t4)-z /4 Xteir7t2 dt
60) Dp(c,rl) = (1
I
-00
Let us see how formula 61) can help us. We would like to show
62) C(z)
t
2
eXsel1 ds
ft
d ese's2 dsI <C
00
Now
j
-1
(1+t4)-z /4 Xt eirlt2 dt
t
f
-1
e'et t2 dt
t
<C(z) r t2dtt<C(z).
J-1
But we already know that
r dt
J T <C.
-1
386 STEPHEN WAINGER
One may finally deal with -00 < t < -1 in the same manner that one
treated 1 < t < 00. Hence 62) is proved.
Let us calculate the kernel, Kz , corresponding to Hy if z = E > 0.
Then
eiCx e myDp(e,rl)dCdrl
KE(x,Y) =fv
00 00
f f
eirlt2
Lt e-177Y (1
+,72t4)-E/4
t
- 00
00 00
f e-'6(x-t)dx = r dt
f
i?7(t2-Y)(1+772t4) -E/4 ei 7t2&(x -t)dt
J J
-00
t
00
00
00
0o
Ir7(x2y)
x3 f e 1
(1+,72)1/4
dr7
-00
x2'-Y2
1p
x3 e/2 x2
(
where PE/2 is a modified Poisson kernel. PE/2 decays exponentially
fast at oo and PE/2(u) ti JuJICE/2 as u -, 0. See [SWE].
Thus KE(x,y) has a singularity near the curve (t,t2) of the form
1 which is just the improvement over the 8-function that we
1e_eo1i-E/2
seek.
A modification of these ideas worked for curves
AVERAGES AND SINGULAR INTEGRALS 387
a'
,ta2,... , tan)
Y(t) = (t
where A is a real nxn matrix such that the real parts of the eigen-
values of A are positive. For example if
y(t) = (t,t2)
TX = exp (A loga) .
Then
where
P(TAX) _ AP(x)
0
A =C1
0 2 )
Then
0 ),
T- ,
and
p(x,y) = (x4+y2)1 /4
66) Hzf=Dz*f
where
69) rl(u) 0.
du)
dI+17
(u) Ajrl
(u)
du1+1
So
n Yd j+t
j=O
aJ duI+1 '7(u) = 0
and
j=0
n
aj a j
duJ
. rr'(u) =0.
d, n'(u) . =0 j = 1 ,2,...,n-1
du3
A
Y '(t) - Y(t)
dJy(t) j = 1,2,...
dyj(t) t= O
span Rn. It turns out that well curved curves can be approximated by
homogeneous curves. We can then prove
dt <ApyIIfIILp, 1 <p<oo.
f f(x-y(t))
-
A general theorem in L2 for curves which are approximately
homogeneous was obtained by Weinberg (We].
began with the study of maximal functions along the curve (t,t2). Thus
we wish to consider averages
h
70) Mhf(x,y) -- r f(x-t,y-t2)dt .
F 0
Mhf(e,r1) = mh(e,r))f(e,rl)
where
h
eit e eit2 'Idt
mh(e,r1) = h
0
f exp Q(x)dx
I. J<F
(' a
Ixl2
+Q(x)
e J e F dx
R
392 STEPHEN WAINGER
71) vh * f(x,y)
f exp( t2)f(x-t,y-t2)dt
h/
.
Then
00
v(he,h277) = f e t2eiheteih2r)t2dt ,
expii h27,expie77
2
vh* f (e,n) _ (i(h,h27l)exp(_ l ((exp )f
rl // i
One might now hope that if one defines a measure vh by the formula
AVERAGES AND SINGULAR INTEGRALS 393
where Br is the ball of radius r centered at the origin, and dur is the
unit rotationally invariant measure on Br. We set, as before,
THEOREM 3:
n-1
IxI O log' IXI
where 1 near the origin and has compact support show that p > nn1
is necessary in order that 77) hold. The situation for n = 2 , p > 2 is
unknown at this time.
I would like to present here Stein's original argument which proved
77) for p = 2 and n = 4 . We define
00 1 /2
Thus
r r
Mrf(x) < n r sn'1 Msf(x) ds + n r sn d Msf(x)ds
0 0
= I(r) + 11(r) .
II(r) < n
rJ f
0
sn-1 /2 s1 /2 Msf(x)ds
r 1 /2
< fr
1 s2n-1 ds g(f) (x)
-rn J 0
<Cg(f)(x).
So if we assume 29), we have
Ig(f)(x)I2dx= J t J IdtMtf(x)I2dxdt
JRn 0 Rn
0
=
J0
r t Jr Mtf(6)12d6dt
Rn
00
_ tJ Imtr(6)I2dedt .
0 Rn
396 STEPHEN WAINGER
and dorm
is bounded. See [SWE]. Thus
Rn
00
00
r tIdm(tIibI2dt<C
0
f
0
1/161
dt <1e12f
0
1 /IeI
tdt<C.
00
dt m(tIeI)I2 dt
t
/IeI
00
Ifi2 f t )n-1
dt < C ,
(t lel
1 /ICI
if n>4.
Let us be more precise about the counterexample in 2-dimensions. We
take
1
x very near 0
IxIlogI1
in Co away from 0 .
80) MBIxIf(x) _ 00
dO
MIxlf(x) _
(a2(1-cos)2+a2sin2ej1 /2 log 1
-n a2[(l-cos)2+sin2O]
ti dO = 00 .
IeI In IeI
-E
398 STEPHEN WAINGER
To prove 83), it suffices to show that for each function j(x) taking
the values
85) provides the inductive step to prove 86). If j(x) is given define
if j(x) is even
if j(x) is odd .
AVERAGES AND SINGULAR INTEGRALS 399
Then, if we set
2k
86) gf(x) _ I IMB f(x)-MB f(x)I2
J=0 k
1+ 1+11
2 2k
we see
(' 2k
J Igf(x)I2 dx = I fMf(x)-MB
1 0 l+2k l+-
2k
=I IMB W) - MB
j=0
f 2
88)
400 STEPHEN WAINGER
and
To prove 88) we use 89) if ICI > 2k and 90) if 161 < 2k
Finally we will show how Stein [SH] proved the maximal function
along the parabola (t,t2) is bounded by using g-functions.
We start with the measure d defined in 24)
2
qS(t,t2)dt
d(O) = J .
We set
Then
dh * f(x,y) = r f(x-ht,y-ht2)dt ,
or
92) dh * f(x,y) _ f 2h
f(x-t,y-t2)dt .
93) 'Ph(x,Y) = h3 0 x , h)
and
Note that
sup
E>0
f Idh*f(x,Y)-h*f(x,Y)Idh
0
sup 1f
E>0 E1/2
E
1 /2
< r Idh *f(x,Y)I2 dh
0
So
If f>0,
E
('
J dh*f(x,Y)dh
0
E 2h
f
0 h
f(x-t,y-t2)dtdh
E E
> f r f(x-t,y-t2)
0
f/2
dh
('E
>
E
f
J
0
f(x-t,y-t2)dt .
fI(gf)(x,y)I2dxdY
= r T J Idh*f-ch*fl2dxdy
0
ff
0
00
f
a*
.77)12
f0,1f( dry.
0 0
00
98) JT
0
1 1
('
99) JJ dh d(he,h27)-cb(he,h27)I2 < C r h2 dh < C
0 0
CN
Also k1;'"?) < for any N, so
(1+e2+772)IJ
00
('
101) 1 dh Ii&(he,h277)I < C .
Now we obtain 98) and hence 96) by combining 100), 101) and 102).
In this section we have emphasized L2 methods. LP results for
p > 1, can be obtained by combining the L2 estimates presented here
with the techniques of section 2. Altogether one can prove the following
theorems :
f
h
!'1
let
!'h
and
axis, and if the curvature of the integral curves of v(x) never vanishes
a2y
will not be zero. It turns out that one can prove the following
at2 t=0
theorems:
ar x
It=o = 0 and a2rlt=ono.
at
Then
and
and
h
lim h1 , f(x-tv(x))dt = f(x) a.e.
h-00 1
0
for f in L2(R2).
These theorems are announced in NSWV. Here we shall give some
discussion of the ideas. Because y(t,x) depends on x, the Fourier
Transform no longer seems like a good tool to study Hy and 59 y. We
must find a different way to employ orthogonality. Here we're motivated
AVERAGES AND SINGULAR INTEGRALS 407
H*f(x,y) = J f(x+t,y+y(t)) dt
HYH*f(x,y) f(x+t-s,y+t2-s2) dt
tss
=ff
where
k(u,v) = C .
IuI u-u u+u
Now k(u,v) does have its support spread out. However the singularities
of k across the curves v = u2 are not locally integrable away from the
408 STEPHEN WAINGER
HYf=Qp*f
where
p(e,r/) _ f e(iet+ir7t2)dt
-00
6
+ better terms.
Lf = ff(x-t,y) dt
L is known to be bounded in L2. This suggests that we try to consider
M = (L-HY)(L*-HY) .
and hence
IIPh(x)f1IL2 <C .
The fact that we had to modify Hy suggests that we should not expect
107) to hold, but we might expect a variant to hold - perhaps involving an
operator
h(x,y)
(x,Y) = f(x-t,Y)dt .
Rh(x,Y)f h(x,y) f
0
(We know
where Bh(x) is some bounded operator. Even 107) is not quite right.
We refer the reader to NSWV for the correct technical modification of 107).
This concludes our discussion of the vector field problem.
V. Recent developments
The positive results of Theorem 2 and Theorem 5 assume that the
curve y(t) has some curvature at the origin. There have been a number
of papers trying to understand what happens when this curvature condition
is dropped. See [C], [CNVWW], [NVWW] 1), 2), 2), and [NE]. Let us note
that we don't have positive results for all C curves.
410 STEPHEN WAINGER
0 for 0<t<1
r(t) =
t-1 for t>1
Then
nHrf(6,71) = my(6,17)f(6,q)
where
00
my(4,i7) = r ei4teirir(t) dt
-00
= r
1
sintt dt + 00
at
0 1
E--0
lim I f(x-y(t)) dt
ItI<E
!'h h
sup I
0
If(x-t,y)1 dt + sup h
f
0
f(x-t,y-t+I)dt t .
if and only if
if and only if
00
and observe
A-2) I(A) _
V"X
where
00
e-t2
A-3) B= dt .
-00
A__t2
('
5e e
At2 dt <
fA21/5<t<1
At2 dt + J e 2
t>1
2 dt
It1,A21/5
<e-AI/5+e,\/2
Thus the main contribution to the integral I(A) comes from the small
interval - 1 < t < 1 . Now if we perturb the integrand in I(A), we
A2/5 - -A2/5
can expand the integrand in a power series in that little interval. For
example we might consider
00
JA) = e Xh(t)dt
J
414 STEPHEN WAINGER
where h(t) > t2 for large t, h(t) = t2 +0(t3) for small t and h(t) > 0
for t > 0.
Then one can easily see that
f dt
Itl>A2/5
e Ah(t)dt
J = f e-At2 . (1+O(at3)
ItIU 2/5 Itl
00
e-'fit2
dt + Ea I /5 e-at2 dt
ltl2/5 -00
B + O(e-AI /5)+0
(1 l
f
= .
K= r eikt2+.k P(t)dt
-00
and P(t) were very negative at infinity and 0(t3) near t = 0 we would
try to write t = a + it and integrate on the line a = r for It l < 1/, 2 /5 ,
and we would expand in a power series in this small interval.
AVERAGES AND SINGULAR INTEGRALS 415
fg(z)eAh(z)dz
b
I= r eiAf(t)dt
a
with f(t) real gets most of its contributions for large A, near a, b, or a
zero of f'(t).
If we had no endpoints for example if f were periodic with period
b - a or if the interval of integration was from -oc to oo and f oscil-
lated very rapidly for large t , we would expect the main contribution to
the integral to come from small neighborhoods of a zero of V.
416 STEPHEN WAINGER
L = 2 [x(t)]2 - V(x(t),t) .
tb
S(x) = r L(x(t))dt .
ta
The path on which the classical particle moves will be a path x(t) such
that z(ta) = xa , z(tb) = xb , and such that
S(x +Sx)IS=o 0.
=
where the integration is an integral over all paths x(t) such that
x(ta) = xa and x(tb) = xb. Leaving aside the question of how such an
integral can be precisely defined, let us try to guess what paths contribute
the most to the integral F . Since h is very small, the principle of
stationary phase would indicate that the main contribution to the integral
F should come from a small neighborhood of a path of which some kind
of a derivative of S was zero. Thus we might expect the main contribu-
tion to the integral F to come from paths which are very close (on a
scale of h ) to the path x defined by *.
There have been many papers in the mathematical and physical litera-
ture dealing with the problem of making sense out of the definition F.
See for example [CS] and references cited there. However, the original
definition in [FH] serves the purpose of making many formal calculations.
We may imagine dividing the t interval into 21 subintervals, Ij , of
equal length
[b_ta]k
Ik = + ta, [tb-ta]2 + to
l nl .
xk = X (ta + [tb-ta])
2
k= We then take
STEPHEN WAINGER
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF WISCONSIN
MADISON, WISCONSIN
REFERENCES
[B] N. DeBruijn, Asymptotic Methods in Analysis, North Holland
Publishing Co., Amsterdam, 1958.
[BF] H. Busemann and W. Feller, "Zur Differentiation des
Lebesguesche Integrale," Fund. Math, Vol. 22, 1934,
pp. 226-256.
[CS] R. Cameron and D. Storvick, A simple definition of the Feynman
integral with applications, Amer. Math. Soc., Providence, 1983.
[CSS] H. Carlsson, P. Sjogren, and J. Stromberg, "Multiparameter
maximal functions along dilation-invariant hypersurfaces" to
appear in Trans. of the A.M.S.
[CW] H. Carlsson and S. Wainger, "Maximal functions related to con-
vex polygonal lines," to appear.
[C] M. Christ, preprint.
[CS] J. L. Clerc and E. M. Stein, "LP multipliers for non-compact
symmetric spaces," Proc. Nat. Acad. Sci., U.S.A., Vol. 71,
1974, pp. 3911-3912.
[CORI] A. Cordoba, "The Kekeya maximal function and the spherical
summation multipliers," Amer. J. of Math., Vol. 99, 1977,
p. 1-22.
423
424 INDEX
maximal functions, 48, 49, 245 Szego kernel, 193, 265, 296, 297
strong, 60
spherical, 359, 393 Transference theorem, 39
on curves, 391, 400
on vector fields, 406
method of layer potentials, 133, 143 Van der Corput lemmas, 309, 370
Mobius transformation, 186
weight norm inequalities, 72
multilinear Fourier analysis, 18
multiparameter differentiation Zygmund conjecture, 67
theory, 57
multipliers, 72
Library of Congress Cataloging-in-Publication Data
Beijing lectures in harmonic analysis.
(Annals of mathematics studies ; no. 112)
Bibliography: p.
Includes index.
1. Harmonic analysis. I. Stein, Elias M.,
1931- . H. Series.
QA403.B34 1986 515'.2433 86-91452
ISBN 0-691-08418-1
ISBN 0-691-08419-X (pbk.)