BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 80, Number 2, March 1974 RESEARCH ANNOUNCEMENTS The purpose of this department is to provide early announcement of significant new results, with some indications of proof. Although ordinarily a research announcement should be a brief summary of a paper to be published in full elsewhere, papers giving complete proofs of results of exceptional interest are also solicited. Manuscripts more than eight typewritten double spaced pages long will not be considered as acceptable. All research announcements are communicated by members of the Council of the American Mathematical Society. An author should send his paper directly to a Council member for consideration as a research announcement. A list of members of the Council for 1974 is given at the end of this issue. SOBOLEV INEQUALITIES FOR RIEMANNIAN BUNDLES BY M. CANTOR Communicated by Murray Protter, August 30, 1973 1. Introduction. Sobolev inequalities play a major role in the study of differential operators and nonlinear functional analysis. The inequalities are the primary tools in the study of the properties of spaces of functions with Sobolev topologies; for example, the Schauder ring theorem. There are theorems involving continuity and closure of composition in such spaces [3, Chapter 2]. The latter theorems involve application of the inequalities to vector fields. It is this case and its generalization which this paper studies, where Rn is replaced by an arbitrary Riemannian manifold satisfying certain geometric conditions. While the Sobolev inequalities over Rn have been known for some time, the usual proofs use transform methods and are therefore hard to generalize. In 1959 Nirenberg [4] presented particularly elegant proofs, due to himself and other authors, which could be generalized. These proofs are the basis for the results of this paper. Throughout, M denotes a complete Riemannian «-dimensional manifold without boundary. The canonical volume form on M is denoted dV. Let 7T:E->M be a vector bundle with a specified smooth metric ( , ). V is a connection on E satisfying d(V, W)xm=(VXmV9 W)+(V, VXmW), where V and W are sections of E and xm e TmM. Vn is the iterated covariant derivative. In most applications E is a tensor bundle over M. AMS (MOS) subject classifications (1970). Primary 58C99, 58D99, 58B20. Key words and phrases. Vector bundles, Sobolev inequalities, co variant derivatives. Copyright © American Mathematical Society 1974 239 240 [March M. CANTOR C™(E) is the space of C00 sections of E with compact support. For x e M, Ve C0°°(£), the quantity |V*K(JC)| is the norm of V*F(X) in the canonical normofL*(rM:£). A more detailed account of the proofs can be found in [3] and shall be published elsewhere. The author wishes to thank Jerrold Marsden for his help and encouragement and E. Calabi for his assistance. 2. Statement of results. 1. Let F G C*(E). Then define (i) (C*norm) | | K L = 2 * s u p | W ( * ) l ; (ii) (LI Sobolev norm) WU=2k ( J M |V'K(*)I' (iii) (Holder norm) for 0 < 6 < 1, DEFINITION rr/1 ^ SU P [^kfc = 2 , S U P V x.yeM CeO(x.v) dvy»; ir(C)VV(x) - v'rooi J, d(X, 7e yf > where G(x,y)={length-minimizing geodesies joining x and y), r(C) is parallel translation along C from ir^iy) to 7r"1(x)9 and d(x, y) is the distance from x to y. Each of the functions defined in Definition 1 is a norm on the vector space C0(E). Also, denote || || 0 =|| ||, | | 3)>0 =| \„, and [ ] M = [ ]0. k k+e DEFINITION 2. (i) C (E) (resp. LUE), C (E)) is the completion of C?(E) with respect to || ||fc (resp. | \Ptk, [ ]$tk). We note that if O < 0 < 1 , then Ck+1(E)<=-Ck+e(E)<=-Ck(E) and the inclusion is continuous. We state the following hypotheses: (CI) The injective radius of M is bounded away from zero. (C2) There is a ô such that for each x e M and V, W e TXM, the sectional curvature \KX(V, W)\<d. THEOREM 1. Let M satisfy CI and C2. Then if p>\ and s>n/p+k, there is a constant C such that, for all feL^{E), we have \\f\\k^C\f\PtS, where C is independent off THEOREM 2. Let M satisfy CI and C2 and assume r>n. Then there is a constant C such that, for each fe Lrs(E), we have [f]i-n/r,s~C\f\r,s+i> where C is independent off THEOREM 3. Let M satisfy CI and C2, 0^j<m, and q,r^l. Then ifmlp=jjr+(m—j)jq, there is a constant C such that, for each f e Um(E)C\ U(E), we have | V y | p < ; C | / | ^ £ | / l i " i / w , where C is independent off In the next theorem, we adopt the following notation, due to Nirenberg [4]. 1974] SOBOLEV INEQUALITIES FOR RIEMANNIAN BUNDLES 241 Notation. For/?>(), the definition of | \P remains as in Definition 1. For/?<0 a n d / e C?(E)9 let h=[-n/p], -u.=h+njp and i ƒ i, = ra«> « > o. 4. Let M satisfy CI and C2, a«J /étf O ^ y ^ m . 7%éw /ƒ m—j—njr is not a nonnegative integer and l//?=l/r+(y—#*)/«, there is a constant C such that, for fe Lrm(E), it follows that | V y | p ^ C | V y | f t t î l ^ where C is independent off. THEOREM Under some circumstances, one can interpolate between these inequalities. We do have the standard interpolation lemma: LEMMA. If — oo<A5^:gv<oo and A^O or v^O, then, for all ue D-i\E)r\L1i\E), ue L^iE), Ml/0 ^ C |tl|1/A |M|l/y In the classical case the assumption that X, \x, and v all have the same sign is unnecessary [4, p. 126], Under what conditions it is necessary is not known to the author. There are several applications of this lemma. For example, 5. Let M satisfy CI and C2 and let 1 ^q, r^ oo lfjln+llr—mln^.0 and,forj\m^a^\, THEOREM 1 j /l m\ - = ^+ a — - p n q then, for u e L m(E)nL (E), r andO^j^m* 1 + (l-a)-, \r n/ it follows that q where C is independent ofu. 3. Discussion of the proofs. The fundamental idea behind all of the theorems is the reduction of the argument to a local one. Using condition C2 and standard comparison results [1, pp. 250-257], one gets the necessary uniform bounds on the exp maps. Note that C2 implies an upper bound in the Ricci curvatures. Thus if 2R is the injective radius of M, for VETXM, \V\<R, it follows that the Jacobian Jexp,. V>A, where A is independent of x and V. PROOF OF THEOREM 1. All constants are denoted C throughout. We assume k=0; the general case follows by induction. Let xeM. With 2R the injective radius of M, let BX(R) be the ball of radius R in TXM. Let (r, d) be spherical coordinates on TXM. 242 M. CANTOR [March Let g:R-+R be a C00 function such that g(t)l for t<R/2 and g(0=0, f2£3u/4, and \g^(t)\<A for i^s. Fix 0 and let r(0) be parallel translation along the geodesic r-^exp^r, 0). From «(*) = - T|-(T(0)g(r)Mexp,(r, 0))</r, Jo ar using integration by parts, we get _ (s(s(-—-1)81)! "1)!1 Jor V |^(T(0)g(r))u(exp,(r,0)) dr. From the standard formula relating T to the covariant derivative, |M(X)| ^ c f V ^ V M ^ C e x p C r , 0))| dr. Integrate with respect to r" -1 dS on the unit sphere in TXM. Thus |u(x)| ^ C f f V " 1 |V8g(r)u(exp,(r, 0))| rn"V dS. Js$ JO Now dV,=rn~1drdS and conclude is the volume in T^M. Apply Holder's inequality r(s-n)v/(v-l)r(n-l) dr X (f dS\ Vv |Vsg(r)W(exp(r, 0))|*dK'ï \jBm(R) I Since s>njp, the first integral is finite and its value is independent of x. Now apply the product rule for covariant derivatives and Minkowski's inequality to the second integral to conclude \u(x)\p<: c[ \Vsu\pdV'. JBm(R) Now if dV is the volume element on M then on exp^i^i*)), we have dF(exp(r, 0))=Jexp(r, 0) dV'. Thus, by the remark preceding the proof \u{x)\* Sc[ Jexv(Bm(R)) \Vsu\p dV <:[ \Vsu\* dV. JM Since this holds for each x e M, the theorem follows. Q.E.D. Theorem 2 is proven similarly to Theorem 1. 19743 SOBOLEV INEQUALITIES FOR RIEMANNIAN BUNDLES 243 Theorems 3 and 4 depend on the existence of a smooth triangulation with known limits on the mesh subordinate to a cover by normal neighborhood. Conditions CI and C2 guarantee such a triangulation [2]. This is used to construct a Ck uniform collection of partition functions. Using these one can reduce the theorems to the case where the sections have compact support in some normal neighborhood in M. In each theorem this case is handled by a suitable generalization of the arguments in [4]. However, since the derivatives of the partition functions cannot be ignored, the right side of the inequality contains entire LI norms rather than the LP norms of the fcth derivative, as in the classical case. The induction argument used in each of the theorems goes through almost exactly as in the classical case, simply noting that if ƒ is a smooth section of E, then V/is a smooth section of the bundle L(TM, E) which has a canonical metric and connection. BIBLIOGRAPHY 1. R. Bishop and R. Crittenden, Geometry of manifolds, Pure and Appl. Math., vol. 15, Academic Press, New York, 1964. MR 29 #6401. 2. E. Calabi (private communication). 3. M. Cantor, Thesis, University of California, Berkeley, Calif., 1973. 4. L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 115-162. MR 22 #823. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, BERKELEY, CALIFORNIA 94720