Abstract
Using limiting interpolation techniques we study the relationship between Besov spaces \(\mathbf B ^{0,-1/q}_{p,q}\) with zero classical smoothness and logarithmic smoothness \(-1/q\) defined by means of differences with similar spaces \(B^{0,b,d}_{p,q}\) defined by means of the Fourier transform. Among other things, we prove that \(\mathbf B ^{0,-1/2}_{2,2}=B^{0,0,1/2}_{2,2}\). We also derive several results on periodic spaces \(\mathbf B ^{0,-1/q}_{p,q}(\mathbb {T})\), including embeddings in generalized Lorentz–Zygmund spaces and the distribution of Fourier coefficients of functions of \(\mathbf B ^{0,-1/q}_{p,q}(\mathbb {T})\).
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Ahmed, I., Edmunds, D.E., Evans, W.D., Karadzhov, G.E.: Reiteration theorems for the \(K\)-interpolation method in limiting cases. Math. Nachr. 284, 421–442 (2011)
Almeida, A.: Wavelet bases in generalized Besov spaces. J. Math. Anal. Appl. 304, 198–211 (2005)
Almeida, A., Caetano, A.M.: Real interpolation of generalized Besov–Hardy spaces and applications. J. Fourier Anal. Appl. 17, 691–719 (2011)
Bennett, C.: Banach function spaces and interpolation methods III. Hausdorff–Young estimates. J. Approx. Theory 13, 267–275 (1975)
Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)
Bergh, J., Löfström, J.: Interpolation Spaces: An Introduction. Springer, Berlin (1976)
Besov, O.V.: On spaces of functions of smoothness zero. Sb. Math. 203, 1077–1090 (2012)
Brudnyĭ, Yu A., Krugljak, N.Ya.: Interpolation Functors and Interpolation Spaces, vol. 1. North-Holland, Amsterdam (1991)
Caetano, A.M., Gogatishvili, A., Opic, B.: Sharp embeddings of Besov spaces involving only logarithmic smoothness. J. Approx. Theory 152, 188–214 (2008)
Caetano, A.M., Gogatishvili, A., Opic, B.: Embeddings and the growth envelope of Besov space involving only slowly varying smoothness. J. Approx. Theory 163, 1373–1399 (2011)
Caetano, A.M., Haroske, D.D.: Continuity envelopes of spaces of generalised smoothness: a limiting case; embeddings and approximation numbers. J. Funct. Spaces Appl. 3, 33–71 (2005)
Cobos, F., Domínguez, O.: Embeddings of Besov spaces of logarithmic smoothness. Stud. Math. 223, 193–204 (2014)
Cobos, F., Domínguez, O.: Approximation spaces, limiting interpolation and Besov spaces. J. Approx. Theory 189, 43–66 (2015)
Cobos, F., Domínguez, O.: On Besov spaces of logarithmic smoothness and Lipschitz spaces. J. Math. Anal. Appl. 425, 71–84 (2015)
Cobos, F., Fernandez, D.L.: Hardy–Sobolev spaces and Besov spaces with a function parameter. Function Spaces and Applications. Lecture Notes in Mathematics, vol. 1302, pp. 158–170. Springer, Berlin (1988)
Cobos, F., Fernández-Cabrera, L.M., Kühn, T., Ullrich, T.: On an extreme class of real interpolation spaces. J. Funct. Anal. 256, 2321–2366 (2009)
Cobos, F., Kühn, T.: Equivalence of \(K\)- and \(J\)-methods for limiting real interpolation spaces. J. Funct. Anal. 261, 3696–3722 (2011)
DeVore, R.A., Riemenschneider, S.D., Sharpley, R.C.: Weak interpolation in Banach spaces. J. Funct. Anal. 33, 58–94 (1979)
Edmunds, D.E., Evans, W.D.: Hardy Operators, Function Spaces and Embeddings. Springer, Berlin (2004)
Evans, W.D., Opic, B.: Real interpolation with logarithmic functors and reiteration. Can. J. Math. 52, 920–960 (2000)
Evans, W.D., Opic, B., Pick, L.: Interpolation of operators on scales of generalized Lorentz–Zygmund spaces. Math. Nachr. 182, 127–181 (1996)
Evans, W.D., Opic, B., Pick, L.: Real interpolation with logarithmic functors. J. Inequal. Appl. 7, 187–269 (2002)
Gogatishvili, A., Opic, B., Tikhonov, S., Trebels, W.: Ulyanov-type inequalities between Lorentz–Zygmund spaces. J. Fourier Anal. Appl. 20, 1020–1049 (2014)
Haroske, D.D., Moura, S.D.: Continuity envelopes of spaces of generalized smoothness, entropy and approximation numbers. J. Approx. Theory 128, 151–174 (2004)
Holmstedt, T.: Interpolation of quasi-normed spaces. Math. Scand. 26, 177–199 (1970)
Leopold, H.-G.: Embeddings and entropy numbers in Besov spaces of generalized smoothness. Function Spaces. Lecture Notes in Pure and Applied Mathematics, vol. 213, pp. 323–336. Marcel Dekker, New York (2000)
Merucci, C.: Applications of interpolation with a function parameter to Lorentz, Sobolev and Besov spaces. Interpolation Spaces and allied Topics in Analysis. Lecture Notes in Mathematics, vol. 1070, pp. 183–201. Springer, Berlin (1984)
Persson, L.E.: Interpolation with a parameter function. Math. Scand. 59, 199–222 (1986)
Pietsch, A.: Approximation spaces. J. Approx. Theory 32, 115–134 (1981)
Pustylnik, E.: Ultrasymmetric sequence spaces in approximation theory. Collect. Math. 57, 257–277 (2006)
Schmeisser, H.-J., Triebel, H.: Topics in Fourier Analysis and Function Spaces. Wiley, Chichester (1987)
Simonov, B., Tikhonov, S.: Sharp Ul’yanov-type inequalities using fractional smoothness. J. Approx. Theory 162, 1654–1684 (2010)
Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978)
Triebel, H.: Theory of Function Spaces. Birkhäuser, Basel (1983)
Triebel, H.: Comments on tractable embeddings and function spaces of smoothness near zero. Report, Jena (2012)
Acknowledgments
The authors would like to thank the referees for their comments. The authors have been supported in part by the Spanish Ministerio de Economía y Competitividad (MTM2013-42220-P). O. Domínguez has also been supported by the FPU Grant AP2012-0779 of the Ministerio de Economía y Competitividad, and F. Cobos by UCM-BS (GR3/14-910348).
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Communicated by Winfried Sickel.
To the memory of Professor Manuel Antonio Fugarolas.
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Cobos, F., Domínguez, Ó. On the Relationship Between Two Kinds of Besov Spaces with Smoothness Near Zero and Some Other Applications of Limiting Interpolation. J Fourier Anal Appl 22, 1174–1191 (2016). https://doi.org/10.1007/s00041-015-9454-6
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DOI: https://doi.org/10.1007/s00041-015-9454-6
Keywords
- Besov spaces
- Logarithmic smoothness
- Limiting interpolation spaces
- Fourier coefficients
- Lorentz–Zygmund spaces