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An extension-restriction theorem for weighted Besov spaces
Part of:
Special classes of linear operators
Linear function spaces and their duals
Harmonic analysis in several variables
Published online by Cambridge University Press: 26 August 2025
Abstract
In this article, the authors establish an extension-restriction theorem between homogeneous weighted Besov spaces and weighted mixed-Riesz potential spaces. This general frame covers both the classical Besov spaces and their logarithmic analogs.
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- © The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society
Footnotes
L. Liu was supported by the National Natural Science Foundation of China (Grant Nos. 12371102 and 12271021). S. Wu was supported by the National Natural Science Foundation of China (Grant No. 12201098) and the Fundamental Research Funds for the Central Universities (Grant No. 3132024199).
References
Adams, D. R., Lectures on
${L}^p$
-potential theory. Univ. Umea, Dept. of Math. 2, 1981.Google Scholar
${L}^p$
-potential theory. Univ. Umea, Dept. of Math. 2, 1981.Google ScholarAdams, D. R. and Xiao, J.,
Strong type estimates for homogeneous Besov capacities
. Math. Ann. 325(2003), 695–709.Google Scholar
Ansorena, J. L. and Blasco, O.,
Characterization of weighted Besov spaces
. Math. Nachr. 171(1995), 5–17.Google Scholar
Aronszajn, N. and Smith, K. T.,
Theory of Bessel potentials
. I. Ann. Inst. Fourier (Grenoble) 11(1961), 385–475.Google Scholar
Benedek, A. and Panzone, R.,
The spaces
${L}^p$
, with mixed norm
. Duke Math. J. 28(1961), 301–324.Google Scholar
${L}^p$
, with mixed norm
. Duke Math. J. 28(1961), 301–324.Google ScholarCaetano, A. M., Gogatishvili, A., and Opic, B.,
Sharp embeddings of Besov spaces involving only logarithmic smoothness
. J. Approx. Theory 152(2008), 188–214.Google Scholar
Caetano, A. M., Gogatishvili, A., and Opic, B.,
Embeddings and the growth envelope of Besov spaces involving only slowly varying smoothness
. J. Approx. Theory 163(2011), 1373–1399.Google Scholar
Chen, T. and Sun, W.,
Extension of multilinear fractional integral operators to linear operators on mixed-norm Lebesgue spaces
. Math. Ann. 379(2021), 1089–1172.Google Scholar
Chen, T. and Sun, W.,
Hardy-Littlewood-Sobolev inequality on mixed-norm Lebesgue spaces
. J. Geom. Anal. 32(2022), Article no. 101, 43 pp.Google Scholar
Cleanthous, G., Georgiadis, A. G., and Nielsen, M.,
Anisotropic mixed-norm Hardy spaces
. J. Geom. Anal. 27(2017), 2758–2787.Google Scholar
Cobos, F. and Domínguez, Ó.,
Embeddings of Besov spaces of logarithmic smoothness
. Studia Math. 223(2014), 193–204.Google Scholar
Cobos, F. and Domínguez, Ó.,
On Besov spaces of logarithmic smoothness and Lipschitz spaces
. J. Math. Anal. Appl. 425(2015), 71–84.Google Scholar
Cobos, F., Domínguez, Ó., and Triebel, H.,
Characterizations of logarithmic Besov spaces in terms of differences, Fourier-analytical decompositions, wavelets and semi-groups
. J. Funct. Anal. 270(2016), 4386–4425.Google Scholar
Di Nezza, E., Palatucci, G., and Valdinoci, E.,
Hitchhiker’s guide to the fractional Sobolev spaces
. Bull. Sci. Math. 136(2012), 521–573.Google Scholar
Domínguez, Ó., Liu, L. and Xiao, J., A functional geometric analysis of the
$\mathit{log}$
-Lipschitz spaces. Submitted.Google Scholar
$\mathit{log}$
-Lipschitz spaces. Submitted.Google ScholarDomínguez, Ó. and Tikhonov, S.,
Function spaces of logarithmic smoothness: embeddings and characterizations
. Mem. Amer. Math. Soc. 282(2023), Article no. 166.Google Scholar
Duoandikoetxea, J., Fourier analysis. American Mathematical Society, Providence, RI, 2001.Google Scholar
Edmunds, D. E. and Evans, W. D., and Operators, H., Function spaces and embeddings. Springer-Verlag, Berlin, 2004.Google Scholar
Edmunds, D. E. and Haroske, D. D.,
Spaces of Lipschitz type, embeddings and entropy numbers
. Dissertationes Math. (Rozprawy Mat.) 380(1999), Article no. 43.Google Scholar
Edmunds, D. E. and Haroske, D. D.,
Embeddings in spaces of Lipschitz type, entropy and approximation numbers, and applications
. J. Approx. Theory 104(2000), 226–271.Google Scholar
Edmunds, D. E. and Triebel, H.,
Spectral theory for isotropic fractal drums
. C. R. Acad. Sci. Paris Sér. I Math. 326(1998), 1269–1274.Google Scholar
Farkas, W. and Leopold, H.-G.,
Characterisations of function spaces of generalised smoothness
. Ann. Mat. Pura Appl. 185(2006), 1–62.Google Scholar
Frazier, M. and Jawerth, B.,
A discrete transform and decompositions of distribution spaces
. J. Funct. Anal. 93(1990), 34–170.Google Scholar
Frazier, M., Jawerth, B., and Weiss, G., Littlewood-Paley theory and the study of function spaces. American Mathematical Society, Providence, RI, 1991.Google Scholar
Gagliardo, E.,
Caratterizzazioni delle trace sulla frontier a relative ad alcune classi difunzioni in
$n$
variabili
. Rend. Sem. Mat. Univ. Padova 27(1957), 284–305.Google Scholar
$n$
variabili
. Rend. Sem. Mat. Univ. Padova 27(1957), 284–305.Google ScholarGol’dman, M. L.,
A description of the trace space for functions of a generalized Hölder class
. Dokl. Akad. Nauk SSSR 231(1976), 525–528.Google Scholar
Gol’dman, M. L.,
A description of the traces of the anisotropic generalized Liouville class
. Dokl. Akad. Nauk SSSR 233(1977), 273–276.Google Scholar
Grafakos, L., Modern Fourier analysis. Third edition, Grad. Texts in Math., 250, Springer, New York, 2014.Google Scholar
Haroske, D. D.,
On more general Lipschitz spaces
. Z. Anal. Anwendungen 19(2000), 781–799.Google Scholar
Haroske, D. D. and Moura, S. D.,
Continuity envelopes of spaces of generalised smoothness, entropy and approximation numbers
. J. Approx. Theory 128(2004), 151–174.Google Scholar
Hörmander, L.,
Estimates for translation invariant operators in
${L}^p$
spaces
. Acta Math. 104(1960), 93–140.Google Scholar
${L}^p$
spaces
. Acta Math. 104(1960), 93–140.Google ScholarHuang, L., Liu, J., Yang, D., and Yuan, W.,
Identification of anisotropic mixed-norm Hardy spaces and certain homogeneous Triebel-Lizorkin spaces
. J. Approx. Theory 258(2020), Article no. 105459, 27 pp.Google Scholar
Kalyabin, G. A.,
A characterization of spaces with generalized Liouville differentiation
. Mat. Sb. (N.S.) 104(1977), no. 146, 42–48, 175 pp.Google Scholar
Kalyabin, G. A.,
Imbedding theorems for generalized Besov and Liouville spaces
. Dokl. Akad. Nauk SSSR 232(1977), 1245–1248.Google Scholar
Leopold, H.-G.,
Embeddings and entropy numbers in Besov spaces of generalized smoothness
. Function spaces (Poznań, 1998), 323-336. Lecture Notes in Pure and Appl. Math., 213, Marcel Dekker, Inc., New York, 2000.Google Scholar
Liu, L., Wu, S., Yang, D., and Yuan, W.,
New characterizations of Morrey spaces and their preduals with applications to fractional Laplace equations
. J. Differ. Equ. 266(2019), 5118–5167.Google Scholar
Peetre, J., New thoughts on Besov spaces, Duke University Mathematics Series, 1. Duke University, Mathematics Department, Durham, NC, 1976.Google Scholar
Silvestre, L.,
Regularity of the obstacle problem for a fractional power of the Laplace operator
. Commun. Pure Appl. Math. 60(2007), 67–112.Google Scholar
Stein, E. M.,
The characterization of functions arising as potentials
. Bull. Amer. Math. Soc. 67(1961), 102–104.Google Scholar
Stein, E. M.,
The characterization of functions arising as potentials II
. Bull. Amer. Math. Soc. 68(1962), 577–582.Google Scholar
Stein, E. M., Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, NJ, 1970.Google Scholar
Stein, E. M. and Weiss, G.,
Fractional integrals on
$n$
-dimensional Euclidean space
. J. Math. Mech. 7(1958), 503–514.Google Scholar
$n$
-dimensional Euclidean space
. J. Math. Mech. 7(1958), 503–514.Google Scholar