Abstract
The purpose of this article is twofold. The first is to strengthen fractional Sobolev type inequalities in Besov spaces via the classical Lorentz space. In doing so, we show that the Sobolev inequality in Besov spaces is equivalent to the fractional Hardy inequality and the iso-capacitary type inequality. Secondly, we will strengthen fractional Sobolev type inequalities in Besov spaces via capacitary Lorentz spaces associated with Besov capacities. For this purpose, we first study the embedding of the associated capacitary Lorentz space to the classical Lorentz space. Then, the embedding of the Besov space to the capacitary Lorentz space is established. Meanwhile, we show that these embeddings are closely related to the iso-capacitary type inequalities in terms of a new-introduced fractional (β,p,q)-perimeter. Moreover, characterizations of more general Sobolev type inequalities in Besov spaces have also been established.
Similar content being viewed by others
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Change history
28 February 2024
A Correction to this paper has been published: https://doi.org/10.1007/s11118-024-10121-z
References
Adams, D.: A Note on Choquet Integrals with Respect to Hausdorff Capacity. Function Spaces and Applications, pp 115–124. Springer, Berlin (1988)
Adams, D.: The classification problem for capacities associated with the Besov and Triebel-Lizorkin spaces. PWN Polish Sci. Publ. 22, 9–24 (1989)
Adams, D.: Besov capacity redux. J. Math. Sci. 162, 307–318 (2009)
Adams, R., Fournier, J.: Sobolev Spaces, 2nd edn. Acadmic Press, New York (2003)
Adams, D., Xiao, J.: Strong type estimates for homogeneous Besov capacities. Math. Ann. 325, 695–709 (2003)
Almgren, F., Lieb, E.: Symmetric decreasing rearrangement is sometimes continuous. J. Amer. Math. Soc. 2, 683–773 (1989)
Ambrosio, L., De Guido, P., Luca, M.: Gamma-convergence of nonlocal perimeter functionals. Manuscripta Mathematica 134, 377–403 (2011)
Beckner, W., Pearson, M.: On sharp Sobolev embedding and the logarithmic Sobolev inequalities. Bull. London Math. Soc. 30, 80–84 (1998)
Brasco, L., Lindgren, E., Parini, E.: The fractional Cheeger problem. Interfaces Free Bound. 16, 419–458 (2014)
Bourgain, J., Brezis, H., Mironescu, P.: Another look at Sobolev spaces. In: Menaldi, J.L., Rofman, E., Sulem, A. (eds.) Optimal Control and Partial Differential Equations, a Volume in Honor of A. Bensoussans 60th birthday, pp 439–455. IOS Press (2001)
Caffarelli, L., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. of Math. 171, 1903–1930 (2010)
Caffarelli, L., Roquejoffre, J.-M., Savin, O.: Nonlocal minimal surfaces. Comm. Pure Appl. Math. 63, 1111–1114 (2010)
Caffarelli, L., Enrico, V.: Uniform estimates and limiting arguments for nonlocal minimal surfaces. Cal. Vari. Partial Differ. Equ. 41, 203–240 (2011)
Carlen, E., Loss, M.: Logarithmic Sobolev inequalities and spectral gaps. In: Recent Advances in the Theory and Applications of Mass Transport. Contemp. Math., vol. 353, pp 53–60. Amer. Math. Soc., Providence (2004)
Cotsiolis, A., Tavoularis, N.: Sharp Sobolev type inequalities for higher fractional derivatives. C. R. Acad. Sci. Paris, Ser. I(335), 801–804 (2002)
Cotsiolis, A, Tavoularis, N: Best constants for Sobolev inequalities for higher order fractional derivatives. J. Math. Anal. Appl. 295, 225–236 (2004)
Cotsiolis, A., Tavoularis, N.: On logarithmic Sobolev inequalities for higher order fractional derivatives. C. R. Acad. Sci. Paris, Ser. I(340), 205–208 (2005)
Dávila, J.: On an open question about functions of bounded variation. Calc. Var. Partial Differ. Equ. 15, 519–527 (2002)
DelPino, M., Dolbeault, J.: The optimal Euclidian Lp −Sobolev logarithmic. Inequal. J. Funct. Anal. 197, 151–161 (2003)
Fusco, N., Vincent, M., Massimiliano, M.: A quantitative isoperimetric inequality for fractional perimeters. J. Funct. Anal. 261, 697–715 (2011)
Frank, R., Lenzmann, E.: Uniqueness of non-linear ground states for fractional Laplacians in \(\mathbb {R}\). Acta Math. 210, 261–318 (2013)
Frank, R., Seiringer, R.: Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255, 3407–3430 (2008)
Gross, L.: Logarithmic Sobolev inequalities. Amer. J. Math. 97, 1061–1083 (1975)
Hajaiej, H., Yu, X., Zhai, Z.: Fractional Gagliardo-Nirenberg and Hardy inequalities under Lorentz norms. J. Math. Anal. Appl. 396, 569–577 (2012)
Hurri-Syrjänen, R., Vähäkangas, A.: Characterizations to the fractional Sobolev inequality Complex analysis and dynamical systems VII (2017)
Li, P., Hu, R., Zhai, Z.: Fractional Besov trace/extension type inequalities via the Caffarelli-Silvestre extension. Submitted
Li, P., Shi, S., Hu, R., Zhai, Z.: Embeddings of function spaces via the Caffarelli-Silvestre extension, capacities and Wolff potentials. Nonlinear Anal. 217, 112758 (2022)
Lieb, E., Loss, M.: Analysis, 2nd. edn. Graduate Studies in Mathematics, vol. 14. AMS, Providence (2001)
Ludwig, M.: Anisotropic fractional Sobolev norms. Adv. Math. 252, 150–157 (2014)
Maz’ya, V.: On capacitary strong type estimates for fractional norms. Zup. Nauch. Sem. Leningrad otel. Math. Inst. Steklov (LOMI) 70, 161–168 (1977)
Merker, J.: Generalizations of logarithmic Sobolev inequalities. Discrete Contin. Dyn. Syst. Ser. S1, 329–338 (2008)
Netrusov, Y.: Estimates of capacities associated with Besov spaces. J. Math. Sci. 78, 199–217 (1996)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchiker’s guide to the fractional Sobolev spaces. Bull des Sci. Math. 136, 521–573 (2012)
Palatucci, G., Pisante, A.: Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces. Cal. Vari. Partial Differ. Equ. 50, 799–829 (2014)
Ponce, A., Spector, D.: A boxing inequalities for the fractional perimeter. Ann. Sc. Norm. Super. Pisa Cl. Sci. XX, 107–141 (2020)
Ponce, A.: A new approach to Sobolev spaces and connections to Γ-convergence. Calc. Var. Partial Differ. Equ. 19, 229–255 (2004)
Xiao, J.: A sharp Sobolev trace Inequalities for the fractional-order derivatives. Bull. Sci. Math. 130, 87–96 (2006)
Xiao, J.: Homogeneous endpoint Besov space embeddings by Hausdorff capacity and heat equation. Adv. Math. 207, 828–846 (2006)
Xiao, J.: A sharp Sobolev trace Inequalities for the fractional-order derivatives. Bull. Sci. Math. 130, 87–96 (2006)
Xiao, J.: The sharp Sobolev and isoperimetric inequalities split twice. Adv. Math. 211, 417–435 (2007)
Xiao, J.: Optimal geometric estimates for fractional Sobolev capacities. C. R. Math. Acad. Sci. Paris 354, 149–153 (2016)
Xiao, J., Zhai, Z.: Fractional Sobolev, Moser-Trudinger, Morrey-Sobolev inequalities under Lorentz norms. J. Math. Science 166, 357–376 (2010)
Xiao, J., Zhai, Z.: C.S.I. for Besov Spaces \(\dot {{\Lambda }}^{p,q}_{{\alpha }} (\mathbb {R}^{n})\) with (α, (p,q)) ∈ (0, 1) × (0, 1] × (0, 1](1, 1). Advanced Lectures in Mathematics Volume 34 Some Topics in Harmonic Analysis and Applications, 407–420 (2016)
Wu, Z.: Strong type estimate and Carleson measures for Lipschitz spaces. Proc. Amer. Math. Soc. 127, 3243–3249 (1999)
Zhai, Z.: Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces. Nonlinear Anal. 73, 2611–2630 (2010)
Funding
Pengtao Li was supported by National Natural Science Foundation of China (No. 11871293, No. 12071272) and Shandong Natural Science Foundation of China (No. ZR2020MA004).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Project supported: Pengtao Li was supported by National Natural Science Foundation of China (No. 11871293, No. 12071272) and Shandong Natural Science Foundation of China (No. ZR2020MA004).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Li, P., Hu, R. & Zhai, Z. Strengthened Fractional Sobolev Type Inequalities in Besov Spaces. Potential Anal 59, 2105–2121 (2023). https://doi.org/10.1007/s11118-022-10030-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-022-10030-z