Abstract
We study Hardy–Littlewood–Sobolev theorem for variable Riesz potentials \(I_{\alpha (\cdot )}f\) in a bounded open set G, including the borderline case \(\alpha ^-:= \inf _{x\in G} \alpha (x) = 0\) and \(p^+:=\sup _{x \in G} p(x) = \infty \).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availibility
All data generated or analysed during this study are included in this published article.
References
Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Springer (1996)
Almeida, A., Hasanov, J., Samko, S.: Maximal and potential operators in variable exponent Morrey spaces. Georg. Math. J. 15, 195–208 (2008)
Capone, C., Cruz-Uribe, D., Fiorenza, A.: The fractional maximal operator and fractional integrals on variable \(L^p\) spaces. Rev. Mat. Iberoamericana 23(3), 743–770 (2007)
Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces. Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis. Birkhauser/Springer, Heidelberg (2013)
Diening, L.: Maximal function on generalized Lebesgue spaces \({L}^{p(\cdot )}\). Math. Inequal. Appl. 7(2), 245–254 (2004)
Diening, L.: Riesz potentials and Sobolev embeddings on generalized Lebesgue and Sobolev spaces \(L^{p(\cdot )}\) and \(W^{k, p(\cdot )}\). Math. Nachr. 263(1), 31–43 (2004)
Diening, L., Harjulehto, P., Hästö, P., R\(\stackrel{\circ }{\rm u}\)žička, M.: Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017. Springer, Heidelberg (2011)
Diening, L., Harjulehto, P., Hästö, P., Mizuta, Y., Shimomura, T.: Maximal functions in variable exponent spaces: limiting cases of the exponent. Ann. Acad. Sci. Fenn. Math. 34, 503–522 (2009)
Edmunds, D.E., Rákosník, J.: Sobolev embedding with variable exponent. Stud. Math. 143, 267–293 (2000)
Edmunds, D.E., Rákosník, J.: Sobolev embedding with variable exponent II. Math. Nachr. 246–247, 53–67 (2002)
Futamura, T., Mizuta, Y.: Continuity properties of Riesz potentials for functions in \(L^{p(\cdot )}\) of variable exponent. Math. Inequal. Appl. 8(4), 619–631 (2005)
Futamura, T., Mizuta, Y., Shimomura, T.: Sobolev embedding for variable exponent Riesz potentials on metric spaces. Ann. Acad. Sci. Fenn. Math. 31, 495–522 (2006)
Futamura, T., Mizuta, Y., Shimomura, T.: Integrability of maximal functions and Riesz potentials in Orlicz spaces of variable exponent. J. Math. Anal. Appl. 366, 391–417 (2010)
Futamura, T., Mizuta, Y., Shimomura, T.: Sobolev embeddings for Riesz potential space of variable exponent. Math. Nachr. 279(13–14), 1463–1473 (2006)
Harjulehto, P., Hästö, P.: Sobolev inequalities for variable exponents attaining the values \(1\) and \(n\). Publ. Mat. 52, 347–363 (2008)
Harjulehto, P., Hästö, P.: A capacity approach to the Poincaré inequality and Sobolev imbeddings in variable exponent Sobolev spaces. Rev. Mater. Comput. 17, 129–146 (2004)
Hästö, P.: Local-to-global results in variable exponent spaces. Math. Res. Lett. 16(2), 263–278 (2009)
Hedberg, L.I.: On certain convolution inequalities. Proc. Am. Math. Soc. 36, 505–510 (1972)
Kokilashvili, V., Samko, S.: Maximal and fractional operators in weighted \(L^{p(x)}\) spaces. Rev. Mat. Iberoamericana 20(4), 493–515 (2004)
Kokilashvili, V., Samko, S.: On Sobolev theorem for Riesz-type potentials in Lebesgue spaces with variable exponent. Zeit. Anal. Anwend. 22, 899–910 (2003)
Maeda, F.-Y., Mizuta, Y., Shimomura, T.: Growth properties of Musielak–Orlicz integral means for Riesz potentials. Nonlinear Anal. 112, 69–83 (2015)
Mizuta, Y.: Potential Theory in Euclidean Spaces. Gakk\(\overline{\rm o}\)tosho, Tokyo (1996)
Mizuta, Y., Nakai, E., Ohno, T., Shimomura, T.: Riesz potentials and Sobolev embeddings on Morrey spaces of variable exponent. Complex Var. Elliptic Equ. 56(7–9), 671–695 (2011)
Mizuta, Y., Nakai, E., Ohno, T., Shimomura, T.: Maximal functions, Riesz potentials and Sobolev embeddings on Musielak–Orlicz–Morrey spaces of variable exponent in \(^n\). Rev. Mater. Comput. 25, 413–434 (2012)
Mizuta, Y., Ohno, T., Shimomura, T.: Sobolev embeddings for Riesz potential spaces of variable exponents near \(1\) and Sobolev’s exponent. Bull. Sci. Math. 134(1), 12–36 (2010)
Mizuta, Y., Shimomura, T.: Sobolev’s inequality for Riesz potentials with variable exponent satisfying a log-Hölder condition at infinity. J. Math. Anal. Appl. 311, 268–288 (2005)
Mizuta, Y., Shimomura, T.: Weighted Sobolev inequality in Musielak–Orlicz space. J. Math. Anal. Appl. 388, 86–97 (2012)
Rafeiro, H., Samko, S.: On the Riesz potential operator of variable order from variable exponent Morrey space to variable exponent Campanato space. Math. Methods Appl. Sci. 43(16), 9337–9344 (2020)
Samko, N., Samko, S., Vakulov, B.G.: Weighted Sobolev theorem in Lebesgue spaces with variable exponent. J. Math. Anal. Appl. 335(1), 560–583 (2007)
Stein, E.M.: Singular Integrals and Differentiability Properties Of Functions. Princeton University Press, Princeton (1970)
Acknowledgements
We would like to express our thanks to the referee for his/her kind comments.
Funding
The authors did not receive support from any organization for the submitted work.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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
Mizuta, Y., Ohno, T. & Shimomura, T. Hardy–Littlewood–Sobolev Theorem for Variable Riesz Potentials. Results Math 78, 100 (2023). https://doi.org/10.1007/s00025-023-01869-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-023-01869-8