Weak-type estimates for variable Riesz potentials with respect to the Hausdorff content over metric measure spaces

Toshihide Futamura; Yoshihiro Sawano; Tetsu Shimomura

doi:10.1515/gmj-2025-2044

Article

Weak-type estimates for variable Riesz potentials with respect to the Hausdorff content over metric measure spaces

Toshihide Futamura , Yoshihiro Sawano and Tetsu Shimomura

Published/Copyright: June 17, 2025

Published by

Become an author with De Gruyter Brill

Author Information Explore this Subject

From the journal Georgian Mathematical Journal

Abstract

We are concerned with weak-type estimates for variable Riesz potentials I α ⁢ ( ⋅ ) , τ ⁢ f with respect to variable dimensional Hausdorff content over bounded metric measure spaces X equipped with lower Ahlfors Q ⁢ ( ⋅ ) -regular measures. As an application, we give an exponential integrability for I α ⁢ ( ⋅ ) , τ ⁢ f with respect to variable dimensional Hausdorff content.

Keywords: Riesz potential; variable exponent; metric measure space; Hausdorff content; exponential integrability; lower Ahlfors regular

MSC 2020: 46E35; 46E30

Funding source: Japan Society for the Promotion of Science

Award Identifier / Grant number: 19K03546

Award Identifier / Grant number: 21K03295

Funding statement: Yoshihiro Sawano was partially supported by Grant-in-Aid for Scientific Research (C) (19K03546), the Japan Society for the Promotion of Science. Tetsu Shimomura was supported by Grant-in-Aid for Scientific Research (C) (21K03295), the Japan Society for the Promotion of Science.

A The Riesz potential U α fails to map L N α ⁢ ( ℝ N ) to L ∞ ⁢ ( ℝ N ) boundedly

Assume that there exists a constant C > 0 such that ∥ U α ⁢ f ∥ L ∞ ⁢ ( ℝ N ) ≤ C ⁢ ∥ f ∥ L N α ⁢ ( ℝ N ) holds for all f ∈ L N α ⁢ ( ℝ N ) . Assuming that f ≥ 0 a.e., we are in the position of using the Fatou lemma to have

∫ ℝ n f ⁢ ( y ) | y | N - α ⁢ 𝑑 y ≤ C ⁢ ∥ f ∥ L N α ⁢ ( ℝ N ) .

Since the function | ⋅ | - N + α is positive,

| ∫ ℝ n f ⁢ ( y ) | y | N - α ⁢ 𝑑 y | ≤ C ⁢ ∥ f ∥ L N α ⁢ ( ℝ N )

holds for all f ∈ L N α ⁢ ( ℝ N ) . By the duality L N α ⁢ ( ℝ N ) - L N N - α ⁢ ( ℝ N ) , we have

∥ | ⋅ | - N + α ∥ L N N - α ⁢ ( ℝ N ) ≤ C .

This is a contradiction.

References

[1] D. R. Adams, A note on Choquet integrals with respect to Hausdorff capacity, Function Spaces and Applications (Lund 1986), Lecture Notes in Math. 1302, Springer, Berlin (1988), 115–124. 10.1007/BFb0078867Search in Google Scholar

[2] D. R. Adams, Choquet integrals in potential theory, Publ. Mat. 42 (1998), no. 1, 3–66. 10.5565/PUBLMAT_42198_01Search in Google Scholar

[3] A. Björn and J. Björn, Nonlinear Potential Theory on Metric Spaces, EMS Tracts Math. 17, European Mathematical Society (EMS), Zürich, 2011. 10.4171/099Search in Google Scholar

[4] L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math. 2017, Springer, Heidelberg, 2011. 10.1007/978-3-642-18363-8Search in Google Scholar

[5] P. Hajłasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, 1–101. 10.1090/memo/0688Search in Google Scholar

[6] P. Harjulehto, P. Hästö and V. Latvala, Sobolev embeddings in metric measure spaces with variable dimension, Math. Z. 254 (2006), no. 3, 591–609. 10.1007/s00209-006-0960-8Search in Google Scholar

[7] P. Harjulehto and R. Hurri-Syrjänen, Estimates for the variable order Riesz potential with applications, Potentials and Partial Differential Equations—The Legacy of David R. Adams, Adv. Anal. Geom. 8, De Gruyter, Berlin (2023), 127–155. 10.1515/9783110792720-006Search in Google Scholar

[8] L. I. Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505–510. 10.2307/2039187Search in Google Scholar

[9] J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer, New York, 2001. 10.1007/978-1-4613-0131-8Search in Google Scholar

[10] V. Kokilashvili, A. Meskhi, H. Rafeiro and S. Samko, Integral Operators in Non-Standard Function Spaces. Vol. 1, Oper. Theory Adv. Appl. 248, Birkhäuser/Springer, Cham, 2016. 10.1007/978-3-319-21015-5_1Search in Google Scholar

[11] V. Kokilashvili, A. Meskhi, H. Rafeiro and S. Samko, Integral Operators in Non-Standard Function Spaces. Vol. 2, Oper. Theory Adv. Appl. 249, Birkhäuser/Springer, Cham, 2016. 10.1007/978-3-319-21018-6Search in Google Scholar

[12] O. Kováčik and J. Rákosník, On spaces L p ⁢ ( x ) and W k , p ⁢ ( x ) , Czechoslovak Math. J. 41(116) (1991), no. 4, 592–618. 10.21136/CMJ.1991.102493Search in Google Scholar

[13] A. D. Martínez and D. Spector, An improvement to the John–Nirenberg inequality for functions in critical Sobolev spaces, Adv. Nonlinear Anal. 10 (2021), no. 1, 877–894. 10.1515/anona-2020-0157Search in Google Scholar

[14] T. Ohno and T. Shimomura, Sobolev embeddings for Riesz potentials of functions in grand Morrey spaces of variable exponents over non-doubling measure spaces, Czechoslovak Math. J. 64(139) (2014), no. 1, 209–228. 10.1007/s10587-014-0095-8Search in Google Scholar

[15] T. Ohno and T. Shimomura, Sobolev’s inequality in central Herz–Morrey–Musielak–Orlicz spaces over metric measure spaces, Complex Var. Elliptic Equ. 67 (2022), no. 5, 1154–1185. 10.1080/17476933.2020.1863382Search in Google Scholar

[16] J. Orobitg and J. Verdera, Choquet integrals, Hausdorff content and the Hardy–Littlewood maximal operator, Bull. Lond. Math. Soc. 30 (1998), no. 2, 145–150. 10.1112/S0024609397003688Search in Google Scholar

[17] N. G. Samko, S. G. Samko and B. G. Vakulov, Weighted Sobolev theorem in Lebesgue spaces with variable exponent, J. Math. Anal. Appl. 335 (2007), no. 1, 560–583. 10.1016/j.jmaa.2007.01.091Search in Google Scholar

[18] Y. Sawano, M. Shigematsu and T. Shimomura, Generalized Riesz potentials of functions in Morrey spaces L ( 1 , φ ; κ ) ⁢ ( G ) over non-doubling measure spaces, Forum Math. 32 (2020), no. 2, 339–359. 10.1515/forum-2019-0140Search in Google Scholar

[19] Y. Sawano and T. Shimomura, Sobolev embeddings for Riesz potentials of functions in Musielak–Orlicz–Morrey spaces over non-doubling measure spaces, Integral Transforms Spec. Funct. 25 (2014), no. 12, 976–991. 10.1080/10652469.2014.955099Search in Google Scholar

[20] K. Stempak, Examples of metric measure spaces related to modified Hardy–Littlewood maximal operators, Ann. Acad. Sci. Fenn. Math. 41 (2016), no. 1, 313–314. 10.5186/aasfm.2016.4119Search in Google Scholar

Received: 2024-11-01

Accepted: 2025-03-18

Published Online: 2025-06-17

You are currently not able to access this content.

https://doi.org/10.1515/gmj-2025-2044

Keywords for this article

Riesz potential; variable exponent; metric measure space; Hausdorff content; exponential integrability; lower Ahlfors regular