Article
| Title: | Weighted inequalities for fractional integrals with Hausdorff capacity (English) |
| Author: | Saito, Hiroki |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 76 |
| Issue: | 3 |
| Year: | 2026 |
| Pages: | 879-904 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | The purpose of this paper is to establish the trace inequality for fractional operators involving Hausdorff capacity and to characterize the Borel measures $\mu $ for which these inequalities hold. We further investigate two-weight norm inequalities for fractional integrals and fractional maximal operators on Choquet spaces with respect to Hausdorff capacities. As an application of the trace inequality, we derive an embedding inequality for Sobolev spaces with respect to the Hausdorff capacity into $L^{q}(\mu )$. This result, in turn, leads to a study of the Hardy inequality. In addition, we establish a generalized Stein-Weiss inequality through the two-weight inequalities. (English) |
| Keyword: | fractional integral operator |
| Keyword: | Hausdorff capacity |
| Keyword: | two-weight inequality |
| Keyword: | trace inequality |
| MSC: | 26D10 |
| MSC: | 42B25 |
| MSC: | 42B35 |
| DOI: | 10.21136/CMJ.2026.0409-25 |
| . | |
| Date available: | 2026-09-11T13:13:15Z |
| Last updated: | 2026-09-14 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153732 |
| . | |
| Reference: | [1] Adams, D. R.: Traces of potentials arising from translation invariant operators.Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 25 (1971), 203-217. Zbl 0219.46027, MR 0287301 |
| Reference: | [2] Adams, D. R.: A trace inequality for generalized potentials.Stud. Math. 48 (1973), 99-105. Zbl 0237.46037, MR 0336316, 10.4064/sm-48-1-99-105 |
| Reference: | [3] Adams, D. R.: A note on the Choquet integrals with respect to Hausdorff capacity.Function Spaces and Applications Lecture Notes in Mathematics 1302. Springer, Berlin (1988), 115-124. Zbl 0658.31009, MR 0942261, 10.1007/BFb0078867 |
| Reference: | [4] Adams, D. R.: Choquet integrals in potential theory.Publ. Mat., Barc. 42 (1998), 3-66. Zbl 0923.31006, MR 1628134, 10.5565/PUBLMAT_42198_01 |
| Reference: | [5] Chen, Y.-W. B., Ooi, K. H., Spector, D.: Capacitary maximal inequalities and applications.J. Funct. Anal. 286 (2024), Article ID 110396, 31 pages. Zbl 1536.42014, MR 4729407, 10.1016/j.jfa.2024.110396 |
| Reference: | [6] Cruz-Uribe, D.: Two weight norm inequalities for fractional integral operators and commutators.Available at https://arxiv.org/abs/1412.4157v1 (2014), 59 pages. 10.48550/arXiv.1412.4157 |
| Reference: | [7] Dahlberg, B. E. J.: Regularity properties of Riesz potentials.Indiana Univ. Math. J. 28 (1979), 257-268. Zbl 0413.31003, MR 523103, 10.1512/iumj.1979.28.28018 |
| Reference: | [8] Duarte, R.: An introduction to pointwise sparse domination.Available at https://arxiv.org/abs/2407.05911v1 (2024), 98 pages. 10.48550/arXiv.2407.05911 |
| Reference: | [9] Frank, R. L., Seiringer, R.: Non-linear ground state representations and sharp Hardy inequalities.J. Funct. Anal. 255 (2008), 3407-3430. Zbl 1189.26031, MR 2469027, 10.1016/j.jfa.2008.05.015 |
| Reference: | [10] Hansson, K.: Imbedding theorems of Sobolev type in potential theory.Math. Scand. 45 (1979), 77-102. Zbl 0437.31009, MR 0567435, 10.7146/math.scand.a-11827 |
| Reference: | [11] Hatano, N., Kawasumi, R., Saito, H., Tanaka, H.: Choquet integrals, Hausdorff content and sparse operators.Arch. Math. 124 (2025), 311-324. Zbl 1557.42038, MR 4860245, 10.1007/s00013-024-02083-w |
| Reference: | [12] Imerlishvili, G., Meskhi, A.: A note on the trace inequality for Riesz potentials.Georgian Math. J. 28 (2021), 739-745. Zbl 1479.26022, MR 4319075, 10.1515/gmj-2020-2077 |
| Reference: | [13] Kerman, R., Sawyer, E. T.: The trace inequality and eigenvalue estimates for Schrödinger operators.Ann. Inst. Fourier 36 (1986), 207-228. Zbl 0591.47037, MR 0867921, 10.5802/aif.1074 |
| Reference: | [14] Korobkov, M. V., Kristensen, J.: The trace theorem, the Luzin $N$- and Morse-Sard properties for the sharp case of Sobolev-Lorentz mappings.J. Geom. Anal. 28 (2018), 2834-2856. Zbl 1430.58005, MR 3833820, 10.1007/s12220-017-9936-7 |
| Reference: | [15] Lerner, A. K.: On pointwise estimates involving sparse operators.New York J. Math. 22 (2016), 341-349. Zbl 1347.42030, MR 3484688 |
| Reference: | [16] Lerner, A. K., Ombrosi, S.: Some remarks on the pointwise sparse domination.J. Geom. Anal. 30 (2020), 1011-1027. Zbl 1434.42020, MR 4058547, 10.1007/s12220-019-00172-9 |
| Reference: | [17] Maz'ya, V. G., Verbitsky, I. E.: Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers.Ark. Mat. 33 (1995), 81-115. Zbl 0834.31006, MR 1340271, 10.1007/BF02559606 |
| Reference: | [18] Mihula, Z., Pick, L., Spector, D.: Potential trace inequalities via a Calderón-type theorem.J. Lond. Math. Soc., II. Ser. 113 (2026), Article ID e70504, 35 pages. Zbl 08179241, MR 5045251, 10.1112/jlms.70504 |
| Reference: | [19] Moen, K.: Weighted inequalities for multilinear fractional integral operators.Collect. Math. 60 (2009), 213-238. Zbl 1172.26319, MR 2514845, 10.1007/BF03191210 |
| Reference: | [20] Muckenhoupt, B., Wheeden, R. L.: Weighted norm inequalities for fractional integrals.Trans. Am. Math. Soc. 192 (1974), 261-274. Zbl 0289.26010, MR 0340523, 10.1090/S0002-9947-1974-0340523-6 |
| Reference: | [21] Orobitg, J., Verdera, J.: Choquet integrals, Hausdorff content and the Hardy-Littlewood maximal operator.Bull. Lond. Math. Soc. 30 (1998), 145-150. Zbl 0921.42016, MR 1489325, 10.1112/S0024609397003688 |
| Reference: | [22] Pérez, C.: Two weight norm inequalities for Riesz potentials and uniform $L^p$-weighted Sobolev inequalities.Indiana Univ. Math. J. 39 (1990), 31-44. Zbl 0736.42015, MR 1052009, 10.1512/iumj.1990.39.39004 |
| Reference: | [23] Saito, H., Tanaka, H.: Dual of the Choquet spaces with general Hausdorff content.Stud. Math. 266 (2022), 323-335. Zbl 1494.42024, MR 4450794, 10.4064/sm210415-29-1 |
| Reference: | [24] Saito, H., Tanaka, H., Watanabe, T.: Abstract dyadic cubes, maximal operators and Hausdorff content.Bull. Sci. Math. 140 (2016), 757-773. Zbl 1350.42037, MR 3543752, 10.1016/j.bulsci.2016.02.001 |
| Reference: | [25] Saito, H., Tanaka, H., Watanabe, T.: Block decomposition and weighted Hausdorff content.Can. Math. Bull. 63 (2020), 141-156. Zbl 1439.42031, MR 4059812, 10.4153/S000843951900033X |
| Reference: | [26] Sawano, Y., Fazio, G. Di, Hakim, D. I.: Morrey Spaces. Introduction and Applications to Integral Operators and PDE's. Volume 1.Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2020). Zbl 1482.42002, MR 4288177, 10.1201/9780429085925 |
| Reference: | [27] Sawyer, E. T.: A characterization of two weight norm inequalities for fractional and Poisson integrals.Trans. Am. Math. Soc. 308 (1988), 533-545. Zbl 0665.42023, MR 0930072, 10.1090/S0002-9947-1988-0930072-6 |
| Reference: | [28] Sawyer, E., Wheeden, R. L.: Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces.Am. J. Math. 114 (1992), 813-874. Zbl 0783.42011, MR 1175693, 10.2307/2374799 |
| Reference: | [29] Sun, L., Wang, L.: Attainability of the best constant of Hardy-Sobolev inequality with full boundary singularities.J. Lond. Math. Soc., II. Ser. 111 (2025), Article ID e70086, 42 pages. Zbl 1564.35017, MR 4866678, 10.1112/jlms.70086 |
| Reference: | [30] Tanaka, H.: A characterization of two-weight trace inequalities for positive dyadic operators in the upper triangle case.Potential Anal. 41 (2014), 487-499. Zbl 1298.42020, MR 3232035, 10.1007/s11118-013-9379-0 |
| Reference: | [31] Turesson, B. O.: Nonlinear Potential Theory and Weighted Sobolev Spaces.Lecture Notes in Mathematics 1736. Springer, Berlin (2000). Zbl 0949.31006, MR 1774162, 10.1007/BFb0103908 |
| Reference: | [32] Verbitsky, I. E.: Nonlinear potentials and trace inequalities.The Maz'ya Anniversary Collection. Volume 2 Operator Theory: Advances and Applications 110. Birkhäuser, Basel (1999), 323-343. Zbl 0941.31001, MR 1747901, 10.1007/978-3-0348-8672-7_18 |
| Reference: | [33] Verbitsky, I. E., Wheeden, R. L.: Weighted trace inequalities for fractinal integrals and applications to semilinear equations.J. Funct. Anal. 129 (1995), 221-241. Zbl 0830.46029, MR 1322649, 10.1006/jfan.1995.1049 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
