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On fractional inequalities on metric measure spaces with polar decomposition

  • Aidyn Kassymov ORCID logo , Michael Ruzhansky ORCID logo and Gulnur Zaur ORCID logo EMAIL logo
Published/Copyright: July 13, 2024

Abstract

In this paper, we prove the fractional Hardy inequality on polarisable metric measure spaces. The integral Hardy inequality for 1 < p q < is playing a key role in the proof. Moreover, we also prove the fractional Hardy–Sobolev type inequality on metric measure spaces. In addition, logarithmic Hardy–Sobolev and fractional Nash type inequalities on metric measure spaces are presented. In addition, we present applications on homogeneous groups and on the Heisenberg group.

MSC 2020: 22E30

Award Identifier / Grant number: G.0H94.18N

Funding source: Universiteit Gent

Award Identifier / Grant number: 01M01021

Award Identifier / Grant number: EP/R003025/2

Award Identifier / Grant number: EP/V005529/1

Award Identifier / Grant number: AP23484106

Funding statement: A. Kassymov and M. Ruzhansky are supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations and by the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant number 01M01021). M. Ruzhansky is also supported by EPSRC grants EP/R003025/2 and EP/V005529/1. A. Kassymov and G. Zaur are supported by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP23484106).

  1. Communicated by: Maria Gordina

References

[1] Adimurthi and A. Mallick, A Hardy type inequality on fractional order Sobolev spaces on the Heisenberg group, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018), no. 3, 917–949. 10.2422/2036-2145.201604_010Search in Google Scholar

[2] Z. Avetisyan and M. Ruzhansky, A note on the polar decomposition in metric spaces, J. Math. Sci. (N. Y.) 280 (2024), no. 1, 73–82. 10.1007/s10958-023-06674-wSearch in Google Scholar

[3] F. Avkhadiev and A. Laptev, Hardy inequalities for nonconvex domains, Around the Research of Vladimir Maz’ya. I, Int. Math. Ser. (N. Y.) 11, Springer, New York (2010), 1–12. 10.1007/978-1-4419-1341-8_1Search in Google Scholar

[4] R. D. Benguria, R. L. Frank and M. Loss, The sharp constant in the Hardy–Sobolev–Maz’ya inequality in the three dimensional upper half-space, Math. Res. Lett. 15 (2008), no. 4, 613–622. 10.4310/MRL.2008.v15.n4.a1Search in Google Scholar

[5] A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer Monogr. Math., Springer, Berlin, 2007. Search in Google Scholar

[6] M. Chatzakou, A. Kassymov and M. Ruzhansky, Logarithmic Sobolev-type inequalities on Lie groups, J. Geom. Anal. 34 (2024), 10.1007/s12220-024-01690-x. 10.1007/s12220-024-01690-xSearch in Google Scholar

[7] P. Ciatti, M. G. Cowling and F. Ricci, Hardy and uncertainty inequalities on stratified Lie groups, Adv. Math. 277 (2015), 365–387. 10.1016/j.aim.2014.12.040Search in Google Scholar

[8] E. B. Davies, A review of Hardy inequalities, The Maz’ya Anniversary Collection, Vol. 2 (Rostock 1998), Oper. Theory Adv. Appl. 110, Birkhäuser, Basel (1999), 55–67. 10.1007/978-3-0348-8672-7_5Search in Google Scholar

[9] P. Drábek, H. P. Heinig and A. Kufner, Higher-dimensional Hardy inequality, General Inequalities. 7 (Oberwolfach 1995), Internat. Ser. Numer. Math. 123, Birkhäuser, Basel (1997), 3–16. 10.1007/978-3-0348-8942-1_1Search in Google Scholar

[10] B. Dyda, A fractional order Hardy inequality, Illinois J. Math. 48 (2004), no. 2, 575–588. 10.1215/ijm/1258138400Search in Google Scholar

[11] B. Dyda and R. L. Frank, Fractional Hardy–Sobolev–Maz’ya inequality for domains, Studia Math. 208 (2012), no. 2, 151–166. 10.4064/sm208-2-3Search in Google Scholar

[12] B. Dyda, J. Lehrbäck and A. V. Vähäkangas, Fractional Poincaré and localized Hardy inequalities on metric spaces, Adv. Calc. Var. 16 (2023), no. 4, 867–884. 10.1515/acv-2021-0069Search in Google Scholar

[13] D. E. Edmunds and W. D. Evans, Hardy Operators, Function Spaces and Embeddings, Springer Monogr. Math., Springer, Berlin, 2004. 10.1007/978-3-662-07731-3Search in Google Scholar

[14] V. Fischer and M. Ruzhansky, Quantization on Nilpotent Lie Groups, Progr. Math. 314, Birkhäuser/Springer, Cham, 2016. 10.1007/978-3-319-29558-9Search in Google Scholar

[15] G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Math. Notes 28, Princeton University, Princeton, 1982. 10.1515/9780691222455Search in Google Scholar

[16] R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal. 255 (2008), no. 12, 3407–3430. 10.1016/j.jfa.2008.05.015Search in Google Scholar

[17] R. L. Frank and R. Seiringer, Sharp fractional Hardy inequalities in half-spaces, Around the Research of Vladimir Maz’ya. I, Int. Math. Ser. (N. Y.) 11, Springer, New York (2010), 161–167. 10.1007/978-1-4419-1341-8_6Search in Google Scholar

[18] S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, 3rd ed., Universitext, Springer, Berlin, 2004. 10.1007/978-3-642-18855-8Search in Google Scholar

[19] N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc. 352 (2000), no. 12, 5703–5743. 10.1090/S0002-9947-00-02560-5Search in Google Scholar

[20] A. Gogatishvili, A. Kufner, L.-E. Persson and A. Wedestig, An equivalence theorem for integral conditions related to Hardy’s inequality, Real Anal. Exchange 29 (2003/04), no. 2, 867–880. 10.14321/realanalexch.29.2.0867Search in Google Scholar

[21] G. H. Hardy, Note on a theorem of Hilbert, Math. Z. 6 (1920), no. 3–4, 314–317. 10.1007/BF01199965Search in Google Scholar

[22] H. P. Heinig, A. Kufner and L.-E. Persson, On some fractional order Hardy inequalities, J. Inequal. Appl. 1 (1997), no. 1, 25–46. 10.1155/S1025583497000039Search in Google Scholar

[23] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Grad. Stud. Math. 34, American Mathematical Society, Providence, 2001. 10.1090/gsm/034Search in Google Scholar

[24] A. Kassymov, M. Ruzhansky and D. Suragan, Reverse integral Hardy inequality on metric measure spaces, Ann. Fenn. Math. 47 (2022), no. 1, 39–55. 10.54330/afm.112455Search in Google Scholar

[25] A. Kassymov, M. Ruzhansky and D. Suragan, Hardy inequalities on metric measure spaces, III: The case q p 0 and applications, Proc. A. 479 (2023), no. 2269, Article ID 20220307. 10.1098/rspa.2022.0307Search in Google Scholar

[26] A. Kassymov and D. Suragan, Lyapunov-type inequalities for the fractional 𝑝-sub-Laplacian, Adv. Oper. Theory 5 (2020), no. 2, 435–452. 10.1007/s43036-019-00037-6Search in Google Scholar

[27] A. Kufner and L.-E. Persson, Weighted Inequalities of Hardy Type, World Scientific, River Edge, 2003. 10.1142/5129Search in Google Scholar

[28] A. Kufner, L.-E. Persson and N. Samko, Weighted Inequalities of Hardy Type, 2nd ed., World Scientific, Hackensack, 2017. 10.1142/10052Search in Google Scholar

[29] B. Opic and A. Kufner, Hardy-Type Inequalities, Pitman Res. Notes Math. Ser. 219, Longman Scientific & Technical, Harlow, 1990. Search in Google Scholar

[30] G. Palatucci and M. Piccinini, Nonlocal Harnack inequalities in the Heisenberg group, Calc. Var. Partial Differential Equations 61 (2022), no. 5, Paper No. 185. 10.1007/s00526-022-02301-9Search in Google Scholar

[31] L. Roncal and S. Thangavelu, Hardy’s inequality for fractional powers of the sublaplacian on the Heisenberg group, Adv. Math. 302 (2016), 106–158. 10.1016/j.aim.2016.07.010Search in Google Scholar

[32] M. Ruzhansky and D. Suragan, Hardy Inequalities on Homogeneous Groups, Progr. Math. 327, Birkhäuser/Springer, Cham, 2019. 10.1007/978-3-030-02895-4Search in Google Scholar

[33] M. Ruzhansky and D. Verma, Hardy inequalities on metric measure spaces, Proc. A. 475 (2019), no. 2223, Article ID 20180310. 10.1098/rspa.2018.0310Search in Google Scholar PubMed PubMed Central

[34] M. Ruzhansky and D. Verma, Hardy inequalities on metric measure spaces, II: The case p > q , Proc. A. 477 (2021), no. 2250, Article ID 20210136. 10.1098/rspa.2021.0136Search in Google Scholar PubMed PubMed Central

[35] J. Yang, Fractional Sobolev–Hardy inequality in R N , Nonlinear Anal. 119 (2015), 179–185. 10.1016/j.na.2014.09.009Search in Google Scholar

Received: 2024-01-30
Revised: 2024-05-11
Published Online: 2024-07-13
Published in Print: 2025-06-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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