Fractional Hardy–Sobolev type inequalities for half spaces and John domains
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- by Bartłomiej Dyda, Juha Lehrbäck and Antti V. Vähäkangas PDF
- Proc. Amer. Math. Soc. 146 (2018), 3393-3402 Request permission
Abstract:
As our main result we prove a variant of the fractional Hardy–Sobolev–Maz’ya inequality for half spaces. This result contains a complete answer to a recent open question by Musina and Nazarov. In the proof we apply a new version of the fractional Hardy–Sobolev inequality that we establish also for more general unbounded John domains than half spaces.References
- Krzysztof Bogdan and Bartłomiej Dyda, The best constant in a fractional Hardy inequality, Math. Nachr. 284 (2011), no. 5-6, 629–638. MR 2663757, DOI 10.1002/mana.200810109
- Bartłomiej Dyda and Rupert L. Frank, Fractional Hardy-Sobolev-Maz’ya inequality for domains, Studia Math. 208 (2012), no. 2, 151–166. MR 2910984, DOI 10.4064/sm208-2-3
- Bartłomiej Dyda, Lizaveta Ihnatsyeva, Juha Lehrbäck, Heli Tuominen, and Antti V. Vähäkangas, Muckenhoupt $A_p$-properties of distance functions and applications to Hardy–Sobolev -type inequalities, Pot. Anal. (accepted for publication), DOI: https://doi.org/10.1007/s11118-017-9674-2
- Bartłomiej Dyda and Moritz Kassmann, Function spaces and extension results for nonlocal Dirichlet problems, arXiv:1612.01628 [math.AP], 2016.
- Bartłomiej Dyda and Antti V. Vähäkangas, A framework for fractional Hardy inequalities, Ann. Acad. Sci. Fenn. Math. 39 (2014), no. 2, 675–689. MR 3237044, DOI 10.5186/aasfm.2014.3943
- S. Filippas, V. G. Maz’ya, and A. Tertikas, Sharp Hardy-Sobolev inequalities, C. R. Math. Acad. Sci. Paris 339 (2004), no. 7, 483–486 (English, with English and French summaries). MR 2099546, DOI 10.1016/j.crma.2004.07.023
- S. Filippas, V. Maz′ya, and A. Tertikas, Critical Hardy-Sobolev inequalities, J. Math. Pures Appl. (9) 87 (2007), no. 1, 37–56 (English, with English and French summaries). MR 2297247, DOI 10.1016/j.matpur.2006.10.007
- Stathis Filippas, Luisa Moschini, and Achilles Tertikas, Sharp trace Hardy-Sobolev-Maz’ya inequalities and the fractional Laplacian, Arch. Ration. Mech. Anal. 208 (2013), no. 1, 109–161. MR 3021545, DOI 10.1007/s00205-012-0594-4
- Stathis Filippas and Georgios Psaradakis, The Hardy-Morrey & Hardy-John-Nirenberg inequalities involving distance to the boundary, J. Differential Equations 261 (2016), no. 6, 3107–3136. MR 3527624, DOI 10.1016/j.jde.2016.05.021
- Rupert L. Frank, Elliott H. Lieb, and Robert Seiringer, Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators, J. Amer. Math. Soc. 21 (2008), no. 4, 925–950. MR 2425175, DOI 10.1090/S0894-0347-07-00582-6
- Rupert L. Frank and Robert Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal. 255 (2008), no. 12, 3407–3430. MR 2469027, DOI 10.1016/j.jfa.2008.05.015
- Rupert L. Frank and Robert Seiringer, Sharp fractional Hardy inequalities in half-spaces, Around the research of Vladimir Maz’ya. I, Int. Math. Ser. (N. Y.), vol. 11, Springer, New York, 2010, pp. 161–167. MR 2723817, DOI 10.1007/978-1-4419-1341-8_{6}
- Ritva Hurri-Syrjänen and Antti V. Vähäkangas, On fractional Poincaré inequalities, J. Anal. Math. 120 (2013), 85–104. MR 3095149, DOI 10.1007/s11854-013-0015-0
- Ritva Hurri-Syrjänen and Antti V. Vähäkangas, Fractional Sobolev-Poincaré and fractional Hardy inequalities in unbounded John domains, Mathematika 61 (2015), no. 2, 385–401. MR 3343059, DOI 10.1112/S0025579314000230
- Michael Loss and Craig Sloane, Hardy inequalities for fractional integrals on general domains, J. Funct. Anal. 259 (2010), no. 6, 1369–1379. MR 2659764, DOI 10.1016/j.jfa.2010.05.001
- Jouni Luukkainen, Assouad dimension: antifractal metrization, porous sets, and homogeneous measures, J. Korean Math. Soc. 35 (1998), no. 1, 23–76. MR 1608518
- Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
- Roberta Musina and Alexander I. Nazarov. Fractional Hardy–Sobolev inequalities on half spaces, arXiv:1707.02710 [math.AP], 2017.
- Carlos Pérez, Two weighted norm inequalities for Riesz potentials and uniform $L^p$-weighted Sobolev inequalities, Indiana Univ. Math. J. 39 (1990), no. 1, 31–44. MR 1052009, DOI 10.1512/iumj.1990.39.39004
- Craig A. Sloane, A fractional Hardy-Sobolev-Maz’ya inequality on the upper halfspace, Proc. Amer. Math. Soc. 139 (2011), no. 11, 4003–4016. MR 2823046, DOI 10.1090/S0002-9939-2011-10818-9
- Jussi Väisälä, Exhaustions of John domains, Ann. Acad. Sci. Fenn. Ser. A I Math. 19 (1994), no. 1, 47–57. MR 1246886
Additional Information
- Bartłomiej Dyda
- Affiliation: Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, ul. Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
- Email: bartlomiej.dyda@pwr.edu.pl
- Juha Lehrbäck
- Affiliation: Department of Mathematics and Statistics, University of Jyvaskyla, P.O. Box 35, FI-40014 University of Jyvaskyla, Finland
- Email: juha.lehrback@jyu.fi
- Antti V. Vähäkangas
- Affiliation: Department of Mathematics and Statistics, University of Jyvaskyla, P.O. Box 35, FI-40014 University of Jyvaskyla, Finland
- Email: antti.vahakangas@iki.fi
- Received by editor(s): September 22, 2017
- Published electronically: April 18, 2018
- Additional Notes: The first author was partially supported by the National Science Centre, Poland, grant no. 2015/18/E/ST1/00239
- Communicated by: Svitlana Mayboroda
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 3393-3402
- MSC (2010): Primary 35A23; Secondary 26D10, 46E35
- DOI: https://doi.org/10.1090/proc/14051
- MathSciNet review: 3803664