An improved Hardy inequality for a nonlocal operator

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2016, 36(3): 1143-1157. doi: 10.3934/dcds.2016.36.1143

An improved Hardy inequality for a nonlocal operator

Boumediene Abdellaoui ^1, and Fethi Mahmoudi ^2,

1.	Laboratoire D'Analyse Nonlinéaire et Mathématiques Appliquées, Département de Mathématiques, Faculté des sciences, Université About Baker Belkad, Tlemcen 13000, Algeria
2.	Centro de Modelamiento Matemático (CMM), Universidad de Chile, Beauchef 851, Santiago, Chile

Received January 2015 Revised June 2015 Published August 2015

Abstract
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Let $0 < s < 1$ and $1< p < 2$ be such that $ps < N$ and let $\Omega$ be a bounded domain containing the origin. In this paper we prove the following improved Hardy inequality:
Given $1 \le q < p$, there exists a positive constant $C\equiv C(\Omega, q, N, s)$ such that $$ \int\limits_{\mathbb{R}^N}\int\limits_{\mathbb{R}^N} \, \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy - \Lambda_{N,p,s} \int\limits_{\mathbb{R}^N} \frac{|u(x)|^p}{|x|^{ps}}\,dx$$$$\geq C \int\limits_{\Omega}\int\limits_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{N+qs}}dxdy $$ for all $u \in \mathcal{C}_0^\infty({\Omega})$. Here $\Lambda_{N,p,s}$ is the optimal constant in the Hardy inequality （1.1）.

Keywords: Fractional Sobolev spaces, nonlocal problems., weighted Hardy inequality.

Mathematics Subject Classification: 49J35, 35A15, 35S1.

Citation: Boumediene Abdellaoui, Fethi Mahmoudi. An improved Hardy inequality for a nonlocal operator. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1143-1157. doi: 10.3934/dcds.2016.36.1143

References:

[1]	B. Abdellaoui and R. Bentiffour, Caffarelli-Kohn-Nirenberg type inequalities of fractional order and applications,, submitted., ().
[2]	B. Abdellaoui, I. Peral and A. Primo, A remark on the fractional Hardy inequality with a remainder term,, C. R. Acad. Sci. Paris, 352* (2014), 299. doi: 10.1016/j.crma.2014.02.003.*
[3]	R. A. Adams, Sobolev Spaces,, Academic Press, (1975).
[4]	F. J. Almgren and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous,, J. Amer. Math. Soc., 2 (1989), 683. doi: 10.1090/S0894-0347-1989-1002633-4.
[5]	B. Barrios, M. Medina and I. Peral, Some remarks on the solvability of non-local elliptic problems with the Hardy potential,, Communications in Contemporary Mathematics, 16 (2014). doi: 10.1142/S0219199713500466.
[6]	B. Barrios, I. Peral and S. Vita, Some remarks about the summability of nonlocal nonlinear problems,, Adv. Nonlinear Anal., 4 (2015), 91. doi: 10.1515/anona-2015-0012.
[7]	H. Brezis, L. Dupaigne and A. Tesei, On a semilinear equation with inverse-square potential,, Selecta Math., 11 (2005), 1. doi: 10.1007/s00029-005-0003-z.
[8]	H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^N$,, Manuscripta Math., 74 (1992), 87. doi: 10.1007/BF02567660.
[9]	L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights,, Compositio Math., 53 (1984), 259.
[10]	L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306.
[11]	A. Di castro, T. Kuusi and G. Palatucci, Nonlocal Harnack inequalities,, J. Funct. Anal., 267* (2014), 1807. doi: 10.1016/j.jfa.2014.05.023.*
[12]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. math., 136* (2012), 521. doi: 10.1016/j.bulsci.2011.12.004.*
[13]	E. B. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations,, Comm. Partial Differential Equations, 7 (1982), 77. doi: 10.1080/03605308208820218.
[14]	M. M. Fall, Semilinear elliptic equations for the fractional Laplacian with Hardy potential,, preprint, ().
[15]	F. Ferrari and I. Verbitsky, Radial fractional Laplace operators and Hessian inequalities,, J. Differential Equations, 253 (2012), 244. doi: 10.1016/j.jde.2012.03.024.
[16]	R. Frank, A simple proof of Hardy-Lieb-Thirring inequalities,, Comm. Math. Phys., 290* (2009), 789. doi: 10.1007/s00220-009-0759-7.*
[17]	R. Frank, E. H. Lieb and R. Seiringer, Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators,, Journal of the American Mathematical Society, 21 (2008), 925. doi: 10.1090/S0894-0347-07-00582-6.
[18]	R. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities,, Journal of Functional Analysis, 255 (2008), 3407. doi: 10.1016/j.jfa.2008.05.015.
[19]	L. Grafakos, Classical Fourier Analysis, Third edition,, Graduate Texts in Mathematics, 249 (2014). doi: 10.1007/978-1-4939-1194-3.
[20]	I. W. Herbst, Spectral theory of the operator $(p^2+m^2)^{1/2}-Ze^2/r$,, Commun. Math. Phys., 53 (1977), 285.
[21]	J. Heinonen, T. Kilpelinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations,, Dover Publications, (2006).
[22]	T. Leonori, I. Peral, A. Primo and F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations,, Discrete and Continuous Dynamical Systems, 35 (2015), 6031. doi: 10.3934/dcds.2015.35.6031.
[23]	E. H. Lieb and M. Loss, Analysis, Second edition,, Graduate Studies in Mathematics, 14 (2001). doi: 10.1090/gsm/014.
[24]	P. Lindqvist, On the equation $\D_p u+ \lambda \|u\|^{p-2}u =0$,, Proc. Amer. Math. Soc., 109 (1990), 157. doi: 10.1090/S0002-9939-1990-1007505-7.
[25]	V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations,, Second edition, 342 (2011). doi: 10.1007/978-3-642-15564-2.

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References:

[1]	B. Abdellaoui and R. Bentiffour, Caffarelli-Kohn-Nirenberg type inequalities of fractional order and applications,, submitted., ().
[2]	B. Abdellaoui, I. Peral and A. Primo, A remark on the fractional Hardy inequality with a remainder term,, C. R. Acad. Sci. Paris, 352* (2014), 299. doi: 10.1016/j.crma.2014.02.003.*
[3]	R. A. Adams, Sobolev Spaces,, Academic Press, (1975).
[4]	F. J. Almgren and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous,, J. Amer. Math. Soc., 2 (1989), 683. doi: 10.1090/S0894-0347-1989-1002633-4.
[5]	B. Barrios, M. Medina and I. Peral, Some remarks on the solvability of non-local elliptic problems with the Hardy potential,, Communications in Contemporary Mathematics, 16 (2014). doi: 10.1142/S0219199713500466.
[6]	B. Barrios, I. Peral and S. Vita, Some remarks about the summability of nonlocal nonlinear problems,, Adv. Nonlinear Anal., 4 (2015), 91. doi: 10.1515/anona-2015-0012.
[7]	H. Brezis, L. Dupaigne and A. Tesei, On a semilinear equation with inverse-square potential,, Selecta Math., 11 (2005), 1. doi: 10.1007/s00029-005-0003-z.
[8]	H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^N$,, Manuscripta Math., 74 (1992), 87. doi: 10.1007/BF02567660.
[9]	L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights,, Compositio Math., 53 (1984), 259.
[10]	L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306.
[11]	A. Di castro, T. Kuusi and G. Palatucci, Nonlocal Harnack inequalities,, J. Funct. Anal., 267* (2014), 1807. doi: 10.1016/j.jfa.2014.05.023.*
[12]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. math., 136* (2012), 521. doi: 10.1016/j.bulsci.2011.12.004.*
[13]	E. B. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations,, Comm. Partial Differential Equations, 7 (1982), 77. doi: 10.1080/03605308208820218.
[14]	M. M. Fall, Semilinear elliptic equations for the fractional Laplacian with Hardy potential,, preprint, ().
[15]	F. Ferrari and I. Verbitsky, Radial fractional Laplace operators and Hessian inequalities,, J. Differential Equations, 253 (2012), 244. doi: 10.1016/j.jde.2012.03.024.
[16]	R. Frank, A simple proof of Hardy-Lieb-Thirring inequalities,, Comm. Math. Phys., 290* (2009), 789. doi: 10.1007/s00220-009-0759-7.*
[17]	R. Frank, E. H. Lieb and R. Seiringer, Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators,, Journal of the American Mathematical Society, 21 (2008), 925. doi: 10.1090/S0894-0347-07-00582-6.
[18]	R. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities,, Journal of Functional Analysis, 255 (2008), 3407. doi: 10.1016/j.jfa.2008.05.015.
[19]	L. Grafakos, Classical Fourier Analysis, Third edition,, Graduate Texts in Mathematics, 249 (2014). doi: 10.1007/978-1-4939-1194-3.
[20]	I. W. Herbst, Spectral theory of the operator $(p^2+m^2)^{1/2}-Ze^2/r$,, Commun. Math. Phys., 53 (1977), 285.
[21]	J. Heinonen, T. Kilpelinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations,, Dover Publications, (2006).
[22]	T. Leonori, I. Peral, A. Primo and F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations,, Discrete and Continuous Dynamical Systems, 35 (2015), 6031. doi: 10.3934/dcds.2015.35.6031.
[23]	E. H. Lieb and M. Loss, Analysis, Second edition,, Graduate Studies in Mathematics, 14 (2001). doi: 10.1090/gsm/014.
[24]	P. Lindqvist, On the equation $\D_p u+ \lambda \|u\|^{p-2}u =0$,, Proc. Amer. Math. Soc., 109 (1990), 157. doi: 10.1090/S0002-9939-1990-1007505-7.
[25]	V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations,, Second edition, 342 (2011). doi: 10.1007/978-3-642-15564-2.

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