An improved Hardy inequality for a nonlocal operator

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March 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., (). Google Scholar
[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. Google Scholar*
[3]	R. A. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^N$,, Manuscripta Math., 74 (1992), 87. doi: 10.1007/BF02567660. Google Scholar
[9]	L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights,, Compositio Math., 53 (1984), 259. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar*
[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. Google Scholar
[14]	M. M. Fall, Semilinear elliptic equations for the fractional Laplacian with Hardy potential,, preprint, (). Google Scholar
[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. Google Scholar
[16]	R. Frank, A simple proof of Hardy-Lieb-Thirring inequalities,, Comm. Math. Phys., 290* (2009), 789. doi: 10.1007/s00220-009-0759-7. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[19]	L. Grafakos, Classical Fourier Analysis, Third edition,, Graduate Texts in Mathematics, 249 (2014). doi: 10.1007/978-1-4939-1194-3. Google Scholar
[20]	I. W. Herbst, Spectral theory of the operator $(p^2+m^2)^{1/2}-Ze^2/r$,, Commun. Math. Phys., 53 (1977), 285. Google Scholar
[21]	J. Heinonen, T. Kilpelinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations,, Dover Publications, (2006). Google Scholar
[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. Google Scholar
[23]	E. H. Lieb and M. Loss, Analysis, Second edition,, Graduate Studies in Mathematics, 14 (2001). doi: 10.1090/gsm/014. Google Scholar
[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. Google Scholar
[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. Google Scholar

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References:

[1]	B. Abdellaoui and R. Bentiffour, Caffarelli-Kohn-Nirenberg type inequalities of fractional order and applications,, submitted., (). Google Scholar
[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. Google Scholar*
[3]	R. A. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^N$,, Manuscripta Math., 74 (1992), 87. doi: 10.1007/BF02567660. Google Scholar
[9]	L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights,, Compositio Math., 53 (1984), 259. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar*
[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. Google Scholar
[14]	M. M. Fall, Semilinear elliptic equations for the fractional Laplacian with Hardy potential,, preprint, (). Google Scholar
[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. Google Scholar
[16]	R. Frank, A simple proof of Hardy-Lieb-Thirring inequalities,, Comm. Math. Phys., 290* (2009), 789. doi: 10.1007/s00220-009-0759-7. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[19]	L. Grafakos, Classical Fourier Analysis, Third edition,, Graduate Texts in Mathematics, 249 (2014). doi: 10.1007/978-1-4939-1194-3. Google Scholar
[20]	I. W. Herbst, Spectral theory of the operator $(p^2+m^2)^{1/2}-Ze^2/r$,, Commun. Math. Phys., 53 (1977), 285. Google Scholar
[21]	J. Heinonen, T. Kilpelinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations,, Dover Publications, (2006). Google Scholar
[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. Google Scholar
[23]	E. H. Lieb and M. Loss, Analysis, Second edition,, Graduate Studies in Mathematics, 14 (2001). doi: 10.1090/gsm/014. Google Scholar
[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. Google Scholar
[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. Google Scholar

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