# American Institute of Mathematical Sciences

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

## An improved Hardy inequality for a nonlocal operator

 1 Laboratoire D'Analyse Nonlinéaire et Mathématiques Appliquées, Département de Mathématiques, Faculté des sciences, Université About Baker Belkad, 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

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）.
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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