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Pure discrete spectrum for a class of one-dimensional substitution tiling systems
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 Belkad, Tlemcen 13000, Algeria |
2. | Centro de Modelamiento Matemático (CMM), Universidad de Chile, Beauchef 851, Santiago, Chile |
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).
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] | |
[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. |
show all references
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] | |
[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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