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A logarithmic Hardy inequality. (English) Zbl 1209.26020

Connections between the Hardy-type inequalities in \(\mathbb R^d\), \(d\geq3\), e.g., \[ \int_{\mathbb R^d}\frac{|u|^2}{|x|^2}\,dx \leq\,\frac4{(d-2)^2}\int_{\mathbb R^d}|\nabla u|^2\,dx, \] (for \(u\) from the class \(\mathcal D(\mathbb R^d)\) of smooth compactly supported functions) and logarithmic Sobolev-type inequalities, e.g., \[ \int_{\mathbb R^d}|u|^2\log|u|^2\,dx\leq \frac d2\log\Big(\frac2{\pi\,d\,e}\int_{\mathbb R^d}|\nabla u|^2\,dx\Big) \] (for \(u\) from the Hardy space \(H^1(\mathbb R^d)\) such that \(\int_{\mathbb R^d}u^2=1\)) are investigated.
Denote by \(\mathcal D^{1,2}(\mathbb R^d)\) the completion of \(\mathcal D(\mathbb R^d)\) under the \(L^2(\mathbb R^d)\)-norm of the gradient of \(u\), and \(\text{S}:=\frac1{\pi d(d-2)}(\Gamma(d)/\Gamma(d/2))^{2/d}\). One of the results is the logarithmic Hardy inequality \[ \int_{\mathbb R^d}\frac{|u|^2}{|x|^2}\log(|x|^{d-2}|u|^2)\,dx \leq\frac d2\log\Big(C_{LH}\int_{\mathbb R^d}|\nabla u|^2\,dx\Big) \] for all \(u\in\mathcal D^{1,2}(\mathbb R^d)\) such that \(\int_{\mathbb R^d}\frac{|u|^2}{|x|^2}\,dx=1\) with a constant \(C_{LH}\in(0,\text{S}]\) independent of \(u\). Under the additional assumption that the functions \(u\) are radially symmetric the constant \(C_{LH}\) can be computed precisely and an extremal function for the inequality is found. Analogous inequalities (for both non-radial and radial cases) with \(|x|^{-2a}\,dx\) instead of \(dx\) are also derived. Using an Emden-Fowler transformation these inequalities can be rewritten as inequalities on the cylinder. Explicit expressions of the sharp constant, as well as extremals, are also established in the radial case.
Reviewer: Petr Gurka (Praha)

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
26D15 Inequalities for sums, series and integrals
46E99 Linear function spaces and their duals
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[1] Abdellaoui, B.; Colorado, E.; Peral, I., Some improved Caffarelli-Kohn-Nirenberg inequalities, Calc. Var. Partial Differential Equations, 23, 327-345 (2005) · Zbl 1207.35114
[2] Adimurthi; Chaudhuri, N.; Ramaswamy, M., An improved Hardy-Sobolev inequality and its application, (Proc. Amer. Math. Soc., vol. 130 (2002)), 489-505 · Zbl 0987.35049
[3] Adimurthi; Filippas, S.; Tertikas, A., On the best constant of Hardy-Sobolev inequalities, Nonlinear Anal., 70, 2826-2833 (2009) · Zbl 1165.35484
[4] Agueh, M., Gagliardo-Nirenberg inequalities involving the gradient \(L^2\)-norm, C. R. Math. Acad. Sci. Paris, 346, 757-762 (2008) · Zbl 1149.35329
[5] Alvino, A.; Ferone, V.; Trombetti, G., On the best constant in a Hardy-Sobolev inequality, Appl. Anal., 85, 171-180 (2006) · Zbl 1087.26010
[6] Ané, C.; Blachère, S.; Chafaï, D.; Fougères, P.; Gentil, I.; Malrieu, F.; Roberto, C.; Scheffer, G., Sur les inégalités de Sobolev logarithmiques, Panor. Synthèses, vol. 10 (2000), Société Mathématique de France: Société Mathématique de France Paris, with a preface by D. Bakry and M. Ledoux · Zbl 0982.46026
[7] Arnold, A.; Bartier, J.-P.; Dolbeault, J., Interpolation between logarithmic Sobolev and Poincaré inequalities, Commun. Math. Sci., 5, 971-979 (2007) · Zbl 1146.60063
[8] Arnold, A.; Markowich, P.; Toscani, G.; Unterreiter, A., On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations, 26, 43-100 (2001) · Zbl 0982.35113
[9] Aubin, T., Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom., 11, 573-598 (1976) · Zbl 0371.46011
[10] Avkhadiev, F. G.; Wirths, K.-J., Unified Poincaré and Hardy inequalities with sharp constants for convex domains, ZAMM Z. Angew. Math. Mech., 87, 632-642 (2007) · Zbl 1145.26005
[11] Badiale, M.; Tarantello, G., A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal., 163, 259-293 (2002) · Zbl 1010.35041
[12] Bakry, D.; Émery, M., Hypercontractivité de semi-groupes de diffusion, C. R. Math. Acad. Sci. Paris, 299, 775-778 (1984) · Zbl 0563.60068
[13] Barthe, F.; Cattiaux, P.; Roberto, C., Concentration for independent random variables with heavy tails, AMRX Appl. Math. Res. Express, 2, 39-60 (2005) · Zbl 1094.60010
[14] Bartier, J.; Blanchet, A.; Dolbeault, J.; Escobedo, M., Improved intermediate asymptotics for the heat equation (2009), preprint
[15] Bartier, J.-P.; Dolbeault, J., Convex Sobolev inequalities and spectral gap, C. R. Math. Acad. Sci. Paris, 342, 307-312 (2006) · Zbl 1086.60013
[16] Beckner, W., Sobolev inequalities, the Poisson semigroup, and analysis on the sphere \(S^n\), Proc. Natl. Acad. Sci. USA, 89, 4816-4819 (1992) · Zbl 0766.46012
[17] Beckner, W., Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. (2), 138, 213-242 (1993) · Zbl 0826.58042
[18] Benguria, R. D.; Frank, R. L.; Loss, M., The sharp constant in the Hardy-Sobolev-Maz’ya inequality in the three dimensional upper half-space, Math. Res. Lett., 15, 613-622 (2008) · Zbl 1173.26011
[19] Benguria, R. D.; Loss, M., Connection between the Lieb-Thirring conjecture for Schrödinger operators and an isoperimetric problem for ovals on the plane, (Partial Differential Equations and Inverse Problems. Partial Differential Equations and Inverse Problems, Contemp. Math., vol. 362 (2004), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 53-61 · Zbl 1087.81042
[20] Berger, M.; Gauduchon, P.; Mazet, E., Le spectre d’une variété riemannienne, Lecture Notes in Math., vol. 194 (1971), Springer-Verlag: Springer-Verlag Berlin · Zbl 0223.53034
[21] Blanchet, A.; Bonforte, M.; Dolbeault, J.; Grillo, G.; Vázquez, J., Asymptotics of the fast diffusion equation via entropy estimates, Arch. Ration. Mech. Anal., 191, 347-385 (2009) · Zbl 1178.35214
[22] Bobkov, S. G.; Götze, F., Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal., 163, 1-28 (1999) · Zbl 0924.46027
[23] Bonforte, M.; Dolbeault, J.; Grillo, G.; Vazquez, J.-L., Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities (2009), preprint
[24] Brezis, H.; Vázquez, J. L., Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10, 443-469 (1997) · Zbl 0894.35038
[25] Byeon, J.; Wang, Z.-Q., Symmetry breaking of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities, Commun. Contemp. Math., 4, 457-465 (2002) · Zbl 1010.35029
[26] Caffarelli, L.; Kohn, R.; Nirenberg, L., First order interpolation inequalities with weights, Compos. Math., 53, 259-275 (1984) · Zbl 0563.46024
[27] Carlen, E.; Loss, M., Logarithmic Sobolev inequalities and spectral gaps, (Recent Advances in the Theory and Applications of Mass Transport. Recent Advances in the Theory and Applications of Mass Transport, Contemp. Math., vol. 353 (2004), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 53-60 · Zbl 1149.46027
[28] Catrina, F.; Wang, Z.-Q., On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54, 229-258 (2001) · Zbl 1072.35506
[29] Chou, K. S.; Chu, C. W., On the best constant for a weighted Sobolev-Hardy inequality, J. London Math. Soc. (2), 48, 137-151 (1993) · Zbl 0739.26013
[30] Cianchi, A.; Ferone, A., Hardy inequalities with non-standard remainder terms, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25, 889-906 (2008) · Zbl 1153.26310
[31] Davies, E. B., Heat Kernels and Spectral Theory, Cambridge Tracts in Math., vol. 92 (1990), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0699.35006
[32] Dávila, J.; Dupaigne, L., Hardy-type inequalities, J. Eur. Math. Soc. (JEMS), 6, 335-365 (2004) · Zbl 1083.26012
[33] Del Pino, M.; Dolbeault, J., Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl. (9), 81, 847-875 (2002) · Zbl 1112.35310
[34] Del Pino, M.; Dolbeault, J., The optimal Euclidean \(L^p\)-Sobolev logarithmic inequality, J. Funct. Anal., 197, 151-161 (2003) · Zbl 1091.35029
[35] Dolbeault, J.; Esteban, M. J.; Loss, M.; Tarantello, G., On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities, Adv. Nonlinear Stud., 9, 713-727 (2009) · Zbl 1182.26031
[36] Dolbeault, J.; Esteban, M. J.; Tarantello, G., The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7, 313-341 (2008) · Zbl 1179.26055
[37] Dolbeault, J.; Nazaret, B.; Savaré, G., A new class of transport distances between measures, Calc. Var. Partial Differential Equations, 34, 193-231 (2009) · Zbl 1157.49042
[38] Felli, V.; Schneider, M., Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type, J. Differential Equations, 191, 121-142 (2003) · Zbl 1088.35023
[39] Filippas, S.; Maz’ya, V.; Tertikas, A., On a question of Brezis and Marcus, Calc. Var. Partial Differential Equations, 25, 491-501 (2006) · Zbl 1121.26014
[40] Filippas, S.; Moschini, L.; Tertikas, A., Sharp two-sided heat kernel estimates for critical Schrödinger operators on bounded domains, Comm. Math. Phys., 273, 237-281 (2007) · Zbl 1172.35013
[41] Filippas, S.; Moschini, L.; Tertikas, A., Improving \(L^2\) estimates to Harnack inequalities, Proc. London Math. Soc., 99, 326-352 (2009) · Zbl 1178.35097
[42] Filippas, S.; Tertikas, A., Optimizing improved Hardy inequalities, J. Funct. Anal.. J. Funct. Anal., J. Funct. Anal., 255, 2095-233 (2008), (Corrigendum) · Zbl 1153.26311
[43] Gidas, B.; Ni, W. M.; Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in \(R^n\), (Mathematical Analysis and Applications, Part A. Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., vol. 7 (1981), Academic Press: Academic Press New York), 369-402
[44] Glaser, V.; Grosse, H.; Martin, A.; Thirring, W., A Family of Optimal Conditions for the Absence of Bound States in a Potential, Studies in Math. Phys. (1976), Princeton University Press: Princeton University Press New Jersey, essays in Honor of Valentine Bargmann · Zbl 0332.31004
[45] Gross, L., Logarithmic Sobolev inequalities, Amer. J. Math., 97, 1061-1083 (1975) · Zbl 0318.46049
[46] Hoffmann-Ostenhof, M.; Hoffmann-Ostenhof, T.; Laptev, A., A geometrical version of Hardy’s inequality, J. Funct. Anal., 189, 539-548 (2002) · Zbl 1012.26011
[47] Horiuchi, T., Best constant in weighted Sobolev inequality with weights being powers of distance from the origin, J. Inequal. Appl., 1, 275-292 (1997) · Zbl 0899.35034
[48] Landau, L.; Lifschitz, E., Physique théorique. Tome III: Mécanique quantique. Théorie non relativiste (1967), Éditions Mir: Éditions Mir Moscow, (in French); translated from Russian by E. Gloukhian · Zbl 0144.47605
[49] Lieb, E. H., Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118, 349-374 (1983) · Zbl 0527.42011
[50] Lin, C.-S.; Wang, Z.-Q., Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities, Proc. Amer. Math. Soc., 132, 1685-1691 (2004) · Zbl 1036.35028
[51] Muckenhoupt, B., Hardy’s inequality with weights, Studia Math., 44, 31-38 (1972), collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I · Zbl 0236.26015
[52] Petersson, J. H., Best constants for Gagliardo-Nirenberg inequalities on the real line, Nonlinear Anal., 67, 587-600 (2007) · Zbl 1119.26021
[53] Talenti, G., Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4), 110, 353-372 (1976) · Zbl 0353.46018
[54] Tertikas, A.; Tintarev, K., On existence of minimizers for the Hardy-Sobolev-Maz’ya inequality, Ann. Mat. Pura Appl. (4) (2007) · Zbl 1206.35113
[55] Toscani, G., Sur l’inégalité logarithmique de Sobolev, C. R. Acad. Sci. Paris Sér. I Math., 324, 689-694 (1997) · Zbl 0905.46018
[56] Vázquez, J. L.; Zuazua, E., The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173, 103-153 (2000) · Zbl 0953.35053
[57] Veling, E. J.M., Lower bounds for the infimum of the spectrum of the Schrödinger operator in \(R^N\) and the Sobolev inequalities, JIPAM. J. Inequal. Pure Appl. Math., 3 (2002), Article 63, 22 pp · Zbl 1330.35093
[58] Veling, E. J.M., Corrigendum on the paper: “Lower bounds for the infimum of the spectrum of the Schrödinger operator in <mml:math altimg=”si7.gif“><mml:mi mathvariant=”double-struck“>RN and the Sobolev inequalities” [JIPAM. J. Inequal. Pure Appl. Math. 3 (4) (2002), Article 63, 22 pp., mr1923362], JIPAM. J. Inequal. Pure Appl. Math., 4 (2003), Article 109, 2 pp · Zbl 1330.35093
[59] Wang, Z.-Q.; Willem, M., Caffarelli-Kohn-Nirenberg inequalities with remainder terms, J. Funct. Anal., 203, 550-568 (2003) · Zbl 1037.26014
[60] Weissler, F. B., Logarithmic Sobolev inequalities for the heat-diffusion semigroup, Trans. Amer. Math. Soc., 237, 255-269 (1978) · Zbl 0376.47019
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