Stability of Hardy-Littlewood-Sobolev inequalities with explicit lower bounds.(English)Zbl 1553.46033
Summary: In this paper, we establish the stability for the Hardy-Littlewood-Sobolev (HLS) inequalities with explicit lower bounds. By establishing the relation between the stability of HLS inequalities and the stability of fractional Sobolev inequalities, we also give the stability of the higher and fractional order Sobolev inequalities with the lower bounds. This extends to some extent the stability of the first order Sobolev inequalities with the explicit lower bounds established by Dolbeault, Esteban, Figalli, Frank and Loss in [J. Dolbeault et al., Camb. J. Math. 13, No. 2, 359–430 (2025;Zbl 08018035)] to the higher and fractional order case. Our proofs are based on the competing symmetries, the continuous Steiner symmetrization inequality for the HLS integral and the dual stability theory. As another application of the stability of the HLS inequality, we also establish the stability ofW. Beckner’s [Acta Math. Sin., Engl. Ser. 31, No. 1, 1–28 (2015;Zbl 1345.46027)]restrictive Sobolev inequalities of fractional order \(s\) with \(0 < s < \frac{n}{2}\) on the flat sub-manifold \(\mathbb{R}^{n - 1}\) and the sphere \(\mathbb{S}^{n - 1}\) with the explicit lower bound. When \(s = 1\), this implies the explicit lower bound for the stability of Escobar’s first order Sobolev trace inequality [J. F. Escobar, Indiana Univ. Math. J. 37, No. 3, 687–698 (1988;Zbl 0666.35014)] which has remained unknown in the literature.
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 26D10 | Inequalities involving derivatives and differential and integral operators |
| 26D15 | Inequalities for sums, series and integrals |
Keywords:
stability of Sobolev inequality;Hardy-Littlewood-Sobolev inequality;restrictive Sobolev inequality;stability of Sobolev trace inequalityCite
References:
| [1] | Aubin, T., Problemes isoperimetriques et espaces de Sobolev, J. Differ. Geom., 11, 573-598, 1976 ·Zbl 0371.46011 |
| [2] | Bartsch, T.; Weth, T.; Willem, M., A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator, Calc. Var. Partial Differ. Equ., 18, 253-268, 2003 ·Zbl 1059.31006 |
| [3] | Beckner, W., Weighted inequalities and Stein-Weiss potentials, Forum Math., 20, 587-606, 2008 ·Zbl 1149.42006 |
| [4] | Beckner, W., Functionals for multilinear fractional embedding, Acta Math. Sin. Engl. Ser., 31, 1-28, 2015 ·Zbl 1345.46027 |
| [5] | Bianchi, G.; Egnell, H., A note on the Sobolev inequality, J. Funct. Anal., 100, 1, 18-24, 1991 ·Zbl 0755.46014 |
| [6] | Brezis, H.; Lieb, E., Sobolev inequalities with remainder terms, J. Funct. Anal., 62, 73-86, 1985 ·Zbl 0577.46031 |
| [7] | Brock, F., Continuous Steiner-symmetrization, Math. Nachr., 172, 25-48, 1995 ·Zbl 0886.49010 |
| [8] | Brock, F., Continuous rearrangement and symmetry of solutions of elliptic problems, Proc. Indian Acad. Sci. Math. Sci., 110, 2, 157-204, 2000 ·Zbl 0965.49002 |
| [9] | Carlen, A. E.; Carrillo, J. A.; Loss, M., Hardy-Littlewood-Sobolev inequalities via fast diffusion flows, Proc. Natl. Acad. Sci., 107, 19696-19701, 2010 ·Zbl 1256.42028 |
| [10] | Carlen, E., Duality and stability for functional inequalities, Ann. Fac. Sci. Toulouse Math. (6), 26, 2, 319-350, 2017 ·Zbl 1387.46031 |
| [11] | Carlen, E.; Loss, M., Extremals of functionals with competing symmetries, J. Funct. Anal., 88, 2, 437-456, 1990 ·Zbl 0705.46016 |
| [12] | Chen, L.; Lu, G.; Tao, C., Existence of extremal functions for the Stein-Weiss inequalities on the Heisenberg group, J. Funct. Anal., 277, 1112-1138, 2019 ·Zbl 1429.26034 |
| [13] | Chen, L.; Lu, G.; Tang, H., Sharp stability of log-Sobolev and Moser-Onofri inequalities on the sphere, J. Funct. Anal., 2023 ·Zbl 1526.46021 |
| [14] | Chen, L.; Lu, G.; Tang, H., Optimal asymptotic lower bound for stability of fractional Sobolev inequality and the global stability of log-Sobolev inequality on the sphere |
| [15] | Chen, L.; Lu, G.; Tang, H., Optimal stability of Hardy-Littlewood-Sobolev and Sobolev inequalities of arbitrary orders with dimension-dependent constants |
| [16] | Chen, S.; Frank, R.; Weth, T., Remainder terms in the fractional Sobolev inequality, Indiana Univ. Math. J., 62, 4, 1381-1397, 2013 ·Zbl 1296.46032 |
| [17] | De Nitti, N.; König, T., Critical functions and blow-up asymptotics for the fractional Brezis-Nirenberg problem in low dimension, 2022 |
| [18] | Dolbeault, J.; Esteban, M.; Figalli, A.; Frank, R.; Loss, M., Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence, 2022 |
| [19] | Escobar, J., Sharp constant in a Sobolev trace inequality, Indiana Univ. Math. J., 37, 687-698, 1988 ·Zbl 0666.35014 |
| [20] | Esposito, P., On some conjectures proposed by Haïm Brezis, Nonlinear Anal., 56, 5, 751-759, 2004 ·Zbl 1134.35045 |
| [21] | Frank, R. L.; Lieb, E. H., Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differ. Equ., 39, 85-99, 2010 ·Zbl 1204.39024 |
| [22] | Frank, R. L.; Lieb, E. H., Sharp constants in several inequalities on the Heisenberg group, Ann. Math. (2), 176, 1, 349-381, 2012 ·Zbl 1252.42023 |
| [23] | Frank, R. L.; Lieb, E. H., A new rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality, (Brown, B. M.E. A., Spectral Theory, Function Spaces and Inequalities. Spectral Theory, Function Spaces and Inequalities, Oper. Theory Adv. Appl., vol. 219, 2012, Birkhauser: Birkhauser Basel), 55-67 ·Zbl 1297.39023 |
| [24] | Han, X., Existence of maximizers for Hardy-Littlewood-Sobolev inequalities on the Heisenberg group, Indiana Univ. Math. J., 62, 737-751, 2013 ·Zbl 1299.39020 |
| [25] | Han, X.; Lu, G.; Zhu, J., Hardy-Littlewood-Sobolev and Stein-Weiss inequalities and integral systems on the Heisenberg group, Nonlinear Anal., 75, 11, 4296-4314, 2012 ·Zbl 1309.42032 |
| [26] | Hardy, G. H.; Littlewood, J. E., Some properties of fractional integrals, Math. Z., 27, 565-606, 1928 ·JFM 54.0275.05 |
| [27] | König, T., On the sharp constant in the Bianchi-Egnell stability inequality ·Zbl 1532.46030 |
| [28] | Lieb, E. H., Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118, 349-374, 1983 ·Zbl 0527.42011 |
| [29] | Lieb, E. H.; Loss, M., Analysis, Graduate Studies in Mathematics, vol. 14, 2001, American Mathematical Society: American Mathematical Society Providence, RI ·Zbl 0966.26002 |
| [30] | Lu, G.; Wei, J., On a Sobolev inequality with remainder terms, Proc. Am. Math. Soc., 128, 75-84, 1999 ·Zbl 0961.35100 |
| [31] | Nazaret, B., Best constant in Sobolev trace inequalities on the half-space, Nonlinear Anal., 65, 1977-1985, 2006 ·Zbl 1119.26020 |
| [32] | Rey, O., The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89, 1-52, 1990 ·Zbl 0786.35059 |
| [33] | Sobolev, S. L., On a theorem in functional analysis, Mat. Sb., 4, 471-497, 1938, (in Russian) ·JFM 64.1100.02 |
| [34] | Stein, E.; Weiss, G., Fractional integrals on n-dimensional Euclidean space, J. Math. Mech., 7, 503-514, 1958 ·Zbl 0082.27201 |
| [35] | Talenti, G., Best constants in Sobolev inequality, Ann. Mat. Pura Appl., 110, 353-372, 1976 ·Zbl 0353.46018 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
