Pitt’s inequality with sharp convolution estimates

Beckner, William

doi:10.1090/S0002-9939-07-09216-7

Pitt’s inequality with sharp convolution estimates
HTML articles powered by AMS MathViewer

by William Beckner PDF

Proc. Amer. Math. Soc. 136 (2008), 1871-1885 Request permission

Abstract:

Sharp $L^p$ extensions of Pitt’s inequality expressed as a weighted Sobolev inequality are obtained using convolution estimates and Stein-Weiss potentials. Optimal constants are obtained for the full Stein-Weiss potential as a map from $L^p$ to itself which in turn yield semi-classical Rellich inequalities on $\mathbb {R}^n$. Additional results are obtained for Stein-Weiss potentials with gradient estimates and with mixed homogeneity. New proofs are given for the classical Pitt and Stein-Weiss estimates.

References

G. Barbatis, Best constants for higher-order Rellich inequalities in $L^p(\Omega )$, Math. Z. 255 (2007), no. 4, 877–896. MR 2274540, DOI 10.1007/s00209-006-0056-5
William Beckner, Inequalities in Fourier analysis, Ann. of Math. (2) 102 (1975), no. 1, 159–182. MR 385456, DOI 10.2307/1970980
William Beckner, Geometric inequalities in Fourier anaylsis, Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991) Princeton Math. Ser., vol. 42, Princeton Univ. Press, Princeton, NJ, 1995, pp. 36–68. MR 1315541
William Beckner, Pitt’s inequality and the uncertainty principle, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1897–1905. MR 1254832, DOI 10.1090/S0002-9939-1995-1254832-9
William Beckner, Sharp inequalities and geometric manifolds, Proceedings of the conference dedicated to Professor Miguel de Guzmán (El Escorial, 1996), 1997, pp. 825–836. MR 1600195, DOI 10.1007/BF02656488
William Beckner, On the Grushin operator and hyperbolic symmetry, Proc. Amer. Math. Soc. 129 (2001), no. 4, 1233–1246. MR 1709740, DOI 10.1090/S0002-9939-00-05630-6
W. Beckner, Weighted inequalities and Stein-Weiss potentials, Forum Math. (in press).
W. Beckner, Grushin symmetry – embedding and potentials (preprint, 2007).
H. J. Brascamp, Elliott H. Lieb, and J. M. Luttinger, A general rearrangement inequality for multiple integrals, J. Functional Analysis 17 (1974), 227–237. MR 0346109, DOI 10.1016/0022-1236(74)90013-5
E. B. Davies and A. M. Hinz, Explicit constants for Rellich inequalities in $L_p(\Omega )$, Math. Z. 227 (1998), no. 3, 511–523. MR 1612685, DOI 10.1007/PL00004389
Stefan Eilertsen, On weighted fractional integral inequalities, J. Funct. Anal. 185 (2001), no. 1, 342–366. MR 1853761, DOI 10.1006/jfan.2001.3777
Gerald B. Folland and Alladi Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl. 3 (1997), no. 3, 207–238. MR 1448337, DOI 10.1007/BF02649110
Filippo Gazzola, Hans-Christoph Grunau, and Enzo Mitidieri, Hardy inequalities with optimal constants and remainder terms, Trans. Amer. Math. Soc. 356 (2004), no. 6, 2149–2168. MR 2048513, DOI 10.1090/S0002-9947-03-03395-6
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395
Ira W. Herbst, Spectral theory of the operator $(p^{2}+m^{2})^{1/2}-Ze^{2}/r$, Comm. Math. Phys. 53 (1977), no. 3, 285–294. MR 436854
V. F. Kovalenko, M. A. Perel′muter, and Ya. A. Semenov, Schrödinger operators with $L^{1/2}_{W}(\textbf {R}^{l})$-potentials, J. Math. Phys. 22 (1981), no. 5, 1033–1044. MR 622855, DOI 10.1063/1.525009
Elliott H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), no. 2, 349–374. MR 717827, DOI 10.2307/2007032
H. R. Pitt, Theorems on Fourier series and power series, Duke Math. J. 3 (1937), no. 4, 747–755. MR 1546029, DOI 10.1215/S0012-7094-37-00363-6
Stefan Samko, Best constant in the weighted Hardy inequality: the spatial and spherical version, Fract. Calc. Appl. Anal. 8 (2005), no. 1, 39–52. MR 2179227
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Elias M. Stein, Analytic continuation of group representations, Advances in Math. 4 (1970), 172–207 (1970). MR 263985, DOI 10.1016/0001-8708(70)90022-8
Charles Fefferman, Robert Fefferman, and Stephen Wainger (eds.), Essays on Fourier analysis in honor of Elias M. Stein, Princeton Mathematical Series, vol. 42, Princeton University Press, Princeton, NJ, 1995. MR 1315539, DOI 10.1515/9781400852949
E. M. Stein and Guido Weiss, Fractional integrals on $n$-dimensional Euclidean space, J. Math. Mech. 7 (1958), 503–514. MR 0098285, DOI 10.1512/iumj.1958.7.57030
D. Yafaev, Sharp constants in the Hardy-Rellich inequalities, J. Funct. Anal. 168 (1999), no. 1, 121–144. MR 1717839, DOI 10.1006/jfan.1999.3462
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776