Pitt’s Inequality with Sharp Convolution Estimates
نویسنده
چکیده
WILLIAM BECKNER Abstract. Sharp Lp 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 Lp to itself which in turn yield semi-classical Rellich inequalities on Rn. 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.
منابع مشابه
Weighted Inequalities and Stein-weiss Potentials
Sharp extensions of Pitt’s inequality and bounds for Stein-Weiss fractional integrals are obtained that incorporate gradient forms and vector-valued operators. Such results include HardyRellich inequalities. Weighted inequalities provide quantitative information to characterize integrability for differential and integral operators and intrinsically are determined by their dilation character. In...
متن کاملm at h . A P ] 3 1 Ju l 2 00 9 PITT ’ S INEQUALITY AND THE FRACTIONAL LAPLACIAN : SHARP ERROR ESTIMATES for Eli Stein
Abstract. Sharp error estimates in terms of the fractional Laplacian and a weighted Besov norm are obtained for Pitt’s inequality by using the spectral representation with weights for the fractional Laplacian due to Frank, Lieb and Seiringer and the sharp Stein-Weiss inequality. Dilation invariance, group symmetry on a non-unimodular group and a nonlinear Stein-Weiss lemma are used to provide s...
متن کاملSome Sharp Weighted Estimates for Multilinear Operators
In[6], Hu and Yang obtain a variant sharp estimate for the multilinear singular integral operators. The main purpose of this paper is to prove a sharp inequality for some multilinear operators related to certain non-convolution type integral operators. In fact, we shall establish the sharp inequality for the multilinear operators only under certain conditions on the size of the integral operato...
متن کامل0 M ay 2 00 9 PITT ’ S INEQUALITY AND THE FRACTIONAL LAPLACIAN : SHARP ERROR ESTIMATES for
Considerable interest exists in understanding the framework of weighted inequalities for differential operators and the Fourier transform, and the application of quantitative information drawn from these inequalities to varied problems in analysis and mathematical physics, including nonlinear partial differential equations, spectral theory, fluid mechanics, stability of matter, stellar dynamics...
متن کاملConvolution Inequalities for the Boltzmann Collision Operator
In this paper we study the integrability properties of a general version of the Boltzmann collision operator that includes inelastic interactions between particles. We prove a Young’s inequality for variable hard potentials, a Hardy-Littlewood-Sobolev inequality for soft potentials, and estimates with Maxwellian weights for variable hard potentials. In addition we obtain sharp constants for Max...
متن کامل