Pitt's inequality with sharp convolution estimates

Abstract

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 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.

The best constant in a fractional Hardy inequality

“…Theorem 1 and the results obtained to date for Laplacian and fractional Laplacian suggest possible strengthenings to weights with additional terms of lower-order boundary asymptotics ( [11,23,4]), and extensions to other specific or more general domains ( [36,4]). To discuss the latter problem, we consider open Ω ⊂ D, and its Hardy constant, κ(Ω), defined as the largest number such that…”

“…We like to note that in some respects, the censored stable process is a better analogue of the killed Brownian motion than the killed stable process is (see [14,7,32], and [37,31]). We suggest the former as a possible setup for studying Dirichlet boundary value problems for non-local integro-differential operators and the corresponding stochastic processes ( [30], [38]) on subdomains of R d ( [3]), beyond the "convolutional" case of the whole of R d ( [21,4]). In this connection, we refer to [25,26,24] for Green-type formulas for the censored process.…”

Mixed norm estimate for Radon transform on weighted L p spaces

“…For proof of the above theorem we refer the readers to [1], [22] and p. 569 of [11]. For 0 < λ < n, let I λ denote the Riesz potential of order λ, that is,…”

Weighted Fourier inequalities: Boas’ conjecture in ℝ n

“…What is more, such inequalities with sharp constants imply the uncertainty principle relations ( [2], [3]). This idea is illustrated at the spectral level by the celebrated Pitt inequality ( [2])…”

“…For n = 1, inequality (2) can be found in [5], [17], [18], [22]; for n ≥ 1, see [3], [5]. W. Beckner [2] found the sharp constant in (2) for p = q = 2 and used this to prove a logarithmic estimate for uncertainty.…”