Pitt’s inequality with sharp convolution estimates

By William Beckner. 2006

115 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/270ad0d7c5bcd317eb2adf040be89fc7611e6ce9

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In Applied Mathematics

Citations in the corpus

Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications ✩ (2012, background): “We refer the reader to [11] for some nonexistence result of extremal functions for Stein–Weiss inequalities on the Heisenberg group by Beckner (see also [10,12] for Stein–Weiss inequalities in the Euclidean spaces).”
Functionals for multilinear fractional embedding (2014, background): “For p = 2, a more explicit realization can be given for the constant (see discussion of Pitt’s inequality in [5], [6]):”
Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group (2015, background): “For instance, the Pitt’s inequality, the Hardy–Littlewood–Sobolev inequality and the logarithmic Sobolev inequality are in connection with the measure of uncertainty [6, 8, 9].”
Some A priori estimates for the homogeneous Landau equation with soft potentials (2013, methodology): “We use Pitt’s inequality [4, 5, 6] to get that”
Hardy and uncertainty inequalities on stratified Lie groups (2013, background): “We were inspired by the work of W. Beckner [2, 3], but unlike Beckner, we do not look for sharp constants.”
The best constant in a fractional Hardy inequality (2008, background): “The fractional Hardy inequality on the whole of R is known as Hardy-Rellich inequality, and the best constant in this inequality was calculated in [28], [39] (see also [4] for Pitt’s inequality).”
Pitt's inequality and the fractional Laplacian: Sharp error estimates (2007, methodology): “In this context, a new Stein–Weiss lemma, unusual in its simplicity, is applied to obtain the Frank–Seiringer “Hardy inequalities” for both Rn and the upper half-space Rn C [22, 23], and corresponding extensions for the Heisenberg group and product spaces with mixed homogeneity.”
Sharp embedding results for spaces of smooth functions with power weights (2011, unknown): “The proof is based on the Stein– Weiss result on fractional integration (see [6] for a short proof).”