Pitt's inequality and the fractional Laplacian: Sharp error estimates

By William Beckner. 2007

49 citations recorded by the source corpus.

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 short proofs of the Frank–Seiringer “Hardy inequalities” where fractional smoothness is measured by a Besov norm.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e915bc3d8504ccd32aac05224798c8eef00b874a

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

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].”
Functionals for multilinear fractional embedding (2014, background): “The sharp L2 embedding constant cn,2 was first noted in [6] (see Theorem 3.”
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).”
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”