Communications in Analysis and Geometry

Volume 30 (2022)

Number 6

Limiting case of an isoperimetric inequality with radial densities and applications

Pages: 1391 – 1411

DOI: https://dx.doi.org/10.4310/CAG.2022.v30.n6.a6

Author

Georgios Psaradakis (Lehrstuhl für Mathematik IV (WIM),Universität Mannheim, Germany; and Department of Mathematics, University of Western Macedonia, Kastoria, Greece)

Abstract

We prove a sharp isoperimetric inequality with radial densities whose functional counterpart corresponds to a limiting case for the exponents of the Il’in (or Caffarelli–Kohn–Nirenberg) inequality in $L^1$. We show how the latter applies to obtain an optimal critical Sobolev weighted norm improvement to one of the $L^1$ weighted Hardy inequalities of [29]. Further applications include an $L^p$ version with the best constant of the functional analogue of this isoperimetric inequality and also a weighted Pólya–Szegö inequality.

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Part of this work was done while visiting the Department of Mathematics and Physics of University of Campania ”Luigi Vanvitelli” through an INdAM/GNAMPA Visiting Professor Program (U-FMBAZ-2018-001525 18- 12-2018).

Received 18 June 2019

Accepted 20 December 2019

Published 26 April 2023