Communications in Analysis and Geometry
Volume 30 (2022)
Limiting case of an isoperimetric inequality with radial densities and applications
Pages: 1391 – 1411
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 . 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.
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