Some sharp Hardy inequalities on spherically symmetric domains

Francesco Chiacchio and Tonia Ricciardi

Vol. 242 (2009), No. 1, 173–187

DOI: 10.2140/pjm.2009.242.173

Abstract

We prove several sharp Hardy inequalities for domains with a spherical symmetry. In particular, we prove an inequality for domains of the unit n-dimensional sphere with a point singularity, and an inequality for functions defined on the half-space ℝ₊ⁿ⁺¹ vanishing on the hyperplane {x_n+1 = 0}, with singularity along the x_n+1-axis. The proofs rely on a one-dimensional Hardy inequality involving a weight function related to the volume element on the sphere, as well as on symmetrization arguments. The one-dimensional inequality is derived in a general form.

Keywords

sharp weighted Hardy inequalities, symmetrization

Mathematical Subject Classification 2000

Primary: 46E35

Secondary: 26D10, 35J25

Milestones

Received: 29 July 2008

Accepted: 20 May 2009

Published: 1 September 2009