Weighted Sobolev spaces of radially symmetric functions

TL;DR

This paper investigates dilation invariant inequalities for radially symmetric functions involving polyharmonic operators and power weights, analyzing extremals and computing optimal constants.

Contribution

It introduces new dilation invariant inequalities for radial functions with polyharmonic operators and determines extremals and best constants in some cases.

Findings

Established dilation invariant inequalities for radial functions

Analyzed existence of extremals for these inequalities

Computed best constants in specific cases

Abstract

We prove dilation invariant inequalities involving radial functions, poliharmonic operators and weights that are powers of the distance from the origin. Then we discuss the existence of extremals and in some cases we compute the best constants.