On Caffarelli-Kohn-Nirenberg inequalities for block-radial functions

TL;DR

This paper extends Caffarelli-Kohn-Nirenberg inequalities to functions with multi-radial symmetry, allowing broader parameter ranges and introducing stronger singularity weights, thereby advancing weighted Sobolev inequality theory.

Contribution

It generalizes CKN inequalities to multi-radial functions, expanding parameter ranges and incorporating stronger singularity weights compared to the radial case.

Findings

Extended parameter ranges for CKN inequalities.

Established pointwise estimates for multi-radial functions.

Introduced stronger singularity weights in inequalities.

Abstract

The paper provides weighted Sobolev inequalities of the Caffarelli-Kohn-Nirenberg type for functions with multi-radial symmetry. Similarly to the previously studied radial case, the range of parameters in CKN inequalities can be extended, sometimes to infinity, providing a pointwise estimate similar to the classical radial estimate. Furthermore, the "multi-radial" weights are a stronger singularity than radial weights of the same homogeneity.