The Caffarelli-Kohn-Nirenberg Inequalities For Radial Functions

TL;DR

This paper extends the Caffarelli-Kohn-Nirenberg inequalities to radial functions in Sobolev and fractional Sobolev spaces, showing a larger parameter range than previously known, with new adaptable proofs and applications to compact embeddings.

Contribution

The paper establishes the full parameter range for radial functions in Sobolev and fractional Sobolev spaces, surpassing previous results limited to specific ranges.

Findings

Larger parameter range for inequalities with radial symmetry

New proof techniques adaptable to other contexts

Applications to compact embedding results

Abstract

We establish the full range of the Caffarelli-Kohn-Nirenberg inequalities for radial functions in the Sobolev and the fractional Sobolev spaces of order $0 < s \le 1$. In particular, we show that the range of the parameters for radial functions is strictly larger than the one without symmetric assumption. Previous known results reveal only some special ranges of parameters even in the case $s=1$. Our proof is new and can be easily adapted to other contexts. Applications on compact embeddings are also mentioned.