Embeddings of weighted Sobolev spaces and generalized Caffarelli-Kohn-Nirenberg inequalities

TL;DR

This paper characterizes the conditions under which weighted Sobolev spaces embed into weighted L^r spaces, linking these embeddings to generalized Caffarelli-Kohn-Nirenberg inequalities and their variants.

Contribution

It provides a complete characterization of embeddings for weighted Sobolev spaces with power weights, extending and unifying known inequalities.

Findings

Characterization of all parameters for continuous embeddings.

Connection between embeddings and multiplicative norm inequalities.

Extensions to radially symmetric functions and specific parameter regimes.

Abstract

We characterize all the real numbers a,b,c and 1<= p,q,r<infty such that the weighted Sobolev space W_{a,b}^(q,p)(R^N\{0}) with power weights |x|^a and |x|^b is continuously embedded into L^{r}(R^N;|x|^cdx). Furthermore, we show that this embedding is (almost) always characterized by a multiplicative norm inequality. When a,b,c>-N and attention is confined to smooth functions with compact support, these inequalities coincide with the Caffarelli-Kohn-Nirenberg inequalities. Variants for radially symmetric functions, or when r<=min{p,q}, are also obtained along the way.