# Compact embeddings of some weighted fractional Sobolev spaces on $\Rn$

## TL;DR

This paper investigates compact embeddings of fractional Sobolev spaces on Rn and bounded domains, providing new results on the conditions under which these spaces embed compactly into Lebesgue spaces, with implications for analysis.

## Contribution

It introduces new compact embedding results for fractional Sobolev spaces on Rn and bounded Lipschitz domains, extending existing theory.

## Key findings

- Established compact embeddings VsqpRn  LqRn for suitable potentials
- Proved embeddings VsqpRn  LlRn under certain conditions
- Extended classical Sobolev embedding results to fractional and weighted spaces

## Abstract

In this paper, we study a family of general fractional Sobolev spaces $\MsqpOm$ when $\Om=\Rn$ or $\Om$ is a bounded domain, having a compact, Lipschitz boundary $\Bdy$, in $\Rn$ for $n\geq2$. Among other results, some compact embedding results of $\MVsqpRn\hookrightarrow\LqRn$ and $\MVsqpRn\hookrightarrow\LlRn$ for suitable potential functions $V(x)$ are described.