On compact embeddings in $\mathbf{L^p}$ and fractional spaces

Abstract

The study of the fractional Laplacian operator $(-\Delta)^s$ in $\mathbb{R}^N$ with Dirichlet boundary conditions gained enormous momentum through its identification with a Neumann operator in $\mathbb{R}^N\times (0, \infty)=\mathbb{R}^{N+1}_+$, a method mainly introduced by Caffarelli and Silvestre. Since then, several other operators have been studied using this method. In general, a crucial question is attached to this method: the embedding (in the trace sense) on the ground space $L^q(\mathbb{R}^{N})$ is compact? This question is very important when dealing with problems of existence of solutions. This paper aims to answer this question for some operators. Passing to an abstract setting, let $X,Y$ be Hilbert spaces and $\mathcal{A}\colon X\to X'$ a continuous and symmetric elliptic operator. We suppose that $X$ is dense in $Y$ and that the embedding $X\subset Y$ is compact. In this paper we show some consequences of this setting for the study of the fractional operator attached to $\mathcal{A}$ in the extension setting $\Omega\times(0,\infty)$ or $\mathbb{R}^{N+1}_+$. Being more specific, we will give some examples where the embedding of the extension domain into $L^2(\Omega)$ is compact, even in the case $\Omega=\mathbb{R}^N$.