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

The study of the fractional Laplacian operator $(-\mathrm{\Delta }{)}^s$ in ${\mathbb{R}}^N$ with Dirichlet boundary conditions gained enormous momentum through its identification with a Neumann operator in ${\mathbb{R}}^N\times (0,\mathrm{\infty })={\mathbb{R}}_{+}^{N+1}$, 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 $\mathcal{A}\text{\hspace{0.17em}⁣}:X\to X^{\mathrm{\prime }}$ 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 $\mathrm{\Omega }\times (0,\mathrm{\infty })$ or ${\mathbb{R}}_{+}^{N+1}$. Being more specific, we will give some examples where the embedding of the extension domain into $L^2(\mathrm{\Omega })$ is compact, even in the case $\mathrm{\Omega }={\mathbb{R}}^N$.