Embeddings of Function Spaces via the Caffarelli-Silvestre Extension, Capacities and Wolff potentials

Let $P_{\alpha} f(x,t)$ be the Caffarelli-Silvestre extension of a smooth function $f(x): \mathbb{R}^n \rightarrow \mathbb{R}^{n+1}_+:=\mathbb{R}^n\times (0,\infty).$ The purpose of this article is twofold. Firstly, we want to characterize a nonnegative measure $\mu$ on $\mathbb{R}^{n+1}_+$ such that $f(x)\rightarrow P_{\alpha} f(x,t)$ induces bounded embeddings from the Lebesgue spaces $L^p(\mathbb{R}^n)$ to the $L^q(\mathbb{R}^{n+1}_+,\mu).$ On one hand, these embeddings will be characterized by using a newly introduced $L^p-$capacity associated with the Caffarelli-Silvestre extension. In doing so, the mixed norm estimates of $P_{\alpha} f(x,t),$ the dual form of the $L^p-$capacity, the $L^p-$capacity of general balls, and a capacitary strong type inequality will be established, respectively. On the other hand, when $p>q>1,$ these embeddings will also be characterized in terms of the Hedberg-Wolff potential of $\mu.$ Secondly, we characterize a nonnegative measure $\mu$ on $\mathbb{R}^{n+1}_+$ such that $f(x)\rightarrow P_{\alpha} f(x,t)$ induces bounded embeddings from the homogeneous Sobolev spaces $\dot{W}^{\beta,p}(\mathbb{R}^n)$ to the $L^q(\mathbb{R}^{n+1}_+,\mu)$ in terms of the fractional perimeter of open sets for endpoint cases and the fractional capacity for general cases.