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

Abstract

Let $P_{\alpha }f(x,t)$ be the Caffarelli-Silvestre extension of a smooth function $f(x):{\mathbb{R}}^n\to {\mathbb{R}}_{+}^{n+1}:={\mathbb{R}}^n\times (0,\infty ).$ The purpose of this article is twofold. Firstly, we want to characterize a nonnegative measure $\mu$ on ${\mathbb{R}}_{+}^{n+1}$ such that $f(x)\to P_{\alpha }f(x,t)$ 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 $f(x)\to P_{\alpha }f(x,t)$ 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.

40 pages. Accepted by Nonlinear Analysis