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

Abstract

Let P α f (x, t) be the Caffarelli-Silvestre extension of a smooth function f (x) : R n → R n+1The purpose of this article is twofold. Firstly, we want to characterize a nonnegative measure+ , µ). 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 α 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 µ. Secondly, we characterize a nonnegative measure µ on R n+1 + such that f (x) → P α f (x, t) induces bounded embedding from the homogeneous Sobolev spaces Ẇβ,p (R n ) to the L q (R n+1 + , µ) in terms of the fractional perimeter of open sets for endpoint cases and the fractional capacity for general cases.

Fractional Besov Trace/Extension Type Inequalities via the Caffarelli-Silvestre extension

“…Xiao-Zhai in [27] explored the embeddings for the homogeneous Besov space Λp,p β (R n ) with p < 1 via the fractional diffusion equations. Li et al in [16] studied the embedding of Sobolev space Ẇβ,p (R n ) into the Lebesgue space L q (R n+1 + , µ) via the Caffarelli-Silvestre extension. This article will be organized as follows.…”