Embeddings of function spaces via the Caffarelli–Silvestre extension, capacities and Wolff potentials. (April 2022) - Li, Pengtao; Shi, Shaoguang; Hu, Rui; Zhai, Zhichun
Abstract: Let P α f ( x, t ) be the Caffarelli–Silvestre extension of a smooth function f ( x ) : R n → R + n + 1 ≔ R n × ( 0, ∞ ) . The purpose of this article is twofold. Firstly, we want to characterize a nonnegative measure μ on R + n + 1 such that f ( x ) → P α f ( x, t ) induces bounded embeddings from the Lebesgue spaces L p ( R n ) to the L q ( R + n + 1, 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 embeddings from the homogeneous Sobolev spaces W ̇ β, 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.