Browsing by Author "Shi, Shaoguang"
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Item Embeddings of function spaces via the Caffarelli–Silvestre extension, capacities and Wolff potentials(2022) Li, Pengtao; Shi, Shaoguang; Hu, Rui; Zhai, ZhichunLet Pαf (x, t) be the Caffarelli–Silvestre extension of a smooth function f (x) Rn → Rn+1+ := Rn × (0, ∞). The purpose of this article is twofold. Firstly, we want to characterize a nonnegative measure μ on Rn+1 + such that f (x) → Pαf (x, t) induces bounded embeddings from the Lebesgue spaces Lp(Rn) to the Lq (Rn+1 + , μ). On one hand, these embeddings will be characterized by using a newly introduced Lp−capacity associated with the Caffarelli–Silvestre extension. In doing so, the mixed norm estimates of Pαf (x, t), the dual form of the Lp−capacity, the Lp−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 Rn+1 + such that f (x) → Pαf (x, t) induces bounded embeddings from the homogeneous Sobolev spaces ̇W β,p(Rn) to the Lq (Rn+1+ , μ) in terms of the fractional perimeter of open sets for endpoint cases and the fractional capacity for general cases.