Li, PengtaoShi, ShaoguangHu, RuiZhai, Zhichun2023-04-212023-04-212022Li, P., Shi, S., Hu, R., & Zhai, Z. Embeddings of function spaces via the Caffarelli–Silvestre extension, capacities and Wolff potentials. Nonlinear Analysis, 217, 112758 (2022). https://doi.org/10.1016/j.na.2021.112758https://hdl.handle.net/20.500.14078/3072Let 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.enAll Rights ReservedFractional LaplacianLebesgue spaceSobolev spacecapacityfractional perimeterArticlehttps://doi.org/10.1016/j.na.2021.112758