Embeddings of function spaces via the Caffarelli–Silvestre extension, capacities and Wolff potentials
Faculty Advisor
Date
2022
Keywords
Fractional Laplacian, Lebesgue space, Sobolev space, capacity, fractional perimeter
Abstract (summary)
Let 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.
Publication Information
Li, 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.2021.112758
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