### Browsing by Author "Li, Pengtao"

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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.Item Fractional Besov trace/extension-type inequalities via the Caffarelli–Silvestre extension(2022) Li, Pengtao; Hu, Rui; Zhai, ZhichunLet u(·, ·) be the Caffarelli–Silvestre extension of f . The first goal of this article is to establish the fractional trace-type inequalities involving the Caffarelli–Silvestre extension u(·, ·) of f . In doing so, firstly, we establish the fractional Sobolev/logarithmic Sobolev/Hardy trace inequalities in terms of ∇(x, t)u(x, t). Then, we prove the fractional anisotropic Sobolev/logarithmic Sobolev/Hardy trace inequalities in terms of ∂tu(x, t) or (−Δ)−γ /2u(x, t) only. Moreover, based on an estimate of the Fourier transform of the Caffarelli–Silvestre extension kernel and the sharp affine weighted L p Sobolev inequality, we prove that the H˙ −β/2(Rn) norm of f (x) can be controlled by the product of the weighted L p-affine energy and the weighted L p-norm of ∂tu(x, t). The second goal of this article is to characterize non-negative measures μ on Rn+1+ such that the embeddings u(·, ·)Lq0,p0 μ (Rn+1) f Λ˙ p,q β (Rn ) hold for some p0 and q0 depending on p and q which are classified in three different cases: (1) p = q ∈ (n/(n + β), 1]; (2) (p, q) ∈ (1, n/β) × (1,∞); (3) (p, q) ∈ (1, n/β) × {∞}. For case (1), the embeddings can be characterized in terms of an analytic condition of the variational capacity minimizing function, the iso-capacitary inequality of open balls, and other weak-type inequalities. For cases (2) and (3), the embeddings are characterized by the iso-capacitary inequality for fractional Besov capacity of open sets.