Fractional Besov trace/extension-type inequalities via the Caffarelli–Silvestre extension
Author
Faculty Advisor
Date
2022
Keywords
Sobolev inequality, Sobolev logarithmic inequality, Hardy inequality, affine energy, Carleson embedding, fractional Laplacian
Abstract (summary)
Let 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.
Publication Information
Li, P., Hu, R. & Zhai, Z. Fractional Besov Trace/Extension-Type Inequalities via the Caffarelli–Silvestre Extension. Journal of Geometric Analysis 32, 236 (2022). https://doi.org/10.1007/s12220-022-00975-3
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