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Fractional Besov trace/extension-type inequalities via the Caffarelli–Silvestre extension

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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).


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