The ∞‐Besov capacity problem

Milman, Mario; Xiao, Jie

doi:10.1002/mana.201600500

Cited by 12 publications

(1 citation statement)

References 30 publications

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“…The Besov capacities C p,q β (•) have been studied in [1,3] for q < ∞ and in [20] for q = ∞. Definition 1.2.…”

Section: Introductionmentioning

confidence: 99%

Fractional Besov Trace/Extension Type Inequalities via the Caffarelli-Silvestre extension

Li¹,

Hu²,

Zhai³

2022

Preprint

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Let u(x, t) be the Caffarelli-Silvestre extension of f (x). The first goal of this article is to establish the fractional trace type inequalities involving the Caffarelli-Silvestre extension u of f (x). 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 ∂ t u(x, t) or (−∆) −γ/2 u(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 Ḣ−β/2 (R n ) norm of f (x) can be controlled by the product of the weighted L p −affine energy and the weighted L p −norm of ∂ t u(x, t). The second goal of this article is to characterize non-negative measures µ on R n+1+ such that the embeddings u(x, t) L q 0 ,p 0 (R n+1 ,µ) f Λp,qhold for some p 0 and q 0 depending on p and q which are classified in three different cases: (1).For case (1), the embeddings can be characterized in terms of an analytic condition of the variational capacity minimizing function, the isocapacitary inequality of open balls, and other weak type inequalities. For cases (2) and ( 3), the embeddings are characterized by the iso-capacitary inequality for fractonal Besov capacity of open sets.

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“…The Besov capacities C p,q β (•) have been studied in [1,3] for q < ∞ and in [20] for q = ∞. Definition 1.2.…”

Section: Introductionmentioning

confidence: 99%