Capacities and embeddings via symmetrization and conductor inequalities

Silvestre, Pilar

doi:10.1090/s0002-9939-2013-11778-8

Cited by 6 publications

(8 citation statements)

References 15 publications

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“…• Adams and Xiao [6] established the capacitary strong estimates forĊ α,p (·, R n ) on homogeneous Besov spaces (see also [29]), and studied the embeddings and dualities of some induced function spaces; • On the basis of the Adams inequality in [2, Theorem B; 3] over R n and a priori estimate for the solution of the homogeneous heat equation, two embeddings of a homogeneous endpoint Besov space were established forĊ α,1 (·, R n ) by Xiao in [30]; • Silvestre [24] proved some isocapacitary inequalities forĊ α,p (·, R n ) by using rearrangements and modulus of continuity to extend [6, Theorem 1] to the Lipschtz-based function spaces; • Xiao [31] discovered some new sharp inequalities relating the fractional Sobolev capacityĊ α,1 (·, R n ) of a set to its standard volume and fractional perimeter respectively, and consequently proved that the sharp fractional Sobolev inequality is equivalent to either the sharp fractional isocapacitary inequality or the sharp fractional isoperimetric inequality. • Continuing from [31], Xiao and Ye [32] introduced the anisotropic Sobolev capacity with fractional order and developed some basic properties for this new object.…”

Section: Existing Resultsmentioning

confidence: 99%

On Fractional Capacities Relative to Bounded Open Lipschitz Sets

Shi

Xiao

2016

Potential Anal

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For 1 ≤ p ≤ ∞ and n/p < α < 1 + n/p, the fractional capacity C α,p (K, ) of a compact subset K with respect to a bounded open Lipschitz set ⊃ K of R n is systematically investigated and effectively applied to the fractional Sobolev space W α,p ( ) which plays an important role in calculus of variations, harmonic analysis, potential theory, partial differential equations, mathematical physics and so forth. In particular, we explore:• the basic properties of C α,p (K, ); the existence of a capacitary potential, and the relationship between C α,p (K, ) and the fractional Laplace equation. • the C α,p (K, )-based nature of a nonnegative Radon measure μ given on that induces an embedding of W α,p ( ) into the Lebesgue space L q ( , μ) as 1 ≤ p ≤ q < ∞ or the exponentially integrable Lebesgue space exp(c|f | 2n 2n−α ) ∈ L 1 ( , μ) as p = 2n/α or the Lebesgue space L ∞ ( , μ) as 2n/α < p ≤ ∞; the geometric characterization of a fractional Sobolev embedding domain via C α,p (K, ).• some of the essential behaviours of K with C α,p (K, ) = 0.

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Section: Existing Resultsmentioning

confidence: 99%

On Fractional Capacities Relative to Bounded Open Lipschitz Sets

Shi

Xiao

2016

Potential Anal

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“…The proof of inequality (17) is similar to that of Theorem 3.1 in [11]. For completeness, we include a brief proof here.…”

Section: Anisotropic Fractional Sobolev Embeddingsmentioning

confidence: 91%

“…Proof. We first prove that inequality (17) holds and is equivalent to inequality (16), and hence inequality (16) holds automatically.…”

Section: Anisotropic Fractional Sobolev Embeddingsmentioning

confidence: 96%

“…For general subset E ⊂ R n , the anisotropic fractional Sobolev capacity (or the homogeneous end-point Besov capacity) of E with respect to K, denoted by cap(E;Λ 1 See also [1,2,3,17,19,22] for special case K = B n 2 . As a natural outcome of exploring some essential links between [19,22] and [12,13], this paper will focus on the above-newly-introduced anisotropic fractional Sobolev capacity, in particular, its immediate applications to the embedding/trace theory of the anisotropic Sobolev space with fractional order.…”

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confidence: 99%

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Anisotropic Sobolev Capacity with Fractional Order

Xiao¹,

Ye²

2017

Can. j. math.

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Abstract. In this paper, we introduce the anisotropic Sobolev capacity with fractional order and develop some basic properties for this new object. Applications to the theory of anisotropic fractional Sobolev spaces are provided. In particular, we give geometric characterizations for a nonnegative Radon measure µ that naturally induces an embedding of the anisotropic fractional Sobolev clasṡ Λ 1,1 α,K into the µ-based-Lebesgue-space L n/β µ with 0 < β ≤ n. Also, we investigate the anisotropic fractional α-perimeter. Such a geometric quantity can be used to approximate the anisotropic Sobolev capacity with fractional order. Estimation on the constant in the related Minkowski inequality, which is asymptotically optimal as α → 0 + , will be provided.

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“…(ii) If we let µ be the n-dimensional Lebesgue measure in Proposition 5 (1) and we have n − αp > 0, then the limiting case of [19,Theorem 2.8] or [27,Corollary 3.3]…”

Section: ∞-Besov Restrictions Extensions and Multipliersmentioning

confidence: 99%

The ∞‐Besov capacity problem

Milman

Xiao

2017

Mathematische Nachrichten

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A theory of ∞-Besov capacities is developed and several applications are provided. In particular, we solve an open problem in the theory of limits of the ∞-Besov semi-norms, we obtain new restriction-extension inequalities, and we characterize the pointwise multipliers acting on the ∞-Besov spaces.where ∈ [1, ∞), −1 is the unit sphere of ℝ ≥1 , is the angle derivation from the vertical, and is the standard surface area measure. In [22]where ∈ [1, ∞), and −1 denotes the surface measure of −1 . A related inequality obtained in [5], and sharpened in [22], can be formulated asMathematische Nachrichten. 2017;290:2961-2976.

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Capacities and embeddings via symmetrization and conductor inequalities

Cited by 6 publications

References 15 publications

On Fractional Capacities Relative to Bounded Open Lipschitz Sets

On Fractional Capacities Relative to Bounded Open Lipschitz Sets

Anisotropic Sobolev Capacity with Fractional Order

The ∞‐Besov capacity problem

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