Strong $A_{\infty}$-weights and scaling invariant Besov capacities

Costea, Şerban

doi:10.4171/rmi/524

Cited by 18 publications

(19 citation statements)

References 12 publications

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“…Vähäkangas [18] studiedĊ α,1 (·, ) and characterized the fractional inequality with fractional relative isocapacitary and isoperimetric inequalities. They gave a sufficient condition and examples so that the fractional relative capacity is bounded by the fractional perimeter of its interior; •Ċ 2n/p,p (·, ) was studied by Costen [13].…”

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 following theorem will show, among other things, that this is true in the case of the (p, q) relative capacity. In our thesis [4] we studied only the case 1 < n = p < ∞. In [6] we extended the results from [4] to the case 1 < p < ∞ and n > 1.…”

Section: Sobolev-lorentz Capacitymentioning

confidence: 99%

“…In our thesis [4] we studied only the case 1 < n = p < ∞. In [6] we extended the results from [4] to the case 1 < p < ∞ and n > 1. The following theorem generalizes Theorem V.23 from [4] and Theorem 4.1.1 from [6] to the case 1 < p < ∞ and n = 1.…”

Section: Sobolev-lorentz Capacitymentioning

confidence: 99%

“…In our recent paper [7] we studied the Sobolev-Lorentz spaces in the Euclidean setting for n ≥ 1. There we extended to the case n = 1 many of the results on Sobolev-Lorentz spaces obtained in [4] and [6] for n ≥ 2. In this paper we extend to n = 1 many on the results on Sobolev-Lorentz capacities obtained in [4] and [6] for n ≥ 2.…”

Section: Introductionmentioning

confidence: 99%

“…The Sobolev-Lorentz spaces have also been studied by Stein in [17], Cianchi-Pick in [2] and [3], by Kauhanen-Koskela-Malý in [14], and by Malý-Swanson-Ziemer in [15]. We studied the Sobolev-Lorentz relative p, q-capacity in the Euclidean setting (see [4], [5] and [6]). See also our joint work [8] with V. Maz'ya.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Sobolev–Lorentz capacity and its regularity in the Euclidean setting

Costea¹

2019

Ann. Acad. Sci. Fenn. Math.

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This paper studies the Sobolev-Lorentz capacity and its regularity in the Euclidean setting for n ≥ 1 integer. We extend here our previous results on the Sobolev-Lorentz capacity obtained for n ≥ 2.Moreover, for n ≥ 2 integer we obtain a few new results concerning the n, 1 relative and global capacities. Specifically, we obtain sharp estimates for the n, 1 relative capacity of the concentric condensers (B(0, r), B(0, 1)) for all r in [0, 1). As a consequence we obtain the exact value of the n, 1 capacity of a point relative to all its bounded open neighborhoods from R n when n ≥ 2. These new sharp estimates concerning the n, 1 relative capacity improve some of our previous results. We also obtain a new result concerning the n, 1 global capacity. Namely, we show that this aforementioned constant is also the value of the n, 1 global capacity of any point from R n , where n ≥ 2 is integer.Computing the aforementioned exact value of the n, 1 relative capacity of a point with respect to all its bounded open neighborhoods from R n allows us to give a new prove of the embedding H

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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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Strong $A_{\infty}$-weights and scaling invariant Besov capacities

Cited by 18 publications

References 12 publications

On Fractional Capacities Relative to Bounded Open Lipschitz Sets

On Fractional Capacities Relative to Bounded Open Lipschitz Sets

Sobolev–Lorentz capacity and its regularity in the Euclidean setting

The ∞‐Besov capacity problem

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