An $L^1$-type estimate for Riesz potentials

Schikorra, Armin; Spector, Daniel; Schaftingen, Jean Van

doi:10.4171/rmi/937

Cited by 43 publications

(39 citation statements)

References 24 publications

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“…For an account on the existing literature on the operator ∇ α , see [31,Section 1]. Here we only refer to [29][30][31][32][33][35][36][37] for the articles tightly connected to the present work and to [27,Section 15.2] for an agile presentation of the fractional operators defined in (1.2) and in (1.4) and of some of their elementary properties. According to [33,Section 1], it is interesting to notice that [20] seems to be the earliest reference for the operator defined in (1.4).…”

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confidence: 99%

A distributional approach to fractional Sobolev spaces and fractional variation: Existence of blow-up

Comi

Stefani

2019

Journal of Functional Analysis

191

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We continue the study of the space BV α (R n ) of functions with bounded fractional variation in R n of order α ∈ (0, 1) introduced in our previous work [10], by dealing with the asymptotic behaviour of the fractional operators involved. After some technical improvements of certain results of [10], we prove that the fractional α-variation converges to the standard De Giorgi's variation both pointwise and in the Γ-limit sense as α → 1 − . We also prove that the fractional β-variation converges to the fractional α-variation both pointwise and in the Γ-limit sense as β → α − for any given α ∈ (0, 1).

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confidence: 99%

A distributional approach to fractional Sobolev spaces and fractional variation: Existence of blow-up

Comi

Stefani

2019

Journal of Functional Analysis

191

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“…In particular, it was first observed by A. Schikorra, the author, and J. Van Schaftingen in [16] that one does not need the L 1 (R d )-norm of f : Let d ≥ 2 and α ∈ (0, d). There exists a constant C = C(α, d) > 0 such that…”

Section: Introductionmentioning

confidence: 99%

A noninequality for the fractional gradient

Spector

2020

Port. Math.

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In this paper we give a streamlined proof of an inequality recently obtained by the author: For every α ∈ (0, 1) there exists a constant C = C(α, d) > 0 such thatWe also give a counterexample which shows that in contrast to the case α = 1, the fractional gradient does not admit an L 1 trace inequality, i.e. D α u L 1 (R d ;R d ) cannot control the integral of u with respect to the Hausdorff content H d−α ∞ . The main substance of this counterexample is a result of interest in its own right, that even a weak-type estimate for the Riesz transforms fails on the spaceIt is an open question whether this failure of a weaktype estimate for the Riesz transforms extends to β ∈ (0, 1).

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“…Secondly, the structure of (1.1) closely resembles the gradient and therefore such a generalization preserves the structural properties of the equation, a point which we will return to later. This aspect has been important in the development of L 1 fractional Sobolev inequalities in terms of (1.1) in [11], as such inequalities are known to be false for the fractional Laplacian.…”

Section: Introductionmentioning

confidence: 99%