Variation for the Riesz transform and uniform rectifiability

Blesa, Albert Mas; Tolsa, Xavier

doi:10.4171/jems/487

Cited by 39 publications

(36 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This article is devoted to obtain L p (1 < p < ∞) and weak-L 1 estimates for the variation for Calderón-Zygmund operators with smooth odd kernel with respect to uniformly rectifiable measures. As a matter of fact, we prove that if the L 2 estimate holds then the L p and weak-L 1 estimates follow; the results in [17] deal with the L 2 case.…”

Section: Introductionmentioning

confidence: 74%

“…The proof of (c) =⇒ (a) in Corollary 1.3 is not as hard as the converse implications. Essentally, a combination of the arguments in [20] with the fact that, in a sense, V ρ • R µ controls R µ * does the job (see [17]). Theorem 1.1 is used to prove that (a) =⇒ (b) in Corollary 1.3, the corresponding result in [17] was only proved for p = 2.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

$L^p$ estimates for the variation for singular integrals on uniformly rectifiable sets

Blesa

Tolsa

2017

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

The L p (1 < p < ∞) and weak-L 1 estimates for the variation for Calderón-Zygmund operators with smooth odd kernel on uniformly rectifiable measures are proven. The L 2 boundedness and the corona decomposition method are two key ingredients of the proof. 1 2 ALBERT MAS AND XAVIER TOLSAConcerning the notion of uniform rectifiability, recall that a Radon measure µ in R d is called n-rectifiable if there exists a countable family of n-dimensional C 1 submanifolds {M i } i∈N in R d such that µ(E \ i∈N M i ) = 0 and µ ≪ H n , where H n stands for the ndimensional Hausdorff measure. Moreover, µ is said to be n-dimensional Ahlfors-David regular, or simply n-AD regular, if there exists some constant C > 0 such thatfor all x ∈ suppµ and 0 < r ≤ diam(suppµ). Note that if diam(suppµ) < +∞ then µ(R d ) < ∞ and so the condition µ(B(x, r)) ≤ Cr n in the definition of AD regularity actually holds for all r > 0. Finally, one says that µ is uniformly n-rectifiable if it is n-AD regular and there exist θ, M > 0 so that, for each x ∈ suppµ and 0 < r ≤ diam(suppµ), there is a Lipschitz mapping g from the n-dimensional ball B n (0, r) ⊂ R n into R d such that Lip(g) ≤ M and µ B(x, r) ∩ g(B n (0, r)) ≥ θr n , where Lip(g) stands for the Lipschitz constant of g. In particular, uniform rectifiability implies rectifiability. A set E ⊂ R d is called n-rectifiable (or uniformly n-rectifiable) if H n | E is n-rectifiable (or uniformly n-rectifiable, respectively).We are ready now to state our main result. In the statement M (R d ) stands for the Banach space of finite real Radon measures in R d equipped with the total variation norm.Theorem 1.1. Let µ be a uniformly n-rectifiable measure in R d . Let K be an odd kernel satisfying (1) and, for ρ > 2, consider the associated variation operator defined in (3). ThenThe variation operator has been studied in different contexts during the last years, being probability, ergodic theory, and harmonic analysis three areas where variational inequalities turned out to be a powerful tool to prove new results or to enhace already known ones (see for example [1,8,9,10,11,13,18], and the references therein). Inspired by the results on variational inequalities for Calderón-Zygmund operators in R n like [2, 3], in [16] we began our study of such type of inequalities when one replaces the underlying space R n and its associated Lebesgue measure by some reasonable measure in R d , being the Hausdorff measure on a Lipschitz graph a first natural candidate. In this regard, Theorem 1.1 should be considered as a natural generalisation of variational inequalities for Calderón-Zygmund operators in R n from a geometric measure-theoretic point of view.A big motivation to prove Theorem 1.

show abstract

Section: Introductionmentioning

confidence: 74%

Section: Introductionmentioning

confidence: 99%

$L^p$ estimates for the variation for singular integrals on uniformly rectifiable sets

Blesa

Tolsa

2017

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Set R μ := {R μ } >0 . By combining some techniques from [9,28], in our forthcoming paper [21], we show that the L 2 (μ) boundedness of V ρ • R μ implies that μ is uniformly n-rectifiable. Moreover, we also prove that V ρ • R μ is bounded in L 2 (μ) for all AD regular uniformly n-rectifiable measures μ.…”

Section: Introductionmentioning

confidence: 89%

“…is bounded We have to prove that there exists a constant C > 0 such that, for any f ∈ L ∞ (H n Γ ) and any cubeD ⊂ R n , there exists some constant c depending on f and D such that yields the boundedness of this operator from L 1 (H n Γ ) to L 1,∞ (H n Γ ). The interested reader may see [21], where a more general result is proved.…”

Section: The Operatormentioning

confidence: 98%

See 1 more Smart Citation

Variation and oscillation for singular integrals with odd kernel on Lipschitz graphs

Blesa¹,

Tolsa

2012

Proceedings of the London Mathematical Society

Self Cite

View full text Add to dashboard Cite

We prove that, for ρ > 2, the ρ-variation and oscillation for the smooth truncations of the Cauchy transform on Lipschitz graphs are bounded in L p for 1 < p < ∞. The analogous result holds for the n-dimensional Riesz transform on n-dimensional Lipschitz graphs, as well as for other singular integral operators with odd kernel. In particular, our results strengthen the classical theorem on the L 2 boundedness of the Cauchy transform on Lipschitz graphs by Coifman, McIntosh, and Meyer.

show abstract

Introduction

Arhancet¹,

Kriegler

2022

Lecture Notes in Mathematics

View full text Add to dashboard Cite

Variation for the Riesz transform and uniform rectifiability

Cited by 39 publications

References 13 publications

$L^p$ estimates for the variation for singular integrals on uniformly rectifiable sets

$L^p$ estimates for the variation for singular integrals on uniformly rectifiable sets

Variation and oscillation for singular integrals with odd kernel on Lipschitz graphs

Introduction

Contact Info

Product

Resources

About