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Benjamin Jaye

Publications and source records attributed to Benjamin Jaye.

24 records · Page 2Linked to original sources
arXiv2015

Reflectionless measures for Calderón-Zygmund operators II: Wolff potentials and rectifiability

We continue our study of the reflectionless measures associated to an $s$-dimensional Calderón-Zygmund operator (CZO) acting in $\mathbb{R}^d$ with $s\in (0,d)$. Here, our focus will be the study of CZOs that are rigid, in the sense that they have few reflectionless measures associated to them. Our goal is to prove that the rigidity properties of a CZO $T$ impose strong geometric conditions upon the support of any measure $μ$ for which $T$ is a bounded operator in $L^2(μ)$. In this way, we shall reduce certain well-known problems at the interface of harmonic analysis and geometric measure theory to a description of reflectionless measures of singular integral operators. What's more, we show that this approach yields promising new results.

math.AP↗
arXiv2014

Reflectionless measures for Calderón-Zygmund Operators I: Basic Theory

We study the properties of reflectionless measures for an $s$-dimensional Calderón-Zygmund operator $T$ acting in $\mathbb{R}^d$, where $s\in (0,d)$. Roughly speaking, these are the measures $μ$ for which $T(μ)$ is constant on the support of the measure. In this series of papers, we develop the basic theory of reflectionless measures, and describe the relationship between the description of reflectionless measures and certain well-known problems in harmonic analysis and geometric measure theory.

math.AP↗
arXiv2013

Reflectionless measures for Calderón-Zygmund operators

We study the properties of reflectionless measures for a Calderón-Zygmund operator T. Roughly speaking, these are measures $μ$ for which T(μ) vanishes (in a weak sense) on the support of the measure. We describe the relationship between certain well-known problems in harmonic analysis and geometric measure theory and the classification of reflectionless measures. As an application of our theory, we give a new proof of a recent theorem of Eiderman, Nazarov, and Volberg, which states that in $\mathbb{R}^d$, the s-dimensional Riesz transform of a non-trivial $s$-dimensional measure is unbounded if $s\in (d-1,d)$.

math.AP↗
arXiv2013

Three revolutions in the kernel are worse than one

An example is constructed of a purely unrectifiable measure $μ$ for which the singular integral operator whose kernel triples and reverses the argument of a complex number is bounded $L^2(μ)$. This is in sharp contrast with the results known for the Cauchy transform, whose kernel reverses the argument of a complex number.

math.CA↗
arXiv2012

The fractional Riesz transform and an exponential potential

In this paper we study the $s$-dimensional Riesz transform of a finite measure $μ$ in $\mathbf{R}^d$, with $s\in (d-1,d)$. We show that the boundedness of the Riesz transform of $μ$ implies that a nonlinear potential of exponential type is finite $μ$-almost everywhere. It appears to be the first result of this type for $s>1$.

math.AP↗