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Benjamin Jaye

Publications and source records attributed to Benjamin Jaye.

At least 19 recordsLinked to original sources
arXiv2026

Quantitative Uniqueness and Rough Damping on $\mathbb T^2$

Motivated by a conjecture of Burq and G\'erard, we investigate quantitative uniqueness principles for functions on $\mathbb T^2$ whose Fourier spectra lie in fixed-width annuli. We obtain observability estimates uniform in the radius, under a mild Sobolev regularity condition, and the generalized geometric control condition (GGCC) on the damping function. This leads to exponential decay for the damped wave equation with rough damping. We also establish an analogous uncertainty principle for spectra near dilates of convex polygonal boundaries, where no Sobolev regularity of the damping is required beyond $L^\infty$.

math.CA↗
arXiv2026

The Usual Square Function on Weakly Flat Sets

We study the usual square function estimate associated with the Cauchy single-layer kernel in the plane, without assuming Ahlfors-David regularity. We prove that a finite Radon measure with positive and finite upper density is rectifiable if it satisfies the usual square function estimate and a weak flatness condition. We also prove that, under the same finiteness and density hypotheses, the weak flatness condition follows when the support is contained in a locally two-sided NTA curve. As a corollary, rectifiability follows when the support is contained in a quasicircle.

math.CA↗
arXiv2025

A High-Frequency Uncertainty Principle for the Fourier-Bessel Transform

Motivated by problems in control theory concerning decay rates for the damped wave equation $$w_{tt}(x,t) + \gamma(x) w_t(x,t) + (-\Delta + 1)^{s/2} w(x,t) = 0,$$ we consider an analogue of the classical Paneah-Logvinenko-Sereda theorem for the Fourier Bessel transform. In particular, if $E \subset \mathbb{R}^+$ is $\mu_\alpha$-relatively dense (where $d\mu_\alpha(x) \approx x^{2\alpha+1}\, dx$) for $\alpha > -1/2$, and $\operatorname{supp} \mathcal{F}_\alpha(f) \subset [R,R+1]$, then we show $$\|f\|_{L^2_\alpha(\mathbb{R}^+)} \lesssim \|f\|_{L^2_\alpha(E)},$$ for all $f\in L^2_\alpha(\mathbb{R}^+)$, where the constants in $\lesssim$ do not depend on $R > 0$. Previous results on PLS theorems for the Fourier-Bessel transform by Ghobber and Jaming (2012) provide bounds that depend on $R$. In contrast, our techniques yield bounds that are independent of $R$, offering a new perspective on such results. This result is applied to derive decay rates of radial solutions of the damped wave equation.

math.CA↗
arXiv2021

The Huovinen transform and rectifiability of measures

For a set $E$ of positive and finite length, we prove that if the Huovinen transform (the convolution operator with kernel $z^k/|z|^{k+1}$ for an odd number $k$) associated to $E$ exists in principal value, then $E$ is rectifiable.

math.CA↗
arXiv2019

Uncertainty Principles Associated to Sets Satisfying the Geometric Control Condition

In this paper, we study forms of the uncertainty principle suggested by problems in control theory. We obtain a version of the classical Paneah-Logvinenko-Sereda theorem for the annulus. More precisely, we show that a function with spectrum in an annulus of a given thickness can be bounded, in $L^2$-norm, from above by its restriction to a neighborhood of a GCC set, with constant independent of the radius of the annulus. We apply this result to obtain energy decay rates for damped fractional wave equations, extending the work of Malhi and Stanislavova to both the higher-dimensional and non-periodic setting.

math.CA↗
arXiv2019

A proof of Carleson's $\varepsilon^2$-conjecture

In this paper we provide a proof of the Carleson $\varepsilon^2$-conjecture. This result yields a characterization (up to exceptional sets of zero length) of the tangent points of a Jordan curve in terms of the finiteness of the associated Carleson $\varepsilon^2$-square function.

math.CA↗
arXiv2019

Remarks on the R\'{e}nyi Entropy of a sum of IID random variables

In this note we study a conjecture of Madiman and Wang which predicted that the generalized Gaussian distribution minimizes the R\'{e}nyi entropy of the sum of independent random variables. Through a variational analysis, we show that the generalized Gaussian fails to be a minimizer for the problem.

cs.IT↗
arXiv2018

Small local action of singular integrals on spaces of non-homogeneous type

Fix $d\geq 2$ and $s\in (0,d)$. In this paper we introduce a notion called small local action associated to a singular integral operator, which is a necessary condition for the existence of principal value integral to exist. Our goal is to understand the geometric properties of a measure for which an associated singular integral has small local action. We revisit Mattila's theory of symmetric measures and relate, under the condition that the measure has finite upper density, the existence of small local action to the cost of transporting the measure to a collection of symmetric measures. As applications, we obtain a soft proof of a theorem of Tolsa and Ruiz-de-Villa on the non-existence of a measure with positive and finite upper density for which the principal value integral associated with the $s$-Riesz transform exists if $s\not\in \mathbb{Z}$. Furthermore, we provide a considerable generalization of this theorem if $s\in (d-1,d)$.

math.CA↗
arXiv2018

On the problem of existence in principal value of a Calder\'{o}n-Zygmund operator on a space of non-homogeneous type

In this paper we study the relationship between two fundamental regularity properties of an $s$-dimensional Calder\'{o}n-Zygmund operator (CZO) acting on a Borel measure $\mu$ in $\mathbb{R}^d$, with $s\in (0,d)$. In the classical case when $s=d$ and $\mu$ is equal to the Lebesgue measure, Calder\'on and Zygmund showed that if a CZO is bounded in $L^2$ then the principal value integral exists almost everywhere. However, there are by now several examples showing that this implication may fail for lower-dimensional kernels and measures, even when the CZO has a homogeneous kernel consisting of spherical harmonics. We introduce sharp geometric conditions on $\mu$, in terms of certain scaled transportation distances, which ensure that an extension of the Calder\'{o}n-Zygmund theorem holds. These conditions are necessary and sufficient in the cases of the Riesz transform and the Huovinen transform. Our techniques build upon prior work by Mattila and Verdera, and incorporate the machinery of symmetric measures, introduced to the area by Mattila.

math.CA↗
arXiv2018

Quantitative uniqueness properties for $L^2$ functions with fast decaying, or sparsely supported, Fourier transform

This paper builds upon two key principles behind the Bourgain-Dyatlov quantitative uniqueness theorem for functions with Fourier transform supported in an Ahlfors regular set. We first provide a characterization of when a quantitative uniqueness theorem holds for functions with very quickly decaying Fourier transform, thereby providing an extension of the classical Paneah-Logvinenko-Sereda theorem. Secondly, we derive a transference result which converts a quantitative uniqueness theorem for functions with fast decaying Fourier transform to one for functions with Fourier transform supported on a fractal set. As well as recovering the result of Bourgain-Dyatlov, we obtain analogous uniqueness results for denser fractals.

math.CA↗
arXiv2016

The measures with an associated square function operator bounded in $L^2$

In this paper we provide an extension of a theorem of David and Semmes ('91) to general non-atomic measures. The result provides a geometric characterization of the non-atomic measures for which a certain class of square function operators, or singular integral operators, are bounded in $L^2(\mu)$. The description is given in terms of a modification of Jones' $\beta$-coefficients.

math.CA↗
arXiv2016

Lower bounds for uncentered maximal functions in any dimension

In this paper we address the following question: given $ p\in (1,\infty)$, $n \geq 1$, does there exists a constant $A(p,n)>1$ such that $\| M f\|_{L^{p}}\geq A(n,p) \| f\|_{L^{p}}$ for any nonnegative $f \in L^{p}(\mathbb{R}^{n})$, where $Mf$ is a maximal function operator defined over the family of shifts and dilates of a centrally symmetric convex body. The inequality fails in general for the centered maximal function operator, but nevertheless we give an affirmative answer to the question for the uncentered maximal function operator and the almost centered maximal function operator. In addition, we also present the Bellman function approach of Melas, Nikolidakis and Stavropoulos to maximal function operators defined over various types of families of sets, and in case of parallelepipeds we will show that $A(n,p)=\left(\frac{p}{p-1}\right)^{1/p}$.

math.AP↗
arXiv2016

The Riesz transform of codimension smaller than one and the Wolff energy

Fix $d\geq 2$, and $s\in (d-1,d)$. We characterize the non-negative locally finite non-atomic Borel measures $\mu$ in $\mathbb{R}^d$ for which the associated $s$-Riesz transform is bounded in $L^2(\mu)$ in terms of the Wolff energy. This extends the range of $s$ in which the Mateu-Prat-Verdera characterization of measures with bounded $s$-Riesz transform is known. As an application, we give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator $(-\Delta)^{\alpha/2}$, $\alpha\in (1,2)$, in terms of a well-known capacity from non-linear potential theory. This result contrasts sharply with removability results for Lipschitz harmonic functions.

math.AP↗