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Laura Prat

Publications and source records attributed to Laura Prat.

15 recordsLinked to original sources
arXiv2026

$L^2$-boundedness of the $n$-th Calder\'on commutator on Lipschitz graphs

This paper investigates the asymptotic behavior of the norm, as a bounded operator in $L^2(\mathbb{R})$, of the $n$-th Calder\'on commutator $T_{A,n}$ on the graph of a Lipschitz function $A:\mathbb{R}\to\mathbb{R}$. We prove the estimate $\|T_{A,n}\|_{L^2\to L^2} \leq Cn\|A'\|_\infty^n$, thus formalizing a claim by Mateu and Verdera via a symmetrization strategy and the $T1$ theorem. We also show that additional regularity on $A$ yields sublinear growth in $n$. Specifically, for $A$ supported in $[0,1]$, the bound improves to a behavior of the form $\sqrt{n}\|A'\|_\infty^n$ under a Dini condition on $A'$, or if $A'$ belongs to the logarithmic Besov space $B^{1,0}_{1,1}(\mathbb{R})$. This space contains all compactly supported functions in the Sobolev spaces $H^s(\mathbb{R})$ for $0<s<1,$ as well as functions of bounded variation. These refined estimates are established through an alternative framework based on H\"ormander-type conditions and interpolation, bypassing the standard $T1$ approach. Counterexamples are provided to demonstrate that the Dini and Sobolev fractional regularity conditions are incomparable.

math.AP↗
arXiv2024

On fractional parabolic $\text{BMO}$ and $\text{Lip}_{\alpha}$ caloric capacities

In the present paper we characterize the removable sets for solutions of the fractional heat equation satisfying some parabolic $\text{BMO}$ or $\text{Lip}_\alpha$ normalization conditions. We do this by introducing associated fractional caloric capacities, that we show to be comparable to a certain parabolic Hausdorff content.

math.AP↗
arXiv2024

On the semi-additivity of the $1/2$-symmetric caloric capacity

In this paper we study properties of a variant of the $1/2$-caloric capacity, called $1/2$-symmetric caloric capacity. The latter is associated simultaneously with the $1/2$-fractional heat equation and its conjugate. We establish its semi-additivity in $\mathbb{R}^{n+1}$ and, moreover, we compute explicitly the $1/2$-symmetric caloric capacity of rectangles, which illustrates its anisotropic behavior.

math.AP↗
arXiv2020

Removable singularities for Lipschitz caloric functions in time varying domains

In this paper we study removable singularities for regular $(1,1/2)$-Lipschitz solutions of the heat equation in time varying domains. We introduce an associated Lipschitz caloric capacity and we study its metric and geometric properties and the connection with the $L^2$ boundedness of the singular integral whose kernel is given by the gradient of the fundamental solution of the heat equation.

math.CA↗
arXiv2019

$L^2$-bounded singular integrals on a purely unrectifiable set in $\mathbb{R}^d$

We construct an example of a purely unrectifiable measure $μ$ in $\mathbb{R}^d$ for which the singular integrals associated to the kernels $\displaystyle{K(x)=\frac{P_{2k+1}(x)}{|x|^{2k+d}}}$, with $k\geq 1$ and $P_{2k+1}$ a homogeneous harmonic polynomial of degree $2k+1$, are bounded in $L^2(μ)$. This contrasts starkly with the results concerning the Riesz kernel $\displaystyle{\frac{x}{|x|^{d}}}$ in $\mathbb{R}^d$.

math.CA↗
arXiv2018

$L^2$-boundedness of gradients of single layer potentials and uniform rectifiability

Let $A(\cdot)$ be an $(n+1)\times (n+1)$ uniformly elliptic matrix with H\"older continuous real coefficients and let $\mathcal E_A(x,y)$ be the fundamental solution of the PDE $\mathrm{div} A(\cdot) \nabla u =0$ in $\mathbb R^{n+1}$. Let $\mu$ be a compactly supported $n$-AD-regular measure in $\mathbb R^{n+1}$ and consider the associated operator $$T_\mu f(x) = \int \nabla_x\mathcal E_A(x,y)\,f(y)\,d\mu(y).$$ We show that if $T_\mu$ is bounded in $L^2(\mu)$, then $\mu$ is uniformly $n$-rectifiable. This extends the solution of the codimension $1$ David-Semmes problem for the Riesz transform to the gradient of the single layer potential. Together with a previous result of Conde-Alonso, Mourgoglou and Tolsa, this shows that, given $E\subset\mathbb R^{n+1}$ with finite Hausdorff measure $\mathcal H^n$, if $T_{\mathcal H^n|_E}$ is bounded in $L^2(\mathcal H^n|_E)$, then $E$ is $n$-rectifiable. Further, as an application we show that if the elliptic measure associated to the above PDE is absolute continuous with respect to surface measure, then it must be rectifiable, analogously to what happens with harmonic measure.

math.CA↗
arXiv2015

Square functions of fractional homogeneity and Wolff potentials

In this paper it is shown that for anymeasure $μ$ in $\mathbb{R}^d$ and for a non-integer $0<s<d$, the Wolff energy $\displaystyle{\iint_0^\infty(\frac{μ(B(x,r))}{r^s})^2\,\frac{dr}{r}dμ(x)}$ is comparable to $$\iint_0^\infty(\frac{μ(B(x,r))}{r^s} - \frac{μ(B(x,2r))}{(2r)^s})^2\,\frac{dr}rdμ(x),$$ unlike in the case when $s$ is an integer. We also study the relation with the $L^2-$norm of $s$-Riesz transforms, $0<s<1$, and we provide a counterexample in the integer case.

math.CA↗
arXiv2015

Non-existence of reflectionless measures for the s-Riesz transform when 0<s<1

A measure $μ$ on $\mathbb{R}^d$ is called reflectionless for the $s$-Riesz transform if the singular integral $R^sμ(x)=\int \frac{y-x}{|y-x|^{s+1}}\,dμ(y)$ is constant on the support of $μ$ in some weak sense and, moreover, the operator defined by $R^s_μ(f)=R^s(f\,μ)$ is bounded in $L^2(μ)$. In this paper we show that the only reflectionless measure for the $s$-Riesz transform is the zero measure when $0<s<1$.

math.FA↗
arXiv2014

Capacities associated with scalar signed Riesz kernels, and analytic capacity

The real and imaginari parts of the Cauchy kernel in the plane are scalar Riesz kernels of homogeneity -1. One can associate with each of them a natural notion of capacity related to bounded potentials. The main result of the paper asserts that these capacities are comparable to classical analytic capacity, thus stressing the real variables nature of analytic capacity. Higher dimensional versions of this result are also considered.

math.CA↗
arXiv2014

Some Calderón-Zygmund kernels and their relations to Wolff capacities and rectifiability

We consider the Calderón-Zygmund kernels $K_ {α,n}(x)=(x_i^{2n-1}/|x|^{2n-1+α})_{i=1}^d$ in $\mathbb{R}^n$ for $0<α\leq 1$ and $n\in\mathbb{N}$. We show that, on the plane, for $0<α<1$, the capacity associated to the kernels $K_{α,n}$ is comparable to the Riesz capacity $C_{\frac23(2-α),\frac 3 2}$ of non-linear potential theory. As consequences we deduce the semiadditivity and bi-Lipschitz invariance of this capacity. Furthermore we show that for any Borel set $E\subset\mathbb{R}^n$ with finite length the $L^2(\mathcal{H}^1 \lfloor E)$-boundedness of the singular integral associated to $K_{1,n}$ implies the rectifiability of the set $E$. We thus extend to any ambient dimension, results previously known only in the plane.

math.CA↗
arXiv2012

Calderón-Zygmund kernels and rectifiability in the plane

Let $E \subset \C$ be a Borel set with finite length, that is, $0<\mathcal{H}^1 (E)<\infty$. By a theorem of David and Léger, the $L^2 (\mathcal{H}^1 \lfloor E)$-boundedness of the singular integral associated to the Cauchy kernel (or even to one of its coordinate parts $x / |z|^2,y / |z|^2,z=(x,y) \in \C$) implies that $E$ is rectifiable. We extend this result to any kernel of the form $x^{2n-1} /|z|^{2n}, z=(x,y) \in \C,n \in \mathbb{N}$. We thus provide the first non-trivial examples of operators not directly related with the Cauchy transform whose $L^2$-boundedness implies rectifiability.

math.CA↗
arXiv2012

Capacities associated with Calderón-Zygmund kernels

Analytic capacity is associated with the Cauchy kernel $1/z$ and the $L^\infty$-norm. For $n\in\mathbb{N}$, one has likewise capacities related to the kernels $K_i(x)=x_i^{2n-1}/|x|^{2n}$, $1\le i\le 2$, $x=(x_1,x_2)\in\mathbb{R}^2$. The main result of this paper states that the capacities associated with the vectorial kernel $(K_1, K_2)$ are comparable to analytic capacity.

math.CA↗
arXiv2004

The capacity associated to signed Riesz kernels, and Wolff potentials

There is a natural capacity associated to any vector valued Riesz kernel of a given homogeneity. If we are in the plane and the kernel is the Cauchy kernel, then this capacity is analytic capacity. Our main result states that if the homogeneity of the kernel is negative and larger than minus one, then the capacity is comparable to one of the well studied capacities of non-linear potential theory.

math.CA↗