arXiv subjects

Joan Hernández

Publications and source records attributed to Joan Hernández.

7 recordsLinked to original sources

arXiv2026

Uniform $L^2$-bounds for the $n$-th Calderón commutator on Lipschitz graphs

We prove that, for every real Lipschitz function $A$ and every integer $n\ge1$, $\|T_{A,n}\|_{L^2(\mathbb{R})\to L^2(\mathbb{R})}\le C\,\|A'\|_\infty^n$, where $T_{A,n}$ is the $n$-th Calderón commutator and $C$ is an absolute constant. The essential tool is the Coifman--McIntosh--Meyer theorem, applied to a holomorphic family of complex curves, together with the Cauchy integral formula on a contour surrounding the interval of secant slopes. We also include a simpler version of the argument that gives $O(\log n)$ and explains why the contour has the shape it has.

math.AP↗

arXiv2026

$L^2$-boundedness of the $n$-th Calderón 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 Calderón 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 Hörmander-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↗

arXiv2026

Removable singularities for Lipschitz fractional caloric functions in time varying domains

In this paper we study removable singularities for regular $(1,\frac{1}{2s})$-Lipschitz solutions of the $s$-fractional heat equation for $1/2<s<1$. To do so, we define a Lipschitz fractional caloric capacity and study its critical dimension and the $L^2$-boundedness of a pair of singular integral operators, whose kernels will be the gradient of the fundamental solution of the fractional heat equation and its conjugate.

math.AP↗

arXiv2025

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↗

arXiv2025

The fractional Lipschitz caloric capacity of Cantor sets

We characterize the s-parabolic Lipschitz caloric capacity of corner-like $s$-parabolic Cantor sets in $\mathbb{R}^{n+1}$ for $1/2<s\leq 1$. Despite the spatial gradient of the s-heat kernel lacking temporal anti-symmetry, we obtain analogous results to those known for analytic and Riesz capacities.

math.AP↗

arXiv2025

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

In the present paper we characterize the removable sets for solutions of the fractional heat equation satisfying some parabolic $\text{BMO}$ or $\text{Lip}_α$ 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↗

arXiv2023

On the $(1/2,+)$-caloric capacity of Cantor sets

In the present paper we characterize the $(1/2,+)$-caloric capacity (associated with the $1/2$-fractional heat equation) of the usual corner-like Cantor set of $\mathbb{R}^{n+1}$. The results obtained for the latter are analogous to those found for Newtonian capacity. Moreover, we also characterize the BMO and $\text{Lip}_α$ variants ($0<α<1$) of the $1/2$-caloric capacity in terms of the Hausdorff contents $\mathcal{H}^n_\infty$ and $\mathcal{H}^{n+α}_\infty$ respectively.

math.AP↗