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Martí Prats

Publications and source records attributed to Martí Prats.

13 recordsLinked to original sources
arXiv2025

The dimension of planar elliptic measures arising from Lipschitz matrices in Reifenberg flat domains

In this paper we show that, given a planar Reifenberg flat domain with small constant and a divergence form operator associated to a real (not necessarily symmetric) uniformly elliptic matrix with Lipschitz coefficients, the Hausdorff dimension of its elliptic measure is at most 1. More precisely, we prove that there exists a subset of the boundary with full elliptic measure and with $σ$-finite one-dimensional Hausdorff measure. For Reifenberg flat domains, this result extends a previous work of Thomas H. Wolff for the harmonic measure.

math.AP↗
arXiv2024

Global smoothness of quasiconformal mappings in the Triebel-Lizorkin scale

We study quasiconformal mappings in planar domains $Ω$ and their regularity properties described in terms of Sobolev, Bessel potential or Triebel-Lizorkin scales. This leads to optimal conditions, in terms of the geometry of the boundary $\partial Ω$ and of the smoothness of the Beltrami coefficient, that guarantee the global regularity of the mappings in these classes. In the Triebel-Lizorkin class with smoothness below $1$, the same conditions give global regularity in $Ω$ for the principal solutions with Beltrami coefficient supported in $Ω$.

math.AP↗
arXiv2023

Triebel-Lizorkin regularity and bi-Lipschitz maps: composition operator and inverse function regularity

We study the stability of Triebel-Lizorkin regularity of bounded functions and Lipschitz functions under bi-Lipschitz changes of variables and the regularity of the inverse function of a Triebel-Lizorkin bi-Lipschitz map in Lipschitz domains. To obtain the results we provide an equivalent norm for the Triebel-Lizorkin spaces with fractional smoothness in uniform domains in terms of the first-order difference of the last weak derivative available averaged on balls.

math.CA↗
arXiv2020

The two-phase problem for harmonic measure in VMO

Let $Ω^+\subset\mathbb R^{n+1}$ be an NTA domain and let $Ω^-= \mathbb R^{n+1}\setminus \overline{Ω^+}$ be an NTA domain as well. Denote by $ω^+$ and $ω^-$ their respective harmonic measures. Assume that $Ω^+$ is a $δ$-Reifenberg flat domain for some $δ>0$ small enough. In this paper we show that $\log\frac{dω^-}{dω^+}\in VMO(ω^+)$ if and only if $Ω^+$ is vanishing Reifenberg flat, $Ω^+$ and $Ω^-$ have joint big pieces of chord-arc subdomains, and the inner unit normal of $Ω^+$ has vanishing oscillation with respect to the approximate normal. This result can be considered as a two-phase counterpart of a more well known related one-phase problem for harmonic measure solved by Kenig and Toro.

math.CA↗
arXiv2019

Minimizers for the thin one-phase free boundary problem

We consider the "thin one-phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in $\mathbb R^{n+1}_+$ plus the area of the positivity set of that function in $\mathbb R^n$. We establish full regularity of the free boundary for dimensions $n \leq 2$, prove almost everywhere regularity of the free boundary in arbitrary dimension and provide content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight. While our results are typical for the calculus of variations, our approach does not follow the standard one first introduced in \cite{AltCaffarelli}. Instead, the nonlocal nature of the distributional measure associated to a minimizer necessitates arguments which are less reliant on the underlying PDE.

math.AP↗
arXiv2018

Measuring Triebel-Lizorkin fractional smoothness on domains in terms of first-order differences

In this note we give equivalent characterizations for a fractional Triebel-Lizorkin space $F^s_{p,q}(Ω)$ in terms of first-order differences in a uniform domain $Ω$. The characterization is valid for any positive, non-integer real smoothness $s\in \mathbb{R}_+\setminus \mathbb{N}$ and {indices $1\leq p<\infty$, $1\leq q \leq \infty$} as long as the fractional part $\{s\}$ is greater than $d/p-d/q$.

math.CA↗
arXiv2017

Sobolev regularity of the Beurling transform on planar domains

Consider a Lipschitz domain $Ω$ and the Beurling transform of its characteristic function $\mathcal{B} χ_Ω(z)= - {\rm p.v.}\frac1{πz^2}*χ_Ω(z) $. It is shown that if the outward unit normal vector $N$ of the boundary of the domain is in the trace space of $W^{n,p}(Ω)$ (i.e., the Besov space $B^{n-1/p}_{p,p}(\partialΩ)$) then $\mathcal{B} χ_Ω\in W^{n,p}(Ω)$. Moreover, when $p>2$ the boundedness of the Beurling transform on $W^{n,p}(Ω)$ follows. This fact has far-reaching consequences in the study of the regularity of quasiconformal solutions of the Beltrami equation.

math.CA↗
arXiv2017

Characterization for stability in planar conductivities

We find a complete characterization for sets of isotropic conductivities with stable recovery in the $L^2$ norm when the data of the Calderón Inverse Conductivity Problem is obtained in the boundary of a disk and the conductivities are constant in a neighborhood of its boundary. To obtain this result, we present minimal a priori assumptions which turn to be sufficient for sets of conductivities to have stable recovery in a bounded and rough domain. The condition is presented in terms of the modulus of continuity of the coefficients involved and their ellipticity bound.

math.AP↗
arXiv2017

Beltrami equations in the plane and Sobolev regularity

New results regarding the Sobolev regularity of the principal solution of the linear Beltrami equation $\bar{\partial} f = μ\partial f + ν\overline{\partial f}$ for discontinuous Beltrami coefficients $μ$ and $ν$ are obtained, using Kato-Ponce commutators, obtaining that $\overline \partial f$ belongs to a Sobolev space with the same smoothness as the coefficients but some loss in the integrability parameter. A conjecture on the cases where the limitations of the method do not work is raised.

math.AP↗
arXiv2016

Sobolev regularity of quasiconformal mappings on domains

Consider a Lipschitz domain $Ω$ and a measurable function $μ$ supported in $\overlineΩ$ with $\left\|μ\right\|_{L^\infty}<1$. Then the derivatives of a quasiconformal solution of the Beltrami equation $\overline{\partial} f =μ\partial f$ inherit the Sobolev regularity $W^{n,p}(Ω)$ of the Beltrami coefficient $μ$ as long as $Ω$ is regular enough. The condition obtained is that the outward unit normal vector $N$ of the boundary of the domain is in the trace space, that is, $N\in B^{n-1/p}_{p,p}(\partialΩ)$.

math.CA↗
arXiv2016

Sharp bounds for composition with quasiconformal mappings in Sobolev spaces

Let $ϕ$ be a quasiconformal mapping, and let $T_ϕ$ be the composition operator which maps $f$ to $f\circϕ$. Since $ϕ$ may not be bi-Lipschitz, the composition operator need not map Sobolev spaces to themselves. The study begins with the behavior of $T_ϕ$ on $L^p$ and $W^{1,p}$ for $1<p<\infty$. This cases are well understood but alternative proofs of some known results are provided. Using interpolation techniques it is seen that compactly supported Bessel potential functions in $H^{s,p}$ are sent to $H^{s,q}$ whenever $0<s<1$ for appropriate values of $q$. The techniques used lead to sharp results and they can be applied to Besov spaces as well.

math.CA↗
arXiv2015

A T(P) theorem for Sobolev spaces on domains

Recently, V. Cruz, J. Mateu and J. Orobitg have proved a T(1) theorem for the Beurling transform in the complex plane. It asserts that given $0 2$ and a Lipschitz domain $Ω\subset \mathbb{C}$, the Beurling transform $Bf=- {\rm p.v.}\frac1{πz^2}*f$ is bounded in the Sobolev space $W^{s,p}(Ω)$ if and only if $Bχ_Ω\in W^{s,p}(Ω)$. In this paper we obtain a generalized version of the former result valid for any $s\in \mathbb{N}$ and for a larger family of Calderón-Zygmund operators in any ambient space $\mathbb{R}^d$ as long as $p>d$. In that case we need to check the boundedness not only over the characteristic function of the domain, but over a finite collection of polynomials restricted to the domain. Finally we find a sufficient condition in terms of Carleson measures for $p\leq d$. In the particular case $s=1$, this condition is in fact necessary, which yields a complete characterization.

math.CA↗