Martí Prats Soler - Papers and preprints

arXiv 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 our 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.

arXiv CPAM 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 by Alt and Caffarelli. Instead, the nonlocal nature of the distributional measure associated to a minimizer necessitates arguments which are less reliant on the underlying PDE.

arXiv to appear in Calc. Var. PDE Let \(\Omega^+\subset\mathbb R^{n+1}\) be an NTA domain and let \(\Omega^-= \mathbb R^{n+1}\setminus \overline{\Omega^+}\) be an NTA domain as well. Denote by \(\omega^+\) and \(\omega^-\) their respective harmonic measures. Assume that \(\Omega^+\) is a \(\delta\)-Reifenberg flat domain, for some \(\delta>0\) small enough. In this paper we show that \(\log\frac{d\omega^-}{d\omega^+}\in VMO(\omega^+)\) if and only if \(\Omega^+\) is vanishing Reifenberg flat, \(\omega^+\) has big pieces of uniformly rectifiable measures, and the inner unit normal of \(\Omega^+\) 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.

arXiv We give sufficient conditions for quasiconformal mappings between simply connected Lipschitz domains to have Hölder, Sobolev and Triebel-Lizorkin regularity in terms of the regularity of the boundary of the domains and the regularity of the Beltrami coefficients of the mappings. The results can be understood as a counterpart for the Kellogg-Warchawski Theorem in the context of quasiconformal mappings.

arXiv JLMS In this note we give equivalent characterizations for a fractional Triebel-Lizorkin space \(F_{p,q}^s(\Omega)\) in terms of first-order differences in a uniform domain \(\Omega\). 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 \(d/p-d/q\).

arXiv J. Differential Equations 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 out 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 integral moduli of continuity of the coefficients involved and their ellipticity bound.

arXiv JMAA Let \( \phi \) be a quasiconformal mapping, and let \( T_\phi \) be the composition operator which maps \( f \) to \( f\circ\phi \). Since \( \phi \) may not be bi-Lipschitz, the composition operator need not map Sobolev spaces to themselves. The study begins with the behavior of \( T_\phi\) 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.

arXiv Commun. Pur. Appl. Anal. Some new results regarding the Sobolev regularity of the principal solution of the linear Beltrami equation \( \bar{\partial} f = \mu \partial f + \nu \overline{\partial f} \) for discontinuous Beltrami coefficients \( \mu \) and \( \nu \) are obtained, using Kato-Ponce commutators. A conjecture on the cases where the limitations of the method do not work is raised.

TDX In this dissertation some new results on the boundedness of Calderón-Zygmund operators on Sobolev spaces on domains in \(R^d\). First a \(T(P)\)-theorem is obtained which is valid for \(W^{n,p} (U)\), where \(U\) is a bounded uniform domain of \(R^d\), \(n\) is a given natural number and \(p>d\). Essentially, the result obtained states that a convolution Calderón-Zygmund operator is bounded on this function space if and only if \(T(P)\) belongs to \(W^{n,p} (U)\) for every polynomial \(P\) of degree smaller than \(n\) restricted to the domain. For indices \(p\) less or equal than \(d\), a sufficient condition for the boundedness in terms of Carleson measures is obtained. In the particular case of \( n=1 \) and \(p \leq d\), this Carleson condition is shown to be necessary in fact. The case where \(n\) is not integer and \(0 < n < 1\) is also studied, and analogous results to the former are obtained for a larger family of function spaces, the so-called Triebel-Lizorkin spaces.

The thesis contains some optimal conditions to establish when the Beurling transform of a polynomial restricted to a domain is contained in a Sobolev space \(W^{n,p}(U)\), where \(U\) is a bounded planar lipschitz domain, in terms of the Besov regularity of the boundary of \(U\). This result, in combination with the one mentioned above, provides a condition to determine whether the Beurling transform is bounded on \(W^{n,p}(U) \) or not for \(p>2\), which is optimal in case \(n=1\).

Finally, an application of the aforementioned results is given for quasiconformal mappings in the complex plane. In particular, it is checked that the regularity \(W^{n,p}(U)\) of the Beltrami coefficient of a quasiconformal mapping for a bounded Lipschitz domain \(U\) with boundary parameterizations in a certain Besov space and \(p>2\), implies that the mapping itself is in \(W^{n+1,p}(U)\).

arXiv J. Geom. Anal. Given any uniform domain \( \Omega \), the Triebel-Lizorkin space \( F^s_{p,q}(\Omega) \) with \( 0 < s < 1 \) and \( 1 < p,q < \infty \) can be equipped with a norm in terms of first order differences restricted to pairs of points whose distance is comparable to their distance to the boundary.

Using this characterization, originally due to Seeger and reproven here, we prove a T(1)-theorem for fractional Sobolev spaces with \( 0 < s < 1 \) for any uniform domain and for a large family of Calderón-Zygmund operators in any ambient space \( \mathbb{R}^d \) as long as \( sp>d \).

arXiv J. Anal. Math. Consider a Lipschitz domain \( \Omega \) and a measurable function \( \mu \) supported in \( \overline\Omega \) with \( \left\|{\mu}\right\|_{L^\infty}< 1 \). Then the derivatives of a quasiconformal solution of the Beltrami equation \( \overline{\partial} f =\mu\, \partial f \) inherit the Sobolev regularity \( W^{n,p}(\Omega) \) of the Beltrami coefficient \( \mu \) as long as \( \Omega \) 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, \( N\in B^{n-1/p}_{p,p}(\partial\Omega) \).

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

2010 Mathematics Subject Classification: 30C62, 42B37, 46E35.

Keywords: Quasiconformal mappings, Sobolev spaces, Lipschitz domains, Beurling transform, David-Semmes betas, Peter Jones' betas.

arXiv J. Funct. Anal. 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 \( 0< s \leq 1 \), \( 1 < p < \infty \) with \(sp>2\) and a Lipschitz domain \(\Omega\subset \mathbb{C}\), the Beurling transform \(Bf=- {\rm p.v.}\frac1{\pi z^2}*f\) is bounded in the Sobolev space \(W^{s,p}(\Omega)\) if and only if \(B\chi_\Omega\in W^{s,p}(\Omega)\).

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.