arXiv subjects

Fernando Ballesta-Yagüe

Publications and source records attributed to Fernando Ballesta-Yagüe.

3 recordsLinked to original sources

arXiv2025

Boundary Value Problems in graph Lipschitz domains in the plane with $A_{\infty}$-measures on the boundary

We prove several results for the Dirichlet, Neumann and Regularity problems for the Laplace equation in graph Lipschitz domains in the plane, considering $A_{\infty}$-measures on the boundary. More specifically, we study the $L^{p,1}$-solvability for the Dirichlet problem, complementing results of Kenig (1980) and Carro and Ortiz-Caraballo (2018). Then, we study $L^p$-solvability of the Neumann problem, obtaining a range of solvability which is empty in some cases, a clear difference with the arc-length case. When it is not empty, it is an interval, and we consider solvability at its endpoints, establishing conditions for Lorentz space solvability when $p>1$ and atomic Hardy space solvability when $p=1$. Solving the Lorentz endpoint leads us to a two-weight Sawyer-type inequality, for which we give a sufficient condition. Finally, we show how to adapt to the Regularity problem the results for the Neumann problem.

math.AP↗

arXiv2025

Sparse domination via the Calderón-Zygmund decomposition: the example of Dini-smooth kernels

In this expository article, we briefly survey the main known schemes of proof of sparse domination principles within harmonic analysis. We then use the one based on the Calderón-Zygmund decomposition to prove a dual sparse domination estimate for Calderón-Zymgund operators with Dini-smooth kernels, with an eye on the difficulties that arise when trying to transfer the argument to spaces with nondoubling measures.

math.CA↗

arXiv2025

Analytic extensions of $A_{\infty}$-weights on Lipschitz curves and their use in weighted Hardy spaces

An $A_{\infty}$-weight on a Lipschitz curve $Λ$ in the plane can be extended analytically to the graph Lipschitz domain $Ω$ above it. This problem was studied by C. Kenig [Ken80], who introduced the class $AE$ of well-behaved analytic extensions. Later, he and D. Jerison [JK82] added a Smirnov-type condition to the definition of this class. In this note, we show that this Smirnov-type condition is equivalent to an $H^1$-integrability condition. As a consequence, one of the conditions in the definition of $AE$ can be dropped. We use this simplification to apply C. Kenig's theory to prove results about weighted Hardy spaces. These are useful to study the Neumann problem in $Ω$ with boundary data in weighted spaces.

math.CA↗