arXiv subjects

Victor Biliatto

Publications and source records attributed to Victor Biliatto.

3 recordsLinked to original sources

arXiv2026

Solvability of elliptic homogeneous linear equations with measure data in weighted Lebesgue spaces

Let $A(D)$ be an elliptic homogeneous linear differential operator with complex constant coefficients, $ μ$ be a vector-valued Borel measure and $w$ be a positive locally integrable function on $\mathbb{R}^N$. In this work, we present sufficient conditions on $μ$ and $w$ for the existence of solutions in the weighted Lebesgue spaces $L^p_w$ for the equation $A^{*}(D)f=μ$, for $ 1\leq p<\infty $. Those conditions are related to a certain control of the Riesz potential of the measure $μ$. We also present sufficient conditions for the solvability when $p=\infty$ adding a canceling condition on the operator. Our method is based on a new weighted $L^1$ Stein-Weiss type inequality on measures for a special class of vector fields.

math.AP↗

arXiv2023

Hausdorff dimension of removable sets for elliptic and canceling homogeneous differential operators in the class of bounded functions

In this note we give an upper bound on the Hausdorff dimension of removable setsfor elliptic and canceling homogeneous differential operators with constant coefficients in the class of bounded functions, using a simple extension of Frostman's lemma in Euclidean space with an additional power decay.

math.AP↗

arXiv2022

A note on Lebesgue solvability of elliptic homogeneous linear equations with measure data

In this work, we present new results on solvability of the equation $A^{*}(D)f=μ$ for $f \in L^{p}$ and positive measure data $μ$ associated to an elliptic homogeneous linear differential operator $A(D)$ of order m. Our method is based on $(m,p)-$energy control of $μ$ giving a natural characterization for solutions when $1\leq p < \infty$. We also obtain sufficient conditions in the limiting case $p=\infty$ using {new $L^{1}$ estimates on measures for elliptic and canceling operators.

math.AP↗