arXiv subjects

Raúl Emilio Vidal

Publications and source records attributed to Raúl Emilio Vidal.

3 recordsLinked to original sources

arXiv2021

Nonlocal diffusion equations in Carnot Groups

Let $G$ be a Carnot group. We study nonlocal diffusion equations in a domain $Ω$ of $G$ of the form $$ u_t^ε(x,t)=\int_{G}\frac{1}{ε^2}K_ε(x,y)(u^ε(y,t)-u^ε(x,t))\,dy, \qquad x\in Ω$$ with $u^ε=g(x,t)$ for $x\notinΩ$. For appropriate rescaled kernel $K_ε$ we prove that solutions $u^ε$, when $ε\rightarrow0$, uniformly approximate the solution of different local Dirichlet problem in $G$. The key tool used is the Taylor series development for a function defined on a Carnot group.

math.AP↗

arXiv2017

Nonlocal heat equations in the Heisenberg group

We study the following nonlocal diffusion equation in the Heisenberg group $\mathbb{H}_n$, \[ u_t(z,s,t)=J\ast u(z,s,t)-u(z,s,t), \] where $\ast$ denote convolution product and $J$ satisfies appropriated hypothesis. For the Cauchy problem we obtain that the asymptotic behavior of the solutions is the same form that the one for the heat equation in the Heisenberg group. To obtain this result we use the spherical transform related to the pair $(U(n),\mathbb{H}_n)$. Finally we prove that solutions of properly rescaled nonlocal Dirichlet problem converge uniformly to the solution of the corresponding Dirichlet problem for the classical heat equation in the Heisenberg group.

math.AP↗

arXiv2013

Weighted inequalities related to a Muckenhoupt and Wheeden problem for one-side singular integrals

In this paper we obtain for $T^+$, a one-sided singular integral given by a Calderón-Zygmund kernel with support in $(-\infty,0)$, a $L^p(w)$ bound when $w\in A_1^+$. A. K. Lerner, S. Ombrosi, and C. Pérez proved in [ "$A_{1}$ Bounds for Calderón-Zygmund operators related to a problem of Muckenhoupt and Wheeden", Math. Res. Lett. \textbf{16} (2009), no. 1, 149-156] that this bound is sharp with respect to $||w||_{A_1} $ and $p$ . We also give a $L^{1,\infty}(w)$ estimate, for a related problem of Muckenhoupt and Wheeden for $w\in A_1^+$ . We improve the classical results, for one-sided singular integrals, by putting in the inequalities a wider class of weights.

math.AP↗