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Fernando Soria

Publications and source records attributed to Fernando Soria.

13 recordsLinked to original sources
arXiv2026

On the Pointwise Convergence of Solutions to the Schr\"odinger Equation Along Certain Highly Tangential Curves

We investigate the Sobolev regularity required for almost everywhere convergence to the initial datum of solutions to the linear Schr\"odinger equation along certain tangential curves. In the regime $\alpha<\tfrac12$, we analyze maximal estimates for expressions of the form $e^{it\Delta}f(x+\gamma(t))$ over specific $\alpha$-H\"older curves $\gamma$ and initial data $f\in H^s(\mathbb{R}^n)$. For the model family $\gamma(t)=(t^{\alpha_1},\ldots,t^{\alpha_n})$, where $\alpha=\min_j \alpha_j$, we show that the critical regularity is $s=\max\left\{\frac{1-2\alpha}{2},\frac{n}{2(n+1)}\right\}.$

math.CA↗
arXiv2025

On the nonlocal heat equation for certain L\'evy operators and the uniqueness of positive solutions

We develop a Widder-type theory for nonlocal heat equations involving quite general L\'evy operators. Thus, we consider nonnegative solutions and look for conditions on the operator that ensure: (i) uniqueness of nonnegative classical and very weak solutions with a given initial trace; (ii) the existence of an initial trace, belonging to certain admissibility class; and (iii) the existence of a solution, given by a representation formula, for any admissible initial trace. Such results are obtained first for purely nonlocal L\'evy operators defined through positive symmetric L\'evy kernels comparable to radial functions with mixed polynomial growth, and then extended to more general operators, including anisotropic ones and operators that have both a local and a nonlocal part.

math.AP↗
arXiv2020

Fractional KPZ equations with critical growth in the gradient respect to Hardy potential

In this work we study the existence of positive solution to the fractional quasilinear problem, $$ \left\{ \begin{array}{rcll} (-Δ)^s u &=&λ\dfrac{u}{|x|^{2s}}+ |\nabla u|^{p}+ μf &\inn Ω,\\ u&>&0 & \innΩ,\\ u&=&0 & \inn(\mathbb{R}^N\setminusΩ), \end{array}\right. $$ where $Ω$ is a $C^{1,1}$ bounded domain in $\mathbb{R}^N$, $N> 2s, μ>0$, $\frac{1}{2} 0$, there exists a critical exponent $p_{+}(λ, s)$ such that for $p> p_{+}(λ,s)$ there is no positive solution. Moreover, $p_{+}(λ,s)$ is optimal in the sense that, if $p<p_{+}(λ,s)$ there exists a positive solution for suitable data and $μ$ sufficiently small.

math.AP↗
arXiv2019

On the KPZ equation with fractional diffusion: global regularity and existence results

In this work we analyze the existence of solutions to the fractional quasilinear problem, $$ (P) \left\{ \begin{array}{rcll} u_t+(-\Delta )^s u &=&|\nabla u|^{\alpha}+ f &\inn \Omega_T\equiv\Omega\times (0,T),\\ u(x,t)&=&0 & \inn(\mathbb{R}^N\setminus\Omega)\times [0,T),\\ u(x,0)&=&u_{0}(x) & \inn\Omega,\\ \end{array}\right. $$ where $\Omega$ is a $C^{1,1}$ bounded domain in $\mathbb{R}^N$, $N> 2s$ and $\frac{1}{2} \dfrac{1}{1-s}$. This behavior clearly exhibits a deep difference with the local case.

math.AP↗
arXiv2017

Principal Eigenvalue of Mixed Problem for the Fractional Laplacian: Moving the Boundary Conditions

We analyze the behavior of the eigenvalues of the following non local mixed problem $\left\{ \begin{array}{rcll} (-Δ)^{s} u &=& λ_1(D) \ u &\innΩ,\\ u&=&0&\inn D,\\ \mathcal{N}_{s}u&=&0&\inn N. \end{array}\right $ Our goal is to construct different sequences of problems by modifying the configuration of the sets $D$ and $N$, and to provide sufficient and necessary conditions on the size and the location of these sets in order to obtain sequences of eigenvalues that in the limit recover the eigenvalues of the Dirichlet or Neumann problem. We will see that the non locality plays a crucial role here, since the sets $D$ and $N$ can have infinite measure, a phenomenon that does not appear in the local case (see for example \cite{D,D2,CP}).

math.AP↗
arXiv2013

Muckenhoupt-Wheeden conjectures in higher dimensions

In recent work by Reguera and Thiele and by Reguera and Scurry, two conjectures about joint weighted estimates for Calderón-Zygmund operators and the Hardy-Littlewood maximal function have been refuted in the one-dimensional case. One of the key ingredients for these results is the construction of weights for which the action of the Hilbert transform is substantially bigger than that of the maximal function. In this work, we show that a similar construction is possible for classical Calderón-Zygmund operators in higher dimensions. This allows us to fully disprove the conjectures.

math.CA↗
arXiv2013

On the dimension dependence of some weighted inequalities

In the context of radial weights we study the dimension dependence of some weighted inequalities for maximal operators. We study the growth of the $A_1$-constants for radial weights and show the equivalence between the uniform boundedness of these constants, a dimension-free weak $L^1$ estimate for the maximal operator on annuli and the condition on the weight to be decreasing and essentially constant over dyadic annuli. Each one of these conditions is shown to provide dimension-free weighted weak type $L^1$ estimates for the centred maximal Hardy-Littlewood operator acting on radial functions. Finally we show that the universal maximal operator is of restricted weak type on weighted $L^n(\real^n)$ with constants uniformly bounded in dimension whenever we consider an $A_1$ weight.

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arXiv2013

Estimates for multiparameter maximal operators of Schrödinger type

Multiparameter maximal estimates are considered for operators of Schrödinger type. Sharp and almost sharp results, that extend work by Rogers and Villarroya, are obtained. We provide new estimates via the integrability of the kernel which naturally appears with a TT* argument and discuss the behavior at the endpoints. We treat in particular the case of global integrability of the maximal operator on finite time for solutions to the linear Schrödinger equation and make some comments on an open problem.

math.AP↗
arXiv2013

Localization and dimension free estimates for maximal functions

In the recent paper [J. Funct. Anal. {\bf 259} (2010)], Naor and Tao introduce a new class of measures with a so-called micro-doubling property and present, via martingale theory and probability methods, a localization theorem for the associated maximal functions. As a consequence they obtain a weak type estimate in a general abstract setting for these maximal functions that is reminiscent of the `$n\log n$ result' of Stein and Strömberg in Euclidean spaces. The purpose of this work is twofold. First we introduce a new localization principle that localizes not only in the time-dilation parameter but also in space. The proof uses standard covering lemmas and selection processes. Second, we show that a uniform condition for micro-doubling in the Euclidean spaces provides indeed dimension free estimates for their maximal functions in all $L^p$ with $p>1$. This is done introducing a new technique that allows to differentiate through dimensions.

math.CA↗
arXiv2013

A Widder's type Theorem for the heat equation with nonlocal diffusion

The main goal of this work is to prove that every non-negative {\it strong solution} $u(x,t)$ to the problem $$ u_t+(-Δ)^{α/2}u=0 \ \quad\mbox{for } (x,t)\in\mathbb{R}^{n}\times(0,T), \quad 0<α<2, $$ can be written as $$u(x,t)=\int_{\mathbb{R}^{n}}{P_{t}(x-y)u(y,0)\, dy},$$ where $$P_{t}(x)=\frac{1}{t^{n/α}}P\left(\frac{x}{t^{1/α}}\right), $$ and $$ P(x):=\int_{\mathbb{R}^{n}}{e^{ix\cdotξ-|ξ|^α}dξ}. $$ This result shows uniqueness in the setting of non-negative solutions and extends some classical results for the heat equation by D. V. Widder in \cite{W0} to the nonlocal diffusion framework.

math.AP↗
arXiv2012

Regularity of solutions of the fractional porous medium flow

We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. More precisely, $$ u_t=\nabla\cdot(u\nabla (-Δ)^{-s}u), \quad \ 0<s<1. $$ The problem is posed in $\{x\in\ren, t\in \re\}$ with nonnegative initial data $u(x,0)$ that are integrable and decay at infinity. A previous paper has established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation. Here we establish the boundedness and $C^α$ regularity of such weak solutions

math.AP↗
arXiv2003

On the Boundedness in $H^{1/4}$ of the Maximal Square Function Associated with the Schroedinger Equation

A long standing conjecture for the linear Schroedinger equation states that 1/4 of derivative in $L^2$, in the sense of Sobolev spaces, suffices in any dimension for the solution to that equation to converge almost everywhere to the initial datum as the time goes to 0. This is only known to be true in dimension 1 by work of Carleson. In this paper we show that the conjecture is true on spherical averages. To be more precise, we prove the $L^2$ boundedness of the associated maximal square function on the Sobolev class $H^{1/4}(R^n)$ in any dimension n.

math.CA↗