Browsing Analysis of Partial Differential Equations (APDE) by Issue Date
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A characterization of two weight norm inequality for LittlewoodPaley $g_{\lambda}^{*}$function
(2017)Let $n\ge 2$ and $g_{\lambda}^{*}$ be the wellknown high dimensional LittlewoodPaley function which was defined and studied by E. M. Stein, $$g_{\lambda}^{*}(f)(x)=\bigg(\iint_{\mathbb R^{n+1}_{+}} \Big(\frac{t}{t+xy ... 
Spectral asymptotics for $\delta$interactions on sharp cones
(2017)We investigate the spectrum of threedimensional Schr\"odinger operators with $\delta$interactions of constant strength supported on circular cones. As shown in earlier works, such operators have infinitely many eigenvalues ... 
Lorentz estimates for the gradient of weak solutions to elliptic obstacle problems with partially BMO coefficients
(2017)We prove global Lorentz estimates for variable power of the gradient of weak solution to linear elliptic obstacle problems with small partially BMO coefficients over a bounded nonsmooth domain. Here, we assume that the ... 
The Calderón problem with corrupted data
(201701)We consider the inverse Calderón problem consisting of determining the conductivity inside a medium by electrical measurements on its surface. Ideally, these measurements determine the DirichlettoNeumann map and, therefore, ... 
Partial regularity and smooth topologypreserving approximations of rough domains
(20170101)For a bounded domain $\Omega\subset\mathbb{R}^m, m\geq 2,$ of class $C^0$, the properties are studied of fields of `good directions', that is the directions with respect to which $\partial\Omega$ can be locally represented ... 
Singularity formation for the 1D cubic NLS and the Schrödinger map on $\mathbb{S}^2$
(20170202)In this note we consider the 1D cubic Schrödinger equation with data given as small perturbations of a Dirac$\delta$ function and some other related equations. We first recall that although the problem for this type of ... 
Quantitative weighted estimates for rough homogeneous singular integrals
(20170311)We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space $L^2(w)$, we obtain a bound ... 
Uniqueness properties for discrete equations and Carleman estimates
(20170325)Using Carleman estimates, we give a lower bound for solutions to the discrete Schrödinger equation in both dynamic and stationary settings that allows us to prove uniqueness results, under some assumptions on the decay of ... 
Sparse domination theorem for multilinear singular integral operators with $L^{r}$Hörmander condition
(20170401)In this note, we show that if $T$ is a multilinear singular integral operator associated with a kernel satisfies the socalled multilinear $L^{r}$Hörmander condition, then $T$ can be dominated by multilinear sparse operators. 
Discretizations of the spectral fractional Laplacian on general domains with Dirichlet, Neumann, and Robin boundary conditions
(20170428)In this work, we propose novel discretisations of the spectral fractional Laplacian on bounded domains based on the integral formulation of the operator via the heatsemigroup formalism. Specifically, we combine suitable ... 
A quantitative approach to weighted Carleson condition
(20170505)Quantitative versions of weighted estimates obtained by F. Ruiz and J.L. Torrea for the operator \[ \mathcal{M}f(x,t)=\sup_{x\in Q,\,l(Q)\geq t}\frac{1}{Q}\int_{Q}f(x)dx \qquad x\in\mathbb{R}^{n}, \, t \geq0 \] are ... 
Gaussian Decay of Harmonic Oscillators and related models
(20170515)We prove that the decay of the eigenfunctions of harmonic oscillators, uniform electric or magnetic fields is not stable under 0order complex perturbations, even if bounded, of these Hamiltonians, in the sense that we can ... 
Some remarks on the $L^p$ regularity of second derivatives of solutions to nondivergence elliptic equations and the Dini condition
(20170530)In this note we prove an endpoint regularity result on the $L^P$ integrability of the second derivatives of solutions to nondivergence form uniformly elliptic equations whose second derivatives are a priori only known ... 
HartreeFock theory with a selfgenerated magnetic field
(20170601)We prove the existence of a ground state within the HartreeFock theory for atoms and molecules, in the presence of selfgenerated magnetic fields, with and without direct spin coupling. The ground state exists provided ... 
Lorentz estimates for asymptotically regular fully nonlinear parabolic equations
(20170620)We prove a global Lorentz estimate of the Hessian of strong solutions to the CauchyDirichlet problem for a class of fully nonlinear parabolic equations with asymptotically regular nonlinearity over a bounded $C^{1,1}$ ... 
Sharp bounds for the ratio of modified Bessel functions
(20170621)Let $I_{\nu }\left( x\right) $ be the modified Bessel functions of the first kind of order $\nu $, and $S_{p,\nu }\left( x\right) =W_{\nu }\left( x\right) ^{2}2pW_{\nu }\left( x\right) x^{2}$ with $W_{\nu }\left( x\right) ... 
Highly rotating fluids with vertical stratification for periodic data and vanishing vertical viscosity
(201707)We prove that the threedimensional, periodic primitive equations with zero vertical diffusivity are globally well posed if the Rossby and Froude number are sufficiently small. The initial data is considered to be of zero ... 
Sharp weighted estimates involving one supremum
(201707)In this note, we study the sharp weighted estimate involving one supremum. In particular, we give a positive answer to an open question raised by Lerner and Moen. We also extend the result to rough homogeneous singular ... 
Weak and strong $A_p$$A_\infty$ estimates for square functions and related operators
(201707)We prove sharp weak and strong type weighted estimates for a class of dyadic operators that includes majorants of both standard singular integrals and square functions. Our main new result is the optimal bound $[w]_{A_p} ... 
El efecto de Talbot: de la óptica a la ecuación de Schrödinger
(201707)El objetivo de este artículo es dar a conocer un bello efecto óptico que se denomina efecto de Talbot. Primero, describiremos el fenómeno y comentaremos su descubrimiento a mediados del siglo XIX. A continuación, analizaremos ...