Ivana Gómez

On dyadic nonlocal Schrödinger equations with Besov initial data

Abstract In this paper we consider the pointwise convergence to the initial data for the Schrodinger–Dirac equation i ∂ u ∂ t = D β u with u ( x , 0 ) = u 0 in a dyadic Besov space. Here D β denotes the fractional derivative of order β associated to the dyadic distance δ on R + . The main tools are a summability formula for the kernel of D β and pointwise estimates of the corresponding maximal operator in terms of the dyadic Hardy–Littlewood function and the Calderon sharp maximal operator.

Heavy Tailed Approximate Identities and σ -stable Markov Kernels

The aim of this paper is to present some results relating the properties of stability, concentration and approximation to the identity of convolution through not necessarily mollification type families of heavy tailed Markov kernels. A particular case is provided by the kernels Kt obtained as the t mollification of Lσ(t) selected from the family ℒ={Lσ:Lσ̂(ξ)=e−|ξ|σ,0<σ<2}

The Dyadic Fractional Diffusion Kernel as a Central Limit

\mathcal {L}=\{L^{\sigma }: \widehat {L^{\sigma }}{(\xi )=e^{-|{\xi }|}}^{\sigma }, 0<\sigma <2\}

Affinity and Distance. On the Newtonian Structure of Some Data Kernels

, by a given function σ with values in the interval (0,2). We show that a basic Harnack type inequality, introduced by C. Calderón in the convolution case, becomes at once natural to the setting and useful to connect the concepts of stability, concentration and approximation of the identity. Some of the general results are extended to spaces of homogeneous type since most of the concepts involved in the theory are given in terms of metric and measure.

Nonlocal Schrödinger equations in metric measure spaces

We obtain the fundamental solution kernel of dyadic diffusions in ℝ+ as a central limit of dyadic mollification of iterations of stable Markov kernels. The main tool is provided by the substitution of classical Fourier analysis by Haar wavelet analysis.

Parabolic Besov Regularity for the Heat Equation

Abstract Let X be a set. Let K(x, y) > 0 be a measure of the affinity between the data points x and y. We prove that K has the structure of a Newtonian potential K(x, y) = φ(d(x, y)) with φ decreasing and d a quasi-metric on X under two mild conditions on K. The first is that the affinity of each x to itself is infinite and that for x ≠ y the affinity is positive and finite. The second is a quantitative transitivity; if the affinity between x and y is larger than λ > 0 and the affinity of y and z is also larger than λ, then the affinity between x and z is larger than ν(λ). The function ν is concave, increasing, continuous from R+ onto R+ with ν(λ) < λ for every λ > 0

Parabolic mean values and maximal estimates for gradients of temperatures

Abstract In this note we consider the pointwise convergence to the initial data for the solutions of some nonlocal dyadic Schrodinger equations on spaces of homogeneous type. We prove the a.e. convergence when the initial data belongs to a dyadic version of an L 2 based Besov space.