On dyadic nonlocal Schr\"{o}dinger equations with Besov initial data

TL;DR

This paper investigates the pointwise convergence of solutions to a dyadic nonlocal Schrödinger equation with initial data in Besov spaces, using kernel sumability and maximal operator estimates.

Contribution

It introduces new pointwise convergence results for dyadic nonlocal Schrödinger equations with Besov initial data, employing novel kernel and maximal operator techniques.

Findings

Established pointwise convergence to initial data in dyadic Besov spaces.

Developed sumability formulas for the fractional derivative kernel.

Derived estimates for the maximal operator using dyadic Hardy-Littlewood and Calderón operators.

Abstract

In this paper we consider the pointwise convergence to the initial data for the Schr\"{o}dinger-Dirac equation $i\tfrac{\partial u}{\partial t}=D^{\beta}u$ with $u(x,0)=u^0$ in a dyadic Besov space. Here $D^{\beta}$ denotes the fractional derivative of order $\beta$ associated to the dyadic distance $\delta$ on $\mathbb{R}^+$. The main tools are a sumability formula for the kernel of $D^{\beta}$ and pointwise estimates of the corresponding maximal operator in terms of the dyadic Hardy-Littlewood function and the Calder\'on sharp maximal operator.