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

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.