On dyadic nonlocal Schrödinger equations with Besov initial data

Abstract

In this paper we consider the pointwise convergence to the initial data for the Schrödinger–Dirac equation i∂u∂t=Dβu with u(x,0)=u0 in a dyadic Besov space. Here Dβ 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 Dβ and pointwise estimates of the corresponding maximal operator in terms of the dyadic Hardy–Littlewood function and the Calderón sharp maximal operator.