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 $i\frac{\partial u}{\partial t}=D^{\beta }u$ 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 Calderón sharp maximal operator.

17 pages, 4 figures. We have changed the setting of the interval by the set of positive real numbers. Reference [8] is new. We have also corrected some typos. Second version has been accepted for publication on JMAA

References in corpus (1)

• Collapse in the nonlocal nonlinear Schrödinger equation

Cited by in corpus (2)

• Nonlocal Schrödinger equations in metric measure spaces
• Fractional uncertainty