On dyadic nonlocal Schrödinger equations with Besov initial data

HUGO AIMAR, BRUNO BONGIOANNI AND IVANA GÓMEZ

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2013 vol. 407 p. 23 - 34

0022-247X

In this paper we consider the pointwise convergence to the initial data for the Schrödinger-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 summability 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.