A counter example on nontangential convergence for oscillatory integrals

Abstract

Consider the solution of the time-dependent Schr{ö}dinger equation with initial data $f$. It is shown in \cite{artikel} that there exists $f$ in the Sobolev space $H^s(\RR), s=n/2$ such that tangential convergence can not be widened to convergence regions. In this paper we show that the corresponding result holds when $-{\Delta }_x$ is replaced by an operator $\varphi (D)$, with special conditions on $\varphi$.