Counterexamples for generalizations of the non-elliptic Schrödinger maximal operator

Abstract

For $P=X_1^2+\cdots +X_n^2$, let $T_t^{P}f(x)$ denote the solution to the linear Schrödinger equation at time $t$. In 1980, Carleson asked for the minimal regularity of an initial data function $f\in H^s({\mathbb{R}}^n)$ that guarantees pointwise convergence of $T_t^{P}f(x)$ to $f(x)$ as $t\to 0$. This was resolved by Bourgain, who constructed counterexamples for the Schrödinger maximal operator to show that $s\ge n/(2(n+1))$ is necessary, and Du and Zhang, who proved that $s>n/(2(n+1))$ is sufficient. Rogers, Vargas, and Vega studied the analogous question for the non-elliptic Schrödinger maximal operator, where $P=X_1^2-X_2^2\pm X_3^2\pm \cdots \pm X_n^2$, and proved that, for all $n\ge 2$, $s\ge 1/2$ is necessary and $s>1/2$ is sufficient. In this paper, we construct counterexamples for generalizations of the non-elliptic case and prove a necessary condition of $s\ge 1/2$ for an infinite class of polynomial symbols $P$.

11 pages