Counterexamples for generalizations of the non-elliptic Schr\"{o}dinger maximal operator

Abstract

For $P=X_1^2+\cdots + X_n^2$, let $T_t^Pf(x)$ denote the solution to the linear Schr\"{o}dinger 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 $T_t^Pf(x)$ to $f(x)$ as $t\rightarrow 0$. This was resolved by Bourgain, who constructed counterexamples for the Schr\"{o}dinger maximal operator to show that $s\geq n/(2(n+1))$ 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 Schr\"{o}dinger 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 $n\geq 2$, $s\geq 1/2$ 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 $s\geq 1/2$ for an infinite class of polynomial symbols $P$.