Sharp convergence for sequences of nonelliptic Schrödinger means

Abstract

We consider pointwise convergence of nonelliptic Schrödinger means $e^{i{t}_n\square }f(x)$ for $f\in H^s({\mathbb{R}}^2)$ and decreasing sequences $\{t_n{\}}_{n=1}^{\infty }$ converging to zero, where \[{e^{it_{n}\square }}f\left( x \right): = \int_{{\mathbb{R}^2}} {{e^{i\left( {x \cdot ξ+ t_{n}{{ ξ_{1}ξ_{2} }}} \right)}}\widehat{f}} \left( ξ\right)dξ.\] We prove that when $0<s<\frac{1}{2}$, \[\mathop {\lim }\limits_{n \to \infty} {e^{it_{n}\square }}f\left( x \right) = f(x) \hspace{0.2cm} a.e.\hspace{0.2cm} x\in \mathbb{R}^2\] holds for all $f\in H^s\left({\mathbb{R}}^2\right)$ if and only if $\{t_n{\}}_{n=1}^{\infty }\in {\ell }^{r(s),\infty }(\mathbb{N})$, $r(s)=\frac{s}{1-s}$. Moreover, our result remains valid in general dimensions.