Regularity and pointwise convergence for dispersive equations on $\mathbb{H}^2$

Abstract

In the prototypical setting of non-Euclidean geometry, the 2-dimensional Real Hyperbolic space $\mathbb{H}^2$, we consider the Carleson's problem for the Schr\"odinger equation and improve the best known result until now by proving that the Sobolev regularity threshold $\beta \ge 1/2$ for the initial data, is sufficient to obtain pointwise convergence of the solution a.e. on $\mathbb{H}^2$. In fact, we prove the same bound for a wide class of dispersive equations that include the fractional Schr\"odinger equations with convex phase, the Boussinesq equation and the Beam equation, also known as the fourth order Wave equation. For the Schr\"odinger equation, we improve the result of Wang-Zhang (Canad J Math 71(4), 983-995, 2019) and for the fractional Schr\"odinger equations with convex phase, we improve the result of Cowling (Lecture Notes Math 992, 83-90, 1983).