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

TL;DR

This paper establishes that a Sobolev regularity of at least 1/2 ensures almost everywhere pointwise convergence of solutions to dispersive equations, including Schrödinger and fractional Schrödinger equations, on hyperbolic space.

Contribution

It improves existing results by proving a lower regularity threshold for pointwise convergence of dispersive equations on hyperbolic space, encompassing a broad class of equations.

Findings

Proves pointwise convergence for Sobolev regularity $eta \,\geq\, 1/2$ on $\\mathbb{H}^2$.

Extends results to fractional Schrödinger, Boussinesq, and Beam equations.

Improves previous bounds established by Wang-Zhang and Cowling.

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).