标题： 色散方程在$\mathbb{H}^2$上的正则性与逐点收敛 显示英文标题

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

摘要： 在非欧几里得几何的典型设置中，二维实双曲空间$\mathbb{H}^2$，我们考虑薛定谔方程的Carleson问题，并通过证明初始数据的Sobolev正则性阈值$\beta \ge 1/2$足以得到解在$\mathbb{H}^2$上几乎处处的点收敛，从而改进了目前最好的结果。事实上，我们证明了对于包括带有凸相位的分数薛定谔方程、Boussinesq方程和Beam方程（也称为四阶波动方程）在内的广泛类别的色散方程，具有相同的界限。对于薛定谔方程，我们改进了Wang-Zhang（Canad J Math 71(4), 983-995, 2019）的结果，而对于带有凸相位的分数薛定谔方程，我们改进了Cowling（Lecture Notes Math 992, 83-90, 1983）的结果。 显示英文摘要

摘要： 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).