I am interested in harmonic analysis and its interactions with partial differential equations, such as the Helmholtz equation and wave equation on compact Riemannian manifolds. Specifically, I am interested in L^p estimates on restrictions of the eigenfunctions of the Laplace-Beltrami operators on compact Riemannian manifolds.

The main idea for my work so far is to convert eigenfunction estimate problems into oscillatory integral operator problems by using the Lax parametrix and/or Hadamard parametrix on compact Riemannian manifolds (or, on their universal covers in the presence of nonpositive sectional curvatures, making use of the Cartan-Hadamard theorem). Once we have oscillatory integral operators in our hands, we can think about harmonic analytic tools, such as, Young's inequality, Hormander's theorem, and/or oscillatory integral estimates related with fold singularities of the canonical relations, and so on.

The list of preprints is as follows:

1. Eigenfunctions restriction estimates for curves with nonvanishing geodesic curvatures in compact Riemannian surfaces with nonpositive curvature Transactions of the AMS, 376 (08), 2023, pp.5809-5855.

2. L^q estimates on the restriction of Schrodinger eigenfunctions with singular potentials with Matthew D. Blair

3. L^p - L^q resolvent restriction estimates for submanifolds with Matthew D. Blair