Decay estimates for one Aharonov-Bohm solenoid in a uniform magnetic field II: Wave equation

This is the second paper of our project exploring the decay estimates for dispersive equations with Aharonov-Bohm solenoids in a uniform magnetic field. In the first paper [36], we have studied the dispersive and Strichartz estimates for the Schrödinger equation with one Aharonov-Bohm solenoid in a uniform magnetic field. The decay estimate for the wave equation in the same setting turns out to be more delicate since the square root of the eigenvalue of the associated Schrödinger operator will prevent the direct construction of the half-wave propagator. To get around this obstacle, we turn to verify the Gaussian boundedness of the related heat kernel via two different approaches. The first one is based on the Davies-Gaffney inequality in this setting and the second one is to obtain an explicit representation of the heat kernel (which contains the full information of both the Aharonov-Bohm solenoid and the uniform magnetic field) with the aid of the Schulman-Sunada formula. As a byproduct, we also establish the Bernstein inequalities and the square function estimates for the involved Schrödinger operator with one Aharonov-Bohm solenoid in a uniform magnetic field.