Strichartz estimates for the Schrödinger and wave equations with a Laguerre potential on the plane

In this paper, we obtain a set of Strichartz inequalities for solutions to the Schrödinger and wave equations with a Laguerre potential on the plane. To obtain the desired inequalities, we intend to prove the dispersive estimates for the involved Schrödinger and wave propagators and then a standard $T{T}^{*}$ argument will enable us to arrive at these inequalities. The proof of the dispersive estimate for the Schödinger propagator relies on a crucial uniform boundedness of a series involving the Bessel functions of the first kind, while the dispersive estimate for the wave equation follows from a sequence of standard steps, such as the Gaussian boundedness of the heat kernel, Bernstein-type inequalities, and Müller–Seeger's subordination formula. We have to verify these classical results in the present setting, which is possible since the spectral properties of the involved Schrödinger operator can be explicitly calculated.