Decay and Strichartz estimates in critical electromagnetic fields

We study the L¹→L^∞-decay estimates for the Klein-Gordon equation in the Aharonov-Bohm magnetic fields, and further prove Strichartz estimates for the Klein-Gordon equation with critical electromagnetic potentials. The novel ingredients are the Schwartz kernels of the spectral measure and heat propagator of the Schrödinger operator in Aharonov-Bohm magnetic fields. In particular, we explicitly construct the representation of the spectral measure and resolvent of the Schrödinger operator with Aharonov-Bohm potentials, and prove that the heat kernel in critical electromagnetic fields satisfies Gaussian boundedness. In future papers, this result on the spectral measure will be used to (i) study the uniform resolvent estimates, and (ii) prove the L^p-regularity property of wave propagation in the same setting.