Pointwise dispersive estimates for Schrödinger and wave equations on conical singular spaces

We study the pointwise decay estimates for the Schrödinger and wave equations on a product cone (X,g), where the metric g=dr²+r²h and X=C(Y)=(0,∞)×Y is a product cone over the closed Riemannian manifold (Y,h) with metric h . Under the assumption that the conjugate radius R_Conj of Y satisfies R_Conj>π, we prove the pointwise dispersive estimates for the Schrödinger and half-wave propagators in this setting. The key ingredient is the modified Hadamard parametrix on Y in which the role of the conjugate points does not come into play if R_Conj>π. A new finding is that a threshold of the conjugate radius of Y for the pointwise dispersive estimates in this setting is the magical number π .