Strichartz estimates and wave equation in a conic singular space

Consider the metric cone X= C(Y) = (0 , ∞) _r× Y with metric g= dr²+ r²h where the cross section Y is a compact (n- 1) -dimensional Riemannian manifold (Y, h). Let Δ _g be the positive Friedrichs extension Laplacian on X and let Δ _h be the positive Laplacian on Y, and consider the operator L_V= Δ _g+ Vr^{- 2} where V∈ C^∞(Y) such that Δ _h+ V+ (n- 2) ²/ 4 is a strictly positive operator on L²(Y). In this paper, we prove global-in-time Strichartz estimates without loss regularity for the wave equation associated with the operator L_V. It verifies a conjecture in Wang (Remark 2.4 in Ann Inst Fourier 56:1903–1945, 2006) for wave equation. The range of the admissible pair is sharp and the range is influenced by the smallest eigenvalue of Δ _h+ V+ (n- 2) ²/ 4. To prove the result, we show a Sobolev inequality and a boundedness of a generalized Riesz transform in this setting. In addition, as an application, we study the well-posed theory and scattering theory for energy-critical wave equation with small data on this setting of dimension n≥ 3.