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We consider the solutions of the Klein–Gordon equation in the2+12{+}1-dimensional space-time which gravity is analyzed, i.e., the manifoldℝt×ℝ+×ℝ/2⁢π⁢α⁢ℤ{\mathbb{R}_{t}\times\mathbb{R}_{+}\times\mathbb{R}/2\pi\alpha\mathbb{Z}}created by a massive point particle. Using the Schwartz kernel of resolvent and spectral measure for Schrödinger operator on the spinless cone, we prove the dispersive estimates and Strichartz estimates for the Klein–Gordon equation. In a future paper, we will consider the problem on the spinning cone.

Decay and Strichartz estimates for Klein–Gordon equation on a cone I: Spinless case