Uniform complex time heat Kernel estimates without Gaussian bounds

Abstract

Abstract The aim of this article is twofold. First, we study the uniform complex time heat kernel estimates of e − z ( − Δ ) α 2 {e}^{-z{\left(-\Delta )}^{\frac{\alpha }{2}}} for α > 0 , z ∈ C + \alpha \gt 0,z\in {{\mathbb{C}}}^{+} . To this end, we establish the asymptotic estimates for P ( z , x ) P\left(z,x) with z z satisfying 0 < ω ≤ ∣ θ ∣ < π 2 0\lt \omega \le | \theta | \lt \frac{\pi }{2} followed by the uniform complex time heat kernel estimates. Second, we studied the uniform complex time estimates of the analytic semigroup generated by H = ( − Δ ) α 2 + V H={\left(-\Delta )}^{\tfrac{\alpha }{2}}+V , where V V belongs to higher-order Kato class.