Advances in Nonlinear Analysis (Nov 2023)
Uniform complex time heat Kernel estimates without Gaussian bounds
Abstract
The aim of this article is twofold. First, we study the uniform complex time heat kernel estimates of e−z(−Δ)α2{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 zz satisfying 0<ω≤∣θ∣<π20\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)α2+VH={\left(-\Delta )}^{\tfrac{\alpha }{2}}+V, where VV belongs to higher-order Kato class.
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