Functional calculus of Laplace transform type on non-doubling parabolic manifolds with ends

Abstract

Let 𝑀 be a non-doubling parabolic manifold with ends and 𝐿 a non-negative self-adjoint operator on 𝐿²(𝑀) which satisfies a suitable heat kernel upper bound named the upper bound of Gaussian type. These operators include the Schrödinger operators 𝐿 = Δ + 𝑉 where Δ is the Laplace–Beltrami operator and 𝑉 is an arbitrary non-negative potential. This paper will investigate the behaviour of the Poisson semi-group kernels of 𝐿 together with its time derivatives and then apply them to obtain the weak type $(1, 1)$ estimate of the functional calculus of Laplace transform type of \sqrt{𝐿} which is defined by 𝔐(\sqrt{𝐿}) 𝑓(𝑥) := ∫₀^∞ [\sqrt{𝐿} 𝑒^{−𝑡 \sqrt{𝐿}} 𝑓(𝑥)] 𝑚(𝑡) 𝑑𝑡 where 𝑚(𝑡) is a bounded function on [0, ∞). In the setting of our study, both doubling condition of the measure on 𝑀 and the smoothness of the operators' kernels are missing. The purely imaginary power 𝐿^𝑖𝑠, 𝑠 ∈ ℝ, is a special case of our result and an example of weak type $(1, 1)$ estimates of a singular integral with non-smooth kernels on non-doubling spaces.