Functional calculus of operators with heat kernel bounds on non-doubling manifolds with ends

Let $\Delta$ be the Laplace--Beltrami operator acting on a non-doubling manifold with two ends $\mathbb R^m \sharp \mathcal R^n$ with $m > n \ge 3$. Let $\frak{h}_t(x,y)$ be the kernels of the semigroup $e^{-t\Delta}$ generated by $\Delta$. We say that a non-negative self-adjoint operator $L$ on $L^2(\mathbb R^m \sharp \mathcal R^n)$ has a heat kernel with upper bound of Gaussian type if the kernel $h_t(x,y)$ of the semigroup $e^{-tL}$ satisfies $h_t(x,y) \le C \frak{h}_{\alpha t}(x,y)$ for some constants $C$ and $\alpha$. This class of operators includes the Schr\"odinger operator $L = \Delta + V$ where $V$ is an arbitrary non-negative potential. We then obtain upper bounds of the Poisson semigroup kernel of $L$ together with its time derivatives and use them to show the weak type $(1,1)$ estimate for the holomorphic functional calculus $\frak{M}(\sqrt{L})$ where $\frak{M}(z)$ is a function of Laplace transform type. Our result covers the purely imaginary powers $L^{is}, s \in \mathbb R$, as a special case and serves as a model case for weak type $(1,1)$ estimates of singular integrals with non-smooth kernels on non-doubling spaces.