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

Let Δ be the Laplace-Beltrami operator acting on a nondoubling manifold with two ends R^m♯Rⁿ with m > n ≥ 3. Let h_t(x,y) be the kernels of the semigroup e^−tΔ generated by Δ. We say that a non-negative self-adjoint operator L on L²(R^m♯Rⁿ) 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) ≤ Ch_αt(x,y) for some constants C and α. This class of operators includes the Schrödinger operator L = Δ + 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 M(√L) where M(z) is a function of Laplace transform type. Our result covers the purely imaginary powers L^is , s ∈ 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.