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Let Δ be the Laplace-Beltrami operator acting on a nondoubling manifold with two ends Rm♯Rn with m > n ≥ 3. Let ht(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(Rm♯Rn) has a heat kernel with upper bound of Gaussian type if the kernel ht(x,y) of the semigroup e−tL satisfies ht(x,y) ≤ Chαt(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 Lis , 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.
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