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Abstract
Let Δ be the LaplaceBeltrami operator acting on a nondoubling manifold with two ends R^{m}♯R^{n} with m > n ≥ 3. Let h_{t}(x,y) be the kernels of the semigroup e^{−tΔ} generated by Δ. We say that a nonnegative selfadjoint operator L on L^{2}(R^{m}♯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) ≤ 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 nonnegative potential. We then obtain upper bounds of the Poisson semigroup kernel of L together with its time derivatives and use them to show the weaktype (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 weaktype (1, 1) estimates of singular integrals with nonsmooth kernels on nondoubling spaces.
Original language  English 

Pages (fromto)  713747 
Number of pages  35 
Journal  Indiana University Mathematics Journal 
Volume  69 
Issue number  3 
DOIs  
Publication status  Published  2020 
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Projects
 1 Finished

Multiparameter Harmonic Analysis: Weighted Estimates for Singular Integrals
Duong, X., Ward, L., Li, J., Lacey, M., Pipher, J. & MQRES, M.
16/02/16 → 30/06/20
Project: Research