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### Abstract

Let Δ be the Laplace-Beltrami 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 non-negative self-adjoint 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 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.Original language | English |
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Pages (from-to) | 713-747 |

Number of pages | 35 |

Journal | Indiana University Mathematics Journal |

Volume | 69 |

Issue number | 3 |

Publication status | Published - 2020 |

## Fingerprint Dive into the research topics of 'Functional calculus of operators with heat kernel bounds on non-doubling manifolds with ends'. Together they form a unique fingerprint.

## 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

## Cite this

Bui, T. A., Duong, X. T., Li, J., & Wick, B. D. (2020). Functional calculus of operators with heat kernel bounds on non-doubling manifolds with ends.

*Indiana University Mathematics Journal*,*69*(3), 713-747.