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July, 2021 Functional calculus of Laplace transform type on non-doubling parabolic manifolds with ends
Hong Chuong DOAN
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J. Math. Soc. Japan 73(3): 703-733 (July, 2021). DOI: 10.2969/jmsj/83348334

Abstract

Let $M$ be a non-doubling parabolic manifold with ends and $L$ a non-negative self-adjoint operator on $L^{2}(M)$ which satisfies a suitable heat kernel upper bound named the upper bound of Gaussian type. These operators include the Schrödinger operators $L = \Delta + V$ where $\Delta$ is the Laplace–Beltrami operator and $V$ is an arbitrary non-negative potential. This paper will investigate the behaviour of the Poisson semi-group kernels of $L$ together with its time derivatives and then apply them to obtain the weak type $(1, 1)$ estimate of the functional calculus of Laplace transform type of $\sqrt{L}$ which is defined by $\mathfrak{M}(\sqrt{L}) f(x) := \int_{0}^{\infty} \bigl[\sqrt{L} e^{-t \sqrt{L}} f(x)\bigr] m(t) dt$ where $m(t)$ is a bounded function on $[0, \infty)$. In the setting of our study, both doubling condition of the measure on $M$ and the smoothness of the operators' kernels are missing. The purely imaginary power $L^{is}$, $s \in \mathbb{R}$, is a special case of our result and an example of weak type $(1, 1)$ estimates of a singular integral with non-smooth kernels on non-doubling spaces.

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Hong Chuong DOAN. "Functional calculus of Laplace transform type on non-doubling parabolic manifolds with ends." J. Math. Soc. Japan 73 (3) 703 - 733, July, 2021. https://doi.org/10.2969/jmsj/83348334

Information

Received: 19 September 2019; Published: July, 2021
First available in Project Euclid: 26 May 2021

Digital Object Identifier: 10.2969/jmsj/83348334

Subjects:
Primary: 42B25

Rights: Copyright ©2021 Mathematical Society of Japan

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Vol.73 • No. 3 • July, 2021
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