Functional calculus of Laplace transform type on non-doubling parabolic manifolds with ends
- 27 July 2021
- journal article
- research article
- Published by Mathematical Society of Japan (Project Euclid) in Journal of the Mathematical Society of Japan
- Vol. 73 (3), 1-31
- https://doi.org/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.
Keywords
This publication has 7 references indexed in Scilit:
- BOUNDEDNESS OF MAXIMAL FUNCTIONS ON NONDOUBLING PARABOLIC MANIFOLDS WITH ENDSJournal of the Australian Mathematical Society, 2020
- Functional calculus of operators with heat kernel bounds on non-doubling manifolds with endsIndiana University Mathematics Journal, 2020
- Riesz transforms on a class of non-doubling manifoldsCommunications in Partial Differential Equations, 2019
- Spectral multipliers via resolvent type estimates on non-homogeneous metric measure spacesMathematische Zeitschrift, 2019
- Heat kernel estimates on connected sums of parabolic manifoldsJournal de Mathématiques Pures et Appliquées, 2018
- Hardy spaces associated with non-negative self-adjoint operatorsStudia Mathematica, 2017
- Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weightsAdvances in Mathematics, 2007