Spectral multipliers for Markov chains. (English) Zbl 1079.22006

Alexopoulos, Georgios K.

J. Math. Soc. Japan 56, No. 3, 833-852 (2004).

Let \((X,\mu)\) be a measure space and \(L\) be a positive self-adjoint operator on \(L^2(X,\mu)\) with spectral resolution \(L=\int_{0}^{+\infty}\lambda \,dE_\lambda.\) Let \(m\) be a bounded Borel measurable function on \(\mathbb R.\) The operator \(m(L)=\int_{0}^{+\infty}m(\lambda) \,dE_\lambda\) is then a bounded operator on \(L^2(X).\) It is interesting to look for conditions on \(m\) ensuring that \(m(L)\) extends to a bounded operator on \(L^p (G)\) for some \(1\leq p\leq \infty\). Such a result is a spectral multiplier theorem. In the classical case where \(X={\mathbb R}^n\) and \(L\) is the Laplacian, the Hörmander-Mikhlin theorem gives such a condition in terms of bounds on the derivatives of \(m.\)
In the paper under review, the author studies operators of the form \(L=I-P\), where \(P\) is a Markov kernel on a metric space \((X,d)\) endowed with a \(\sigma\)-finite measure. Assuming that \(d\) has the doubling volume property and that \(P\) satisfies a Gaussian type estimate, he obtains a spectral multiplier theorem for \(I-P.\) Moreover, he shows how to adapt his arguments in order to obtain a spectral multiplier theorem when \(L\) is a symmetric differential operator on a Riemannian manifold for which the kernel of the semigroup \(T_t= e^{-tL}\) satisfies a Gaussian type estimate.

Reviewer: Bachir Bekka (Rennes)

MSC:

22E30	Analysis on real and complex Lie groups
42B15	Multipliers for harmonic analysis in several variables
35P99	Spectral theory and eigenvalue problems for partial differential equations
22E25	Nilpotent and solvable Lie groups
43A80	Analysis on other specific Lie groups

Keywords:

spectral multiplier; Markov chain; heat kernel