Spectral multipliers for Markov chains
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.
scientific article; zbMATH DE number 2111277