Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers

Let L be a non-negative self-adjoint operator acting on L²(X) where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e−tL whose kernels pt(x,y) have Gaussian upper bounds but there is no assumption on the regularity in variables x and y. In this article, we study weighted Lp-norm inequalities for spectral multipliers of L. We show that sharp weighted Hörmander-type spectral multiplier theorems follow from Gaussian heat kernel bounds and appropriate L² estimates of the kernels of the spectral multipliers. These results are applicable to spectral multipliers for large classes of operators including Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces, elliptic operators on compact manifolds and Schrödinger operators with non-negative potentials.