Spectral multipliers from to

Let L be a non-negative self-adjoint operator acting on L 2 ( X ) , where X is a space of homogeneous type. Assume that the heat kernels p t ( x , y ) corresponding to the semigroup e − t L satisfy Gaussian upper bounds but possess no regularity in variables x and y. In this article, we prove a spectral multiplier theorem for F ( L ) from H L 1 ( X ) to L q ( X ) for some 1 ⩽ q ⩽ 2 , if the function F possesses the Sobolev norm of order s with suitable bounds and s > n ( 1 q − 1 2 ) where n is a measure of the dimension of the space. We also study the weighted L p – L q estimates for spectral multiplier theorem.