Spectral multipliers from to Lq(X)

Let L be a non-negative self-adjoint operator acting on L2(X), where X is a space of homogeneous type. Assume that the heat kernels pt(x,y) corresponding to the semigroup e−tL satisfy Gaussian upper bounds but possess no regularity in variables x and y. In this article, we prove a spectral multiplier theorem for from to Lq(X) for some 1⩽q⩽2, if the function F possesses the Sobolev norm of order s with suitable bounds and where n is a measure of the dimension of the space. We also study the weighted Lp–Lq estimates for spectral multiplier theorem.