Georgia Institute of Technology College of Sciences

In this talk,we study weighted L^p-norm inequalities for general spectralmultipliersfor self-adjoint positive definite operators on L^2(X), where X is a space of homogeneous type. We show that the sharp weighted Hormander-type spectral multiplier theorems follow from the appropriate estimatesof the L^2 norm of the kernel of spectral multipliers and the Gaussian boundsfor the corresponding heat kernels. These results are applicable to spectral multipliersfor group invariant Laplace operators acting on Lie groups of polynomialgrowth and elliptic operators on compact manifolds. This is joint work with Adam Sikora and Lixin Yan.