In this talk,
we study weighted $L^p$-norm inequalities for general spectral multipliers
for self-adjoint positive definite operators on $L^2(X)$, where $X$ is
a space of homogeneous type. We show that the sharp weighted H\"ormander-type
spectral multiplier theorems follow from the appropriate estimates of the $L^2$
norm of the kernel of spectral multipliers and the Gaussian bounds for the corresponding
heat kernels. These results are applicable to spectral multipliers for group
invariant Laplace operators acting on Lie groups of polynomial growth and elliptic
operators on compact manifolds.