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Let $L$ be a non-negative self adjoint operator acting on $L^2(X)$ where $X$is a space of homogeneous type. Assume that $L$ generates a holomorphicsemigroup $e^{-tL}$ whose kernels $p_t(x,y)$ have Gaussian upper bounds butpossess no regularity in variables $x$ and $y$. In this article, we studyweighted $L^p$-norm inequalities for spectral multipliers of $L$. We show sharpweighted H\"ormander-type spectral multiplier theorems follow from Gaussianheat kernel bounds and appropriate $L^2$ estimates of the kernels of thespectral multipliers. These results are applicable to spectral multipliers forlarge classes of operators including Laplace operators acting on Lie groups ofpolynomial growth or irregular non-doubling domains of Euclidean spaces,elliptic operators on compact manifolds and Schr\"odinger operators withnon-negative potentials on complete Riemannian manifolds.

Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers