Sharp weighted $L^p$ estimates for spectral multipliers

Let $L$ be a non-negative self-adjoint operator on $L^2(X)$. Assume that operator $L$ generates the analytic semigroup $\{e^{-tL}{\}}_{t>0}$ whose kernels satisfy the standard Gaussian upper bounds. This paper investigates the sharp weighted inequalities for spectral multipliers $F(L)$ and their commutators with BMO functions. The obtained results in this paper can be considered to be improvements of those in \cite{DSY}[Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers, {\it J. Funct. Anal.}, {\bf 260} (2011), 1106-1131].