Let $X$ be a metric space with a doubling measure satisfying $\mu(B)\gtrsim r_{B}^n$ for any ball $B$ with radius $r_{B}>0$.  Let $L$ be a non negative self-adjoint operator on $L^{2}(X)$. We assume that the semigroup $e^{-tL}$ satisfies a Gaussian upper bound and that the flow $e^{itL}$ satisfies a typical $L^{1}-L^{\infty}$ dispersive estimate of the form
   

\begin{equation*}
      \|e^{itL}\|_{L^{1}\to L^{\infty}} \lesssim |t|^{-\frac n2}.
\end{equation*}
 

Then we prove a similar $L^{1}-L^{\infty}$ dispersive estimate for a general class of flows $e^{it \phi (L)}$, with $\phi (r)$ of power type near 0 and near $\infty$. In the case of fractional powers $\phi(L)=L^{\nu}$, $\nu\in (0,1)$, we deduce dispersive estimates for $e^{itL^{\nu}}$ with data in Sobolev, Besov or Hardy spaces $H^{p}_{L}$ with $p\in(0,1]$, associated to the operator $L$.

This is a joint work with The Anh Bui, Piero D'Ancona and Detlef M\"uller.