On sharp estimates for Schrödinger groups of fractional powers of nonnegative self-adjoint operators

Let $L$ be a non negative, selfadjoint operator on $L^{2}(X)$, where $X$ is a metric space endowed with a doubling measure. Consider the Schr\"{o}dinger group for fractional powers of $L$. If the heat flow $e^{-tL}$ satisfies suitable conditions of Davies--Gaffney type, we obtain the following estimate in Hardy spaces associated to $L$: \begin{equation*} \big\|(I+L)^{-\beta /2}e^{i\tau L^{\gamma /2}}f\big\|_{H^p_L(X)} \leq C (1 + |\tau|)^{n\mathfrak s_p}\|f\|_{H^p_L(X)} \end{equation*} where $p\in(0,1]$, $\gamma\in(0,1]$, $\beta /\gamma = n|\frac 12-\frac 1p|= n\mathfrak s_p$ and $\tau\in \mathbb{R}$. If in addition $e^{-itL}$ satisfies a localized $L^{p_{0}}\to L^{2}$ polynomial estimate for some $p_{0}\in[1,2)$, we obtain \begin{equation*} \big\|(I+L)^{-\beta /2}e^{i\tau L^{\gamma /2}}f\big\|_{p_0,\vc} \leq C (1 + |\tau|)^{n\mathfrak s_{p_0}}\|f\|_{p_0}, \quad \forall \tau \in \mathbb{R}. \end{equation*} provided $0<\gamma \ne 1$, $\beta /\gamma = n|\frac 12-\frac 1p|= n\mathfrak s_p$ and $\tau\in \mathbb{R}$. By interpolation, the second estimate implies also, for all $p\in(p_{0},p_{0}')$, the strong $(p,p)$ type estimate \begin{equation*} \big\|(I+L)^{-\beta /2}e^{i\tau L^{\gamma /2}}f\big\|_{p} \leq C (1 + |\tau|)^{n\mathfrak s_{p_0}}\|f\|_{p}. \end{equation*}