Assume that \((X,d,\mu )\) is a metric space endowed with a non-negative Borel measure \(\mu \) satisfying the doubling condition and the additional condition that \(\mu (B(x,r))\gtrsim r^n\) for any \(x\in X, \,r>0\) and some \(n\ge 1\) . Let L be a non-negative self-adjoint operator on \(L^2(X,\mu )\) . We assume that \(e^{-tL}\) satisfies a Gaussian upper bound and the Schrödinger operator \(e^{itL}\) satisfies an \(L^1\rightarrow L^\infty \) decay estimate of the form \(\begin{aligned} \Vert e^{itL}\Vert _{L^1\rightarrow L^\infty } \lesssim |t|^{-\frac{n}{2}}. \end{aligned}\) Then for a general class of dispersive semigroup \(e^{it\phi (L)}\) , where \(\phi : {\mathbb {R}}^+ \rightarrow {\mathbb {R}}\) is smooth, we establish a similar \(L^1\rightarrow L^\infty \) decay estimate by a suitable subordination formula connecting it with the Schrödinger operator \(e^{itL}\) . As applications, we derive new Strichartz estimates for several dispersive equations related to Hermite operators, twisted Laplacians and Laguerre operators.
