Sharp Estimates for Schrödinger Groups on Hardy Spaces for $$0<p\le 1$$

Abstract Let X be a space of homogeneous type with the doubling order n . L nonnegative self-adjoint operator on $$L^2(X)$$ L 2 ( X ) and suppose that kernel $$e^{-tL}$$ e - t satisfies Gaussian upper bound. This paper shows for $$0<p\le 1$$ 0 < p ? 1 $$s=n(1/p-1/2)$$ s = n / , $$\begin{aligned}\Vert (I+L)^{-s}e^{itL}f\Vert _{H^p_L(X)} \lesssim (1+|t|)^{s}\Vert f\Vert \end{aligned}$$ ? I + itL f H ? | all $$t\in {\mathbb {R}}$$ ? R where $$H^p_L(X)$$ is Hardy associated to recovers classical results in particular case when $$L=-\Delta $$ ? extends number known results.