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

Abstract

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