Sharp endpoint estimates for Schrödinger groups on Hardy spaces.

Let L be a non-negative self-adjoint operator acting on L 2 (X) where X is a space of homogeneous type with a dimension n. Suppose that the heat kernel of L satisfies the Davies-Gaffney estimates of order m ≥ 2. Let H L 1 (X) be the Hardy space associated with L. In this paper we obtain the sharp endpoint estimate for the Schrödinger group e i t L associated with L such that ‖ (I + L) − n / 2 e i t L f ‖ L 1 (X) + ‖ (I + L) − n / 2 e i t L f ‖ H L 1 (X) ≤ C (1 + | t |) n / 2 ‖ f ‖ H L 1 (X) , t ∈ R for some constant C = C (n , m) > 0 independent of t. We further apply our result to provide the sharp estimate for Schrödinger group of the Kohn Laplacian □ b on polynomial model domains treated by Nagel–Stein [41] , where e − t □ b satisfies only the second order Davies-Gaffney estimates. Moreover, when the heat kernel of L satisfies a Gaussian upper bound, by a duality and interpolation argument, it gives a new proof of a recent result of [13] for sharp endpoint L p -Sobolev bound for e i t L : ‖ (I + L) − s e i t L f ‖ L p (X) ≤ C (1 + | t |) s ‖ f ‖ L p (X) , t ∈ R , s ≥ n | 1 2 − 1 p | for every 1 < p < ∞ , which extends the classical results due to Miyachi ([39,40]) for the Laplacian on the Euclidean space R n. [ABSTRACT FROM AUTHOR]