Sharp endpoint estimates for Schrodinger groups on Hardy spaces

Let L be a non-negative self-adjoint operator acting on L2(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 HL1 (X)be the Hardy space associated with L. In this paper we obtain the sharp endpoint estimate for the Schrodinger group eitL associated with L such that (I +L)-n/2eitL f L1(X)+ (I+L)-n/2eitL fHL1(X)1+|t|)n/2IIfIIHL1(X), te R for some constant C = C(n, m) > 0 independent of t. We further apply our result to provide the sharp estimate for Schrodinger 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 Lp-Sobolev bound for eitL: (I + L)-seitLf Lp(X)1+|t|)sIIfIILp(X), teR, s>n ?? 1 2- 1 ?? p for every 1 < p < oo, which extends the classical results due to Miyachi ([39,40]) for the Laplacian on the Euclidean space Rn.& COPY; 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by /4 .0/).