Sharp endpoint estimates for Schrödinger 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 Schrödinger group eitL associated with L such that‖(I+L)−n/2eitLf‖L1(X)+‖(I+L)−n/2eitLf‖HL1(X)≤C(1+|t|)n/2‖f‖HL1(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.