Sharp endpoint L^p estimates for Schrödinger groups

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 operator e^{- t L} satisfies the generalized Gaussian (p₀,p′₀)-estimates of order m for some 1 ≤ p0< 2. In this paper we prove sharp endpoint L^p-Sobolev bound for the Schrödinger group e^itL, that is for every p∈(p₀,p′₀) there exists a constant C= C(n, p) > 0 independent of t such that

∥(I+L)^-se^itLf∥_p≤C(1+|t|)^s‖f‖_p, t∈R, s≥n|1/2-1/p|.

As a consequence, the above estimate holds for all 1 < p< ∞ when the heat kernel of L satisfies a Gaussian upper bound. This extends classical results due to Feffermann and Stein, and Miyachi for the Laplacian on the Euclidean spaces Rⁿ. We also give an application to obtain an endpoint estimate for L^p-boundedness of the Riesz means of the solutions of the Schrödinger equations.