Sharp estimates for the imaginary power operators

Let (X,d,μ) be a metric space with a metric d and a doubling measure μ. Assume that L is a nonnegative self-adjoint operator on L²(X) satisfying the Davies–Gaffney estimates. In this paper, we prove the sharp estimates for the imaginary power operator L^iα, α ∈ R on the Hardy space associated with the operator L. In addition, if L satisfies an additional L^p0 − L² estimate for p0 ∈ [1, 2), we prove the sharp weak type (p0, p0) estimate for L^iα.