Imaginary power operators on Hardy spaces

Let (X, d, μ) be an Ahlfors n-regular metric measure space. Let L be a non-negative self-adjoint operator on L²(X) with heat kernel satisfying Gaussian estimate. Assume that the kernels of the spectral multiplier operators F(L) satisfy an appropriate weighted L² estimate. By the spectral theory, we can define the imaginary power operator L^is, s ∈ R, which is bounded on L²(X). The main aim of this paper is to prove that for any p ∈ (0, ∞), ||L^isf||||_HL p ₍X₎ ≤ C(1 + |s|)ⁿ|1/p−1/2^|||f||_HL p ₍X), s ∈ R, where H_L^p (X) is the Hardy space associated to L, and C is a constant independent of s. Our result applies to sub-Laplaicans on stratified Lie groups and Hermite operators on Rⁿ with n ≥ 2.