Imaginary power operators on Hardy spaces.

Let (X,d, \mu) be an Ahlfors n-regular metric measure space. Let \mathcal {L} be a non-negative self-adjoint operator on L^2(X) with heat kernel satisfying Gaussian estimate. Assume that the kernels of the spectral multiplier operators F(\mathcal {L}) satisfy an appropriate weighted L^2 estimate. By the spectral theory, we can define the imaginary power operator \mathcal {L}^{is}, s\in \mathbb R, which is bounded on L^2(X). The main aim of this paper is to prove that for any p \in (0,\infty), \begin{equation*} \big \|\mathcal {L}^{is} f\big \|_{H^p_{\mathcal {L}}(X)} \leq C (1+|s|)^{n|1/p-1/2|} \|f\|_{H^p_{\mathcal {L}}(X)}, \quad s \in \mathbb {R}, \end{equation*} where H^p_\mathcal {L}(X) is the Hardy space associated to \mathcal {L}, and C is a constant independent of s. Our result applies to sub-Laplaicans on stratified Lie groups and Hermite operators on \mathbb {R}^n with n\ge 2. [ABSTRACT FROM AUTHOR]