Imaginary power operators on Hardy spaces

Abstract:

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 $n\ge 2$.