Imaginary power operators on Hardy spaces

Abstract

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