Boundedness of Generalized Riesz Transforms on Orlicz–Hardy Spaces Associated to Operators

Abstract

Let \({\Phi}\) be a continuous, strictly increasing and concave function on (0, ∞) of critical lower type index \({p_\Phi^- \in(0,\,1]}\). Let L be an injective operator of type ω having a bounded H ∞ functional calculus and satisfying the k-Davies–Gaffney estimates with \({k \in {\mathbb Z}_+}\). In this paper, the authors first introduce an Orlicz–Hardy space \({H^{\Phi}_{L}(\mathbb{R}^n)}\) in terms of the non-tangential L-adapted square function and then establish its molecular characterization. As applications, the authors prove that the generalized Riesz transform \({D_{\gamma}L^{-\delta/(2k)}}\) is bounded from the Orlicz–Hardy space \({H^{\Phi}_{L}(\mathbb{R}^n)}\) to the Orlicz space \({L^{\widetilde{\Phi}}(\mathbb{R}^n)}\) when \({p_\Phi^- \in (0, \frac{n}{n+ \delta - \gamma}]}\), \({0 0 }}\) is bounded on \({L^p(\mathbb{R}^n)}\).