H^p → H^p boundedness implies H^p → L^p boundedness

In this paper, we explore a general method to derive H^p → L^p boundedness from H^p → H^p boundedness of linear operators, an idea originated in the work of Han and Lu in dealing with the multiparameter flag singular integrals ([19]). These linear operators include many singular integral operators in one parameter and multiparameter settings. In this paper, we will illustrate further that this method will enable us to prove the H^p → L^p boundedness on product spaces of homogeneous type in the sense of Coifman and Weiss ([5]) where maximal function characterization of Hardy spaces is not available. Moreover, we also provide a particularly easy argument in those settings such as one parameter or multiparameter Hardy spaces H^p(ℝⁿ) and H ^p(ℝⁿ × ℝ^m) where the maximal function characterization exists. The key idea is to prove ∥f∥L ^p ≤ C ∥f∥H^p for f ∈ L^q ∩ H^p (1 < q < ∞, 0 < p ≤ 1). It is surprising that this simple result even in this classical setting has been absent in the literature.