Hardy Spaces on Open Subsets of Rⁿ

The theory of Hardy spaces over Rⁿ , originated by C. Fefferman and Stein [1], was generalized several decades ago to the case of subsets of Rⁿ . The pioneering work of generalization was done by Jonsson, Sjögren, and Wallin [2] for the case of suitable closed subsets and by Miyachi [3] for the case of proper open subsets. In this article, we study Hardy spaces on proper open Ω ⊂ Rⁿ , where Ω satisfies a doubling condition and | Ω | = ∞ . We first establish a variant of the Calderón–Zygmund decomposition, and then explore the relationship among Hardy spaces by means of atomic decomposition, radial maximal function, and grand maximal function.