LUSIN AREA FUNCTION AND MOLECULAR CHARACTERIZATIONS OF MUSIELAK–ORLICZ HARDY SPACES AND THEIR APPLICATIONS

Abstract

Let φ : ℝ n × [0,∞) → [0,∞) be a growth function such that φ(x, ⋅) is nondecreasing, φ(x, 0) = 0, φ(x, t) > 0 when t > 0, lim t→∞ φ(x, t) = ∞, and φ(⋅, t) is a Muckenhoupt A ∞ (ℝ n ) weight uniformly in t. In this paper, the authors establish the Lusin area function and the molecular characterizations of the Musielak–Orlicz Hardy space H φ (ℝ n ) introduced by Luong Dang Ky via the grand maximal function. As an application, the authors obtain the φ-Carleson measure characterization of the Musielak–Orlicz BMO-type space BMO φ (ℝ n ), which was proved to be the dual space of H φ (ℝ n ) by Luong Dang Ky.