Boundedness of Marcinkiewicz integrals with rough kernels on Musielak-Orlicz Hardy spaces

摘要

Let φ:ℝⁿ × [0, ∞) → [0, ∞) satisfy that φ(x,  ⋅ ), for any given x ∈ ℝⁿ, is an Orlicz function and φ( ⋅ , t) is a Muckenhoupt A∞ weight uniformly in t ∈ (0, ∞). The Musielak-Orlicz Hardy space H^φ(ℝⁿ) is defined to be the set of all tempered distributions such that their grand maximal functions belong to the Musielak-Orlicz space L^φ(ℝⁿ). In this paper, the authors establish the boundedness of Marcinkiewicz integral μΩ from H^φ(ℝⁿ) to L^φ(ℝⁿ) under weaker smoothness conditions assumed on Ω. This result is also new even when φ(x, t): = ϕ(t) for all (x, t) ∈ ℝⁿ × [0, ∞), where ϕ is an Orlicz function.