Some Estimates for Maximal Bochner—Riesz Means on Musielak—Orlicz Hardy Spaces

Abstract

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 Hφ(ℝn) is defined to be the space of all tempered distributions whose grand maximal functions belong to the Musielak–Orlicz space Lφ(ℝn). In this paper, the authors establish the boundedness of maximal Bochner–Riesz means T*δ from Hφ(ℝn) to WLφ(ℝn) or Lφ(ℝn). These results are also new even when φ(x, t):= Φ(t) for all (x, t) ∈ ℝn × [0, ∞), where Φ is an Orlicz function.