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Two boundedness criteria for a class of operators on Musielak–Orlicz Hardy spaces and applications

Part of: Linear function spaces and their duals Harmonic analysis in several variables

Published online by Cambridge University Press:  16 July 2019

Xiaoli Qiu ,
Baode Li ,
Xiong Liu  and
Bo Li
Xiaoli Qiu
Affiliation:
College of Mathematics and System Sciences, Xinjiang University, Urumqi830046, P. R. China (2237424863@qq.com; baodeli@xju.edu.cn; 1394758246@qq.com)
Baode Li
Affiliation:
College of Mathematics and System Sciences, Xinjiang University, Urumqi830046, P. R. China (2237424863@qq.com; baodeli@xju.edu.cn; 1394758246@qq.com)
Xiong Liu
Affiliation:
College of Mathematics and System Sciences, Xinjiang University, Urumqi830046, P. R. China (2237424863@qq.com; baodeli@xju.edu.cn; 1394758246@qq.com)
Bo Li*
Affiliation:
Center for Applied Mathematics, Tianjin University, Tianjin300072, P. R. China (bli.math@outlook.com)
*
*Corresponding author.
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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 (weak) Musielak–Orlicz Hardy space Hφ(ℝn) (WHφ(ℝn)) generalizes both the weighted (weak) Hardy space and the (weak) Orlicz Hardy space and hence has wide generality. In this paper, two boundedness criteria for both linear operators and positive sublinear operators from Hφ(ℝn) to Hφ(ℝn) or from Hφ(ℝn) to WHφ(ℝn) are obtained. As applications, we establish the boundedness of Bochner–Riesz means from Hφ(ℝn) to Hφ(ℝn), or from Hφ(ℝn) to WHφ(ℝn) in the critical case. These results are new even when φ(x, t): = Φ(t) for all (x, t) ∈ ℝn × [0, ∞), where Φ is an Orlicz function.

Keywords

Hardy spaceMusielak–Orlicz functionMuckenhoupt weightBochner–Riesz means

MSC classification

Primary: 42B30: $H^p$-spaces
Secondary: 42B15: Multipliers 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 1 , February 2020 , pp. 13 - 35
Copyright
Copyright © Edinburgh Mathematical Society 2019

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