Banach Journal of Mathematical Analysis

Parametric Marcinkiewicz integrals with rough kernels acting on weak Musielak–Orlicz Hardy spaces

Bo Li

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Let φ:Rn×[0,)[0,) satisfy that φ(x,), for any given xRn, is an Orlicz function and that φ(,t) is a Muckenhoupt A weight uniformly in t(0,). The weak Musielak–Orlicz Hardy space WHφ(Rn) is defined to be the set of all tempered distributions such that their grand maximal functions belong to the weak Musielak–Orlicz space WLφ(Rn). For parameter ρ(0,) and measurable function f on Rn, the parametric Marcinkiewicz integral μΩρ related to the Littlewood–Paley g-function is defined by setting, for all xRn,

μΩρ(f)(x):=(0||xy|tΩ(xy)|xy|nρf(y)dy|2dtt2ρ+1)1/2, where Ω is homogeneous of degree zero satisfying the cancellation condition.

In this article, we discuss the boundedness of the parametric Marcinkiewicz integral μΩρ with rough kernel from weak Musielak–Orlicz Hardy space WHφ(Rn) to weak Musielak–Orlicz space WLφ(Rn). These results are new even for the classical weighted weak Hardy space of Quek and Yang, and probably new for the classical weak Hardy space of Fefferman and Soria.

Article information

Banach J. Math. Anal., Volume 13, Number 1 (2019), 47-63.

Received: 18 March 2018
Accepted: 7 May 2018
First available in Project Euclid: 30 October 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 42B30: $H^p$-spaces 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Marcinkiewicz integral weak Hardy space Muckenhoupt weight Musielak–Orlicz function


Li, Bo. Parametric Marcinkiewicz integrals with rough kernels acting on weak Musielak–Orlicz Hardy spaces. Banach J. Math. Anal. 13 (2019), no. 1, 47--63. doi:10.1215/17358787-2018-0015.

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