Banach Journal of Mathematical Analysis

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

Bo Li

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1540865070

Digital Object Identifier
doi:10.1215/17358787-2018-0015

Mathematical Reviews number (MathSciNet)
MR3892337

Zentralblatt MATH identifier
07002031

Subjects
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.)

Keywords
Marcinkiewicz integral weak Hardy space Muckenhoupt weight Musielak–Orlicz function

Citation

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. https://projecteuclid.org/euclid.bjma/1540865070


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