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January 2019 Parametric Marcinkiewicz integrals with rough kernels acting on weak Musielak–Orlicz Hardy spaces
Bo Li
Banach J. Math. Anal. 13(1): 47-63 (January 2019). DOI: 10.1215/17358787-2018-0015

Abstract

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

μ Ω ρ ( f ) ( x ) : = ( 0 | | x y | t Ω ( x y ) | x y | n ρ f ( y ) d y | 2 d t t 2 ρ + 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 μ Ω ρ with rough kernel from weak Musielak–Orlicz Hardy space WH φ ( R n ) to weak Musielak–Orlicz space WL φ ( R n ) . 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.

Citation

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Bo Li. "Parametric Marcinkiewicz integrals with rough kernels acting on weak Musielak–Orlicz Hardy spaces." Banach J. Math. Anal. 13 (1) 47 - 63, January 2019. https://doi.org/10.1215/17358787-2018-0015

Information

Received: 18 March 2018; Accepted: 7 May 2018; Published: January 2019
First available in Project Euclid: 30 October 2018

zbMATH: 07002031
MathSciNet: MR3892337
Digital Object Identifier: 10.1215/17358787-2018-0015

Subjects:
Primary: 42B25
Secondary: 42B30, 46E30

Rights: Copyright © 2019 Tusi Mathematical Research Group

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Vol.13 • No. 1 • January 2019
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