Boundedness of multilinear fractional type operators on Hardy spaces with variable exponents
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Boundedness of multilinear fractional type operators on Hardy spaces with variable exponents Jian Tan1 Received: 24 August 2019 / Revised: 12 September 2020 / Accepted: 18 October 2020 © Springer Nature Switzerland AG 2020
Abstract In this article, we show that multilinear fractional type operators are bounded from product of Hardy spaces with variable exponents into Lebesgue spaces with variable exponents via the atomic decomposition theory. Keywords Multilinear fractional type operators · Hardy spaces · Variable exponents · Atomic decomposition Mathematics Subject Classification Primary 42B20; Secondary 42B30 · 46E30
1 Introduction The study of Hardy spaces began in the early 1900s in the context of Fourier series and complex analysis in one variable. It was not until 1960 when the groundbreaking work in Hardy space theory in Rn came from Coifman [1], Coifman and Weiss [2] and Fefferman and Stein [9]. The classical Hardy space can be characterized by the Littlewood–Paley–Stein square functions, maximal functions and atomic decompositions. Especially, atomic decomposition is a significant tool in harmonic analysis and wavelet analysis for the study of function spaces and the operators acting on these spaces. Atomic decomposition was first introduced by Coifman [1] in one dimension in 1974 and later was extended to higher dimensions by Latter [18]. As we all know, atomic decompositions of Hardy spaces play an important role in the boundedness of operators on Hardy spaces and it is commonly sufficient to check that atoms are mapped into bounded elements of quasi-Banach spaces. Another stage in the progress of the theory of Hardy spaces was done by Nakai and Sawano [21] and Cruz-Uribe and Wang [7] when they independently considered
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Jian Tan [email protected] School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, People’s Republic of China 0123456789().: V,-vol
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Hardy spaces with variable exponents. It is quiet different to obtain the boundedness of operators on Hardy spaces with variable exponents. It is not sufficient to show the H p(·) -boundedness merely by checking the action of the operators on H p(·) -atoms. In the linear theory, the boundedness of some operators on variable Hardy spaces have been established in [7,13,21,25,33] as applications of the corresponding atomic decompositions theories. In more recent years, the study of multilinear operators on Hardy space theory has received increasing attention by many authors, see for example [12,14,15]. While the multilinear operators worked well on the product of Hardy spaces, it is surprising that these similar results in the setting of variable exponents were unknown for a long time. The boundedness of some multilinear operators on products of classical Hardy spaces was investigated by Grafakos and Kalton [12] and Li et al. [19]. In Tan et al. [29] studied some multilinear operators are bounded on variable Lebesgue spaces L p(·) . However, there are some subtle difficulties in proving the bounde
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