Martingale transforms and fractional integrals on rearrangement-invariant martingale Hardy spaces
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Martingale transforms and fractional integrals on rearrangement-invariant martingale Hardy spaces Kwok-Pun Ho1
© Akadémiai Kiadó, Budapest, Hungary 2020
Abstract We establish an interpolation result for the rearrangement-invariant martingale Hardy spaces. By using this interpolation result, we extend the mapping properties of the martingale transforms and the fractional integrals on martingale function spaces. In particular, we obtain the mapping properties on the martingale Hardy–Orlicz spaces, the grand martingale Hardy spaces and the martingale Hardy–Lorentz–Karamata spaces. Keywords Martingale transforms · Fractional integrals · Rearrangement-invariant spaces · Interpolation · Martingale · Hardy spaces · Orlicz spaces · Grand Lebesgue spaces · Lorentz–Karamata spaces Mathematics Subject Classification 60G42 · 60G46 · 46E30 · 42B20
1 Introduction This paper establishes the mapping properties of the martingale transforms and the fractional integrals on rearrangement-invariant martingale Hardy spaces. The rearrangement-invariant martingale Hardy spaces include the martingale Hardy–Lorentz spaces [27,36,37], the martingale Hardy–Lorentz–Karamata spaces [12,26], the grand martingale Hardy spaces [11] and the martingale Hardy–Orlicz spaces [28]. Therefore, the results obtained in this paper also give the mapping properties of the martingale transforms and the fractional integrals on these martingale Hardy spaces. We briefly describe our method. We first establish an interpolation result for the rearrangement-invariant martingale Hardy spaces by using an interpolation functor used in [14]. In [14], we see that this interpolation functor can explicitly generate a given rearrangement-invariant quasi-Banach function spaces. In this paper, we show that this functor can also generate a given rearrangement-invariant martingale Hardy spaces.
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Kwok-Pun Ho [email protected] Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, Hong Kong, China
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K.-P. Ho
Once we have this interpolation result, we can apply it to study the mapping properties for the martingale transforms and the fractional integrals on rearrangement-invariant martingale Hardy spaces. As applications of the main results of this paper, we obtain the mapping properties for the martingale transforms and the fractional integrals on the martingale Hardy–Orlicz spaces, the grand martingale Hardy spaces and the martingale Hardy–Lorentz–Karamata spaces. This paper is organized as follows. The basic properties of rearrangement-invariant quasiBanach function spaces and the interpolation functor used in this paper are presented in Sect. 2. The interpolation result for the rearrangement-invariant martingale Hardy spaces is established in Sect. 3. The studies of the martingale transforms and the fractional integrals on rearrangement-invariant martingale Hardy spaces are presented in Sects. 4 and 5, respectively. The applications of our main results on the martingale Hardy–Orlicz spaces, the grand mart
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