Variable martingale Hardy spaces and their applications in Fourier analysis

“…Moreover, Musielak-Orlicz-Hardy spaces were studied in Yang et al [44]. These results were also investigated for martingale Hardy spaces in Jiao et al [26,27] and Xie et al [42]. The mixed-norm classical Hardy spaces were intensively studied by Huang et al [22,23] and Huang and Yang [24].…”

Section: Introductionmentioning

confidence: 99%

Mixed Martingale Hardy Spaces

Szarvas

Weisz

2020

J Geom Anal

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In this paper, we consider the martingale Hardy spaces defined with the help of the mixed $$L_{\overrightarrow{p}}$$ L p → -norm. Five mixed martingale Hardy spaces will be investigated: $$H_{\overrightarrow{p}}^{s}$$ H p → s , $$H_{\overrightarrow{p}}^S$$ H p → S , $$H_{\overrightarrow{p}}^M$$ H p → M , $$\mathcal {P}_{\overrightarrow{p}}$$ P p → , and $$\mathcal {Q}_{\overrightarrow{p}}$$ Q p → . Several results are proved for these spaces, like atomic decompositions, Doob’s inequality, boundedness, martingale inequalities, and the generalization of the well-known Burkholder–Davis–Gundy inequality.

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“…If pðÁÞ is a constant, we get back the usual L p space. This topic needs essentially new ideas and is investigated very intensively in the literature nowadays (see e.g., Cruz-Uribe and Fiorenza [1], Diening et al [2], Nakai and Sawano [9,10], Jiao et al [4][5][6], Liu et al [7,8]). Interest in the variable Lebesgue spaces has increased since the 1990s because of their use in a variety of applications (see the references in Jiao et al [4]).…”

Section: Introductionmentioning

confidence: 99%

“…Usually, we suppose that pðÁÞ satisfies the Hölder continuity condition. In [4,5,12] as well as in this paper, we suppose a slightly more general condition. It is known (see [4,5]) that the usual dyadic (or martingale) maximal operator is bounded on L pðÁÞ with 1\p À p þ \1.…”

Section: Introductionmentioning

confidence: 99%