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Littlewood–Paley–Rubio De Francia Inequality for the Two-Parameter Walsh System

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A version of Littlewood–Paley–Rubio de Francia inequality for the two-parameter Walsh system is proved: for any family of disjoint rectangles \( {I}_k={I}_k^1\times {I}_k^2 \) in ℤ+ × ℤ+ and a family of functions fk with Walsh spectrum inside Ik the following is true:

$$ {\left\Vert \sum_k{f}_k\right\Vert}_{L^p}\le {C}_p{\left\Vert {\left(\sum_k{\left|{f}_k\right|}^2\right)}^{1/2}\right\Vert}_{L^p},\kern2em 1<p\le 2, $$

where Cp does not depend on the choice of rectangles {Ik} or functions {fk}. The arguments are based on the atomic theory of two-parameter martingale Hardy spaces. In the course of the proof, a two-parameter version of Gundy’s theorem on the boundedness of operators taking martingales to measurable functions is formulated, which might be of independent interest.

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Authors and Affiliations

  1. St.Petersburg Department of Steklov Mathematical Institute, St.Petersburg, Russia

    V. Borovitskiy

  2. St.Petersburg State University, St.Petersburg, Russia

    V. Borovitskiy

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  1. V. Borovitskiy
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Correspondence to V. Borovitskiy.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 491, 2020, pp. 27–42.

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Borovitskiy, V. Littlewood–Paley–Rubio De Francia Inequality for the Two-Parameter Walsh System. J Math Sci 261, 746–756 (2022). https://doi.org/10.1007/s10958-022-05785-0

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