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arXiv · 2107.06384

Bellman function method for general operators on martingales

Viacheslav Borovitskiy·Nikolay N. Osipov·Anton Tselishchev
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Abstract

It is shown that the Bellman function method can be applied to study the $L^p$-norms of general operators on martingales, i.e., of operators that are not necessarily martingale transforms. Informally, we provide a single Bellman-type function that "encodes" the $L^p$-boundedness of "almost all" operators from Gundy's extrapolation theorem. As examples of such operators, we consider the Haar transforms and the operator whose $L^p$-boundedness underlies Rubio de Francia's inequality for the Walsh system.

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Viacheslav Borovitskiy, Nikolay N. Osipov, Anton Tselishchev. 2021-07-13. Bellman function method for general operators on martingales. https://doi.org/10.1016/j.aim.2022.108746

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