Burkholder meets Gundy: Bellman function method for general operators on martingales

It is shown that the Bellman function method can be applied to study the Lp-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 Lp-boundedness of “almost all” operators...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2022-12, Vol.410, p.108746, Article 108746
Main Authors: Borovitskiy, Viacheslav, Osipov, Nikolay N., Tselishchev, Anton
Format: Article
Language:English
Subjects:
Burkholder method
Gundy theorem
Haar transform
Rubio de Francia inequality
Walsh system
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Summary:It is shown that the Bellman function method can be applied to study the Lp-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 Lp-boundedness of “almost all” operators from Gundy's extrapolation theorem. As examples of such operators, we consider the Haar transforms and the operator whose Lp-boundedness underlies Rubio de Francia's inequality for the Walsh system.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2022.108746