Sharp Lorentz-norm estimates for differentially subordinate martingales and applications
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Abstract:
Let $1<p<q\leq 2$. The paper contains the identification of the best constant $C_{p,q}$ such that the following holds. If $X$, $Y$ are Hilbert-space valued martingales such that $Y$ is differentially subordinate to $X$, then we have \begin{equation*} \|Y\|_{p,\infty }\leq C_{p,q}\|X\|_{p,q}. \end{equation*} The proof rests on the careful combination of Burkholder’s method and optimization arguments. As an application, related sharp Lorentz-norm inequalities for a wide class of Fourier multipliers are obtained.References
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Additional Information
- Adam Osękowski
- Affiliation: Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
- ORCID: 0000-0002-8905-2418
- Email: A.Osekowski@mimuw.edu.pl
- Received by editor(s): November 4, 2020
- Received by editor(s) in revised form: March 10, 2021
- Published electronically: July 27, 2021
- Additional Notes: The research was supported by Narodowe Centrum Nauki (Poland), grant 2018/30/Q/ST1/ 00072
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 7235-7262
- MSC (2020): Primary 60G44, 60G42; Secondary 46E30
- DOI: https://doi.org/10.1090/tran/8460
- MathSciNet review: 4315603