The $\boldsymbol {\mathrm {BMO}}\boldsymbol {\to }\boldsymbol {\mathrm {BLO}}$ action of the maximal operator on $\boldsymbol \alpha$-trees
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A. Osȩkowski, L. Slavin and V. Vasyunin
Translated by: V. Vasyunin - St. Petersburg Math. J. 31 (2020), 831-863
- DOI: https://doi.org/10.1090/spmj/1625
- Published electronically: September 3, 2020
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Abstract:
The explicit upper Bellman function is found for the natural dyadic maximal operator acting from $\mathrm {BMO}(\mathbb {R}^n)$ into $\mathrm {BLO}(\mathbb {R}^n)$. As a consequence, it is shown that the $\mathrm {BMO}\to \mathrm {BLO}$ norm of the natural operator equals $1$ for all $n$, and so does the norm of the classical dyadic maximal operator. The main result is a partial consequence of a theorem for the so-called $\alpha$-trees, which generalize dyadic lattices. The Bellman function in this setting exhibits an interesting quasiperiodic structure depending on $\alpha$, but also allows a majorant independent of $\alpha$, hence a dimension-free norm constant. Also, the decay of the norm is described with respect to the growth of the difference between the average of a function on a cube and the infimum of its maximal function on that cube. An explicit norm-optimizing sequence is constructed.References
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Bibliographic Information
- A. Osȩkowski
- Affiliation: Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
- ORCID: 0000-0002-8905-2418
- Email: ados@mimuw.edu.pl
- L. Slavin
- Affiliation: University of Cincinnati; St. Petersburg State University
- MR Author ID: 121075
- ORCID: 0000-0002-9502-8852
- Email: leonid.slavin@uc.edu
- V. Vasyunin
- Affiliation: St. Petersburg Department of the V. A. Steklov Mathematical Institute, RAS; St. Petersburg State University
- Email: vasyunin@pdmi.ras.ru
- Received by editor(s): November 12, 2018
- Published electronically: September 3, 2020
- Additional Notes: The second and third authors research was supported by the Russian Science Foundation grant 14-41-00010.
- © Copyright 2020 American Mathematical Society
- Journal: St. Petersburg Math. J. 31 (2020), 831-863
- MSC (2010): Primary 42A05, 42B35, 49K20
- DOI: https://doi.org/10.1090/spmj/1625
- MathSciNet review: 4022004