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This article is cited in 1 scientific paper (total in 1 paper)
Research Papers
The $ \mathrm{BMO}\rightarrow\mathrm{BLO}$ action of the maximal operator on $\alpha$-trees
V. Vasyuninab, A. Osękowskic, L. Slavindb a St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
b Saint Petersburg State University
c Faculty of Mathematics, Informatics and Mechanics, University of Warsaw
d University of Cincinnati
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.
Keywords:
BMO, BLO $\alpha$-trees, maximal functions, explicit Bellman function, sharp constants.
Received: 12.11.2018
Citation:
V. Vasyunin, A. Osȩkowski, L. Slavin, “The $ \mathrm{BMO}\rightarrow\mathrm{BLO}$ action of the maximal operator on $\alpha$-trees”, Algebra i Analiz, 31:5 (2019), 106–153; St. Petersburg Math. J., 31:5 (2020), 831–863
Linking options:
https://www.mathnet.ru/eng/aa1670 https://www.mathnet.ru/eng/aa/v31/i5/p106
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