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The 𝐵𝑀𝑂→𝐵𝐿𝑂 action of the maximal operator on 𝛼-trees
St. Petersburg Mathematical Journal ( IF 0.800 ) Pub Date : 2020-09-03 , DOI: 10.1090/spmj/1625
A. Osȩkowski; L. Slavin; V. Vasyunin

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.
更新日期:2020-09-18

 

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