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2021 Embedding $BMO$ into weighted $BMO$
Adam Osękowski
Publ. Mat. 65(1): 335-361 (2021). DOI: 10.5565/PUBLMAT6512112

Abstract

A classical result of harmonic analysis asserts that if a weight $w$ satisfies Muckenhoupt's condition $A_\infty$, then the unweighted class $\mathit{BMO}$ is contained in the weighted space $\mathit{BMO}(w)$. The paper identifies the norm of this embedding in the one-dimensional setting. Specifically, for any function $f\in \mathit{BMO}(\mathbb{R})$ and any weight $w\in A_\infty(\mathbb{R})$ of characteristic $[w]_{A_\infty}$, we have the estimate $$ \|f\|_{\mathit{BMO}(w)}\leq e\sqrt{2}[w]_{A_\infty}\|f\|_{\mathit{BMO}}. $$ The constant $e\sqrt{2}=3.8442\dots$ is the best possible. We also prove a sharp version of this result in which the characteristic $[w]_{A_\infty}$ of the weight is fixed. Further extensions to the theory of martingales are obtained.

Funding Statement

Research supported by Narodowe Centrum Nauki (Poland), grant no. DEC-2014 14/E/ST1/00532.

Citation

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Adam Osękowski. "Embedding $BMO$ into weighted $BMO$." Publ. Mat. 65 (1) 335 - 361, 2021. https://doi.org/10.5565/PUBLMAT6512112

Information

Received: 24 October 2019; Revised: 23 September 2020; Published: 2021
First available in Project Euclid: 11 December 2020

MathSciNet: MR4185837
Digital Object Identifier: 10.5565/PUBLMAT6512112

Subjects:
Primary: 42A05 , 42B35 , 49K20

Keywords: $\mathit{BMO}$ , Bellman function , martingale , weight

Rights: Copyright © 2021 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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Vol.65 • No. 1 • 2021
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