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Muckenhoupt Class Weight Decomposition and BMO Distance to Bounded Functions

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  25 March 2019

Morten Nielsen  and
Hrvoje Šikić
Morten Nielsen
Affiliation:
Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, DK-9220 Aalborg East, Denmark (mnielsen@math.aau.dk)
Hrvoje Šikić
Affiliation:
Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, HR-10000 Zagreb, Croatia (hsikic@math.hr)
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Abstract

We study the connection between the Muckenhoupt Ap weights and bounded mean oscillation (BMO) for general bases for ℝd. New classes of bases are introduced that allow for several deep results on the Muckenhoupt weights–BMO connection to hold in a very general form. The John–Nirenberg type inequality and its consequences are valid for the new class of Calderón–Zygmund bases which includes cubes in ℝd, but also the basis of rectangles in ℝd. Of particular interest to us is the Garnett–Jones theorem on the BMO distance, which is valid for cubes. We prove that the theorem is equivalent to the newly introduced A2-decomposition property of bases. Several sufficient conditions for the theorem to hold are analysed as well. However, the question whether the theorem fully holds for rectangles remains open.

Keywords

Muckenhoupt conditionBMOCalderón–Zygmund decompositionweightsGarnett–Jones distanceJones factorization

MSC classification

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 42B35: Function spaces arising in harmonic analysis
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 4 , November 2019 , pp. 1017 - 1031
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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