Revista Matemática Iberoamericana

Translation averages of dyadic weights are not always good weights

Lesley A. Ward

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The process of translation averaging is known to improve dyadic BMO to the space BMO of functions of bounded mean oscillation, in the sense that the translation average of a family of dyadic BMO functions is necessarily a BMO function. The present work investigates the effect of translation averaging in other dyadic settings. We show that translation averages of dyadic doubling measures need not be doubling measures, translation averages of dyadic Muckenhoupt weights need not be Muckenhoupt weights, and translation averages of dyadic reverse Hölder weights need not be reverse Hölder weights. All three results are proved using the same construction.

Article information

Rev. Mat. Iberoamericana, Volume 18, Number 2 (2002), 379-407.

First available in Project Euclid: 28 April 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B35: Function spaces arising in harmonic analysis 42B25: Maximal functions, Littlewood-Paley theory

Doubling measures dyadic weights $A_p$ weights reverse Hölder weights Muckenhoupt weights translation average


Ward, Lesley A. Translation averages of dyadic weights are not always good weights. Rev. Mat. Iberoamericana 18 (2002), no. 2, 379--407.

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