Revista Matemática Iberoamericana

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Volume 18, Issue 2, 2002, pp. 379–407
DOI: 10.4171/RMI/323

Published online: 2002-08-31

Translation averages of dyadic weights are not always good weights

Lesley A. Ward[1]

(1) University of South Australia, Mawson Lakes, Australia

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.

Keywords: Doubling measures, dyadic weights, $A_p$ weights, reverse Hölder weights, Muckenhoupt weights, translation average

Ward Lesley: Translation averages of dyadic weights are not always good weights. Rev. Mat. Iberoam. 18 (2002), 379-407. doi: 10.4171/RMI/323