Translation averages of dyadic weights are not always good weights

Abstract

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.