Construction of an exotic measure: Dyadic doubling and triadic doubling does not imply doubling

Abstract

What kinds of properties of measures, and of functions, must necessarily hold in a continuous setting if they hold in certain martingale settings? And for which martingale settings? Consider the doubling property enjoyed by some measures, of importance in both harmonic analysis and complex analysis. It is known that if a measure is (dyadic) doubling with respect to the standard lattice of dyadic intervals in the real line, and also with respect to certain translations of that dyadic lattice, then the measure has the doubling property on all intervals. Analogous results hold for functions of bounded mean oscillation (BMO), and also for Muckenhoupt's Ap weights, for reverse Hölder weights, and for functions of vanishing mean oscillation (VMO), both on Euclidean spaces and on one-parameter or product spaces of homogeneous type in the sense of Coifman and Weiss. By contrast, as we show here, a measure that is doubling both with respect to the standard lattice of dyadic intervals and also with respect to a lattice of triadic intervals need not be doubling on all intervals. In other words, the doubling property is not necessarily inherited from the dyadic and triadic martingale settings to the continuous setting. The proof is by means of an intricate construction, yielding measures that are both dyadic and triadic doubling but not doubling, and making use of some number-theoretic properties of dyadic and triadic rational numbers.