Disproof of Wright's Problems 14 and 15 on lacunary maximal functions

Abstract

We disprove the proposed weak endpoint bounds in James Wright's Problems 14 and 15 from the 2017 collection of harmonic-analysis problems. For every $d\ge1$, we exhibit a fixed compactly supported singular probability measure with Fourier decay of every order $0<\varepsilon<1/2$, for which the dyadic convolution maximal operator has infinite weak $L^1$ and infinite $H^1$ to weak $L^1$ operator norms. For every finite nonempty set of dyadic scales, both optimal weak constants equal the number of scales, with the Hardy constant measured in the specified atomic norm. Increasingly separated renewal digits provide uniform real-frequency Fourier decay while allowing the disjoint output regions needed for the positive and mean-zero tests. We first establish the fixed measure, then prove the two endpoint estimates using separate normalized tests. Finally we prove the two Fourier lemmas used in the construction.