Some remarks on the dyadic Rademacher maximal function

TL;DR

This paper investigates the dyadic Rademacher maximal function, establishing key inequalities and estimates, including weighted L^p, weak type, and BMO bounds, and explores extensions across various dyadic systems and measures.

Contribution

It provides new equivalences between inequalities and estimates for the dyadic Rademacher maximal function, including extensions to different dimensions and measures.

Findings

Proves equivalence between weighted L^p inequalities and weak type estimates.

Derives a BMO estimate to address the lack of an L^ inequality.

Considers dyadic systems in different dimensions and measures.

Abstract

Properties of a maximal function for vector-valued martingales were studied by the author in an earlier paper. Restricting here to the dyadic setting, we prove the equivalence between (weighted) L^p inequalities and weak type estimates, and discuss an extension to the case of locally finite Borel measures on R^n. In addition, to compensate for the lack of an L^\infty inequality, we derive a suitable BMO estimate. Different dyadic systems in different dimensions are also considered.