Full divergence and maximal functions with cancellation
Volume 152 / 2018
Abstract
We consider the maximal functions $f^*_\mathcal {I}= \sup_n T_n|f|$ and $f^*_\mathcal {O}=\sup_n|T_nf|$ for a variety of sequences $(T_n)$ of positive $L_1$-$L_\infty $ contractions. There are well-known cases where we have functions $f \in L_1(X)$ such that $\| f^*_\mathcal {O}\| _1 \lt \infty $, but $\| f^*_\mathcal {I}\| _1 = \infty $. We seek to describe as wide a class of examples as possible where this phenomenon occurs. We also consider this more generally for $L_p$-norms. As part of this project, it is important that in some non-trivial cases for all $f \in L_p(X) \setminus L_\infty (X)$, we have $\| f^*_\mathcal {O}\| _p = \infty $. Indeed, actually for all $f \in L_p(X) \setminus L_\infty (X)$ we have $f^*_\mathcal {O}=\infty $ a.e. We call this phenomenon full divergence. Understanding when full divergence occurs is an additional focus in this article.
