Full divergence and maximal functions with cancellation
Tom 152 / 2018
Streszczenie
We consider the maximal functions 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.
