On the Rademacher maximal function

TL;DR

This paper investigates a Banach space-valued maximal operator, extending its Euclidean definition to general measure spaces, and establishes conditions for its boundedness using martingale techniques and concave functions.

Contribution

It generalizes the Rademacher maximal function to sigma-finite measure spaces and characterizes its boundedness independently of the measure space or filtration.

Findings

L^p-boundedness depends on the Banach space's type and cotype.

Weak type inequality suffices for L^p-boundedness.

Characterization via concave functions is provided.

Abstract

This paper studies a new maximal operator introduced by Hyt\"onen, McIntosh and Portal in 2008 for functions taking values in a Banach space. The L^p-boundedness of this operator depends on the range space; certain requirements on type and cotype are present for instance. The original Euclidean definition of the maximal function is generalized to sigma-finite measure spaces with filtrations and the L^p-boundedness is shown not to depend on the underlying measure space or the filtration. Martingale techniques are applied to prove that a weak type inequality is sufficient for L^p-boundedness and also to provide a characterization by concave functions.