On the Rademacher maximal function

Mikko Kemppainen¹

¹ Department of Mathematics and Statistics University of Helsinki Gustaf Hällströmin katu 2b FI-00014 Helsinki, Finland

Studia Mathematica, Tome 203 (2011) no. 1, pp. 1-31

Résumé

This paper studies a new maximal operator introduced by Hytönen, 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.

Détail

DOI : 10.4064/sm203-1-1

Keywords: paper studies maximal operator introduced hyt nen mcintosh portal functions taking values banach space p boundedness operator depends range space certain requirements type cotype present instance original euclidean definition maximal function generalized sigma finite measure spaces filtrations p boundedness shown depend underlying measure space filtration martingale techniques applied prove weak type inequality sufficient p boundedness provide characterization concave functions