Maximal Operators Associated With Walsh‐Paley Systems on Dyadic Hardy Spaces

ABSTRACT The concept of a critical point of the maximum operator associated with the Walsh‐Paley system is the focus of this study. Namely, a point is called critical with respect to , if is bounded from to , for all and it is not bounded from to . The main result of the this paper states that if serves as a critical point, then is bounded from to . Additionally, we demonstrate that for every value , there exists a sequence of operators even over subset that is not uniformly bounded from to .