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Modular function spaces and control functions of almost everywhere convergence

Kita H., Miyamoto T., Yoneda K.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2001

[Z] 41

99-133

The modular spaces have been studied by many authors. In this paper we consider the modular function spaces which provide a generalization of Banach function spaces, and discuss some relations between properties of these spaces and control functions of almost everywhrere convergence of functions in modular function spaces. We give a necessary and sufficient condition for a control function to be in the same modular function space as functions appearing in almost everywhere convergence.