THE CRITERIA OF WEAK GENERALIZED LOCALIZATION FOR MULTIPLE WALSH–FOURIER SERIES OF FUNCTIONS IN ORLICZ CLASSES

Abstract

For the wide class of measurable sets [Formula: see text], [Formula: see text], N≥1, the criteria are found (in terms of structural and geometric characteristics of sets [Formula: see text] called [Formula: see text] and [Formula: see text] properties) for validity of the weak generalized localization almost everywhere (WGL) for multiple Walsh–Fourier series of functions equal zero on [Formula: see text], in the Orlicz classes Φ(L)(IN) "lying between" L1 and Lp, p>1. In particular, it is found that in the class L( log +L)2WGL holds on the set [Formula: see text] iff [Formula: see text] has the [Formula: see text] property and in any class L( log + log +L)1-ε, 0<ε<1, WGL holds on [Formula: see text] iff [Formula: see text] has the [Formula: see text] property.