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

摘要

For the wide class of measurable sets u, u (is contained in) I{sup}N, N ≥ 1, the criteria are found (in terms of structural and geometric characteristics of sets u called B{sub}1 and B{sub}2 properties) for validity of the weak generalized localization almost everywhere (WGL) for multiple Walsh-Fourier series of functions equal zero on u in the Orlicz classes Φ(L)(I{sup}N) "lying between" L{sub}1 and L{sub}p,p > 1. In particular, it is found that in the class L(log{sup}+L){sup}2 WGL holds on the set u iff u has the B{sub}2 property and in any class L(log{sup}+ log{sup}+ L){sup}(1-ε), 0 < ε < 1, WGL holds on u iff u has the B{sub}1 property.