Characterizations of the boundedness of generalized fractional maximal functions and related operators in Orlicz spaces

Given 0 < α < n and a Young function η, we consider the generalized fractional maximal operator M α , ηdefined byM α , η f ( x ) = sup B ∋ x| B | α / n| | f | | η , B , where the supremum is taken over every ball B contained in R n . In this article, we give necessary and sufficient Dini type conditions on the functions A , B and η such that M α , ηis bounded from the Orlicz spaceL A ( R n )into the Orlicz spaceL B ( R n ) . We also present a version of this result for open subsets of R n with finite measure. Both results generalize those contained in [6][A. Cianchi, 1999] and [14][E. Harboure, 2002] when η ( t ) = t , respectively. As a consequence, we obtain a characterization of the functions involved in the boundedness of the higher order commutators of the fractional integral operator with BMO symbols. Moreover, we give sufficient conditions that guarantee the continuity in Orlicz spaces of a large class of fractional integral operators of convolution type with less regular kernels and their commutators, which are controlled by M α , η .