Boundedness of the Maximal and Singular Operators on Generalized Orlicz–Morrey Spaces

Boundedness of the maximal operator and its commutators on vanishing generalized Orlicz-Morrey spaces

“…On boundedness of the maximal operator and their commutators in Orlicz spaces. The following lemma was proved in [7].…”

“…The definition of Orlicz-Morrey spaces introduced in [7] and used here is different from that of the papers [24], [34] and other papers.…”

“…Such spaces were studied in [24] first, see also [7,13,25,26,34]. The most general spaces of such a type, MusielakOrlicz-Morrey spaces, unifying the classical and variable exponent approaches, were an object of study in [22], where potential operators were studied.…”

“…The Orlicz-Morrey spaces we work with are precisely defined in Section 2.3. The weakest assumptions on the functions Φ and ϕ, defining the space M Φ,ϕ (R n ), for the boundedness of the maximal operator, were obtained in [7], up to authors' knowledge, together with weak estimates.…”

“…Conditions for the boundedness of the maximal operator in Orlicz spaces are known, see [10], [20]; we use such a result in the theorem below in the form proved in [7].…”