Abstract
We study the boundedness of Φ-admissible singular operators and their commutators on vanishing generalized Orlicz-Morrey spaces including their weak versions. These conditions are satisfied by most of the operators in harmonic analysis, such as the Hardy-Littlewood maximal operator, the Calderon-Zygmund singular integral operator and so on. In all the cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on weights without assuming any monotonicity property of on r. MSC:42B20, 42B25, 42B35, 46E30.
Highlights
1 Introduction As is well known Morrey [ ] introduced the classical Morrey spaces to investigate the local behavior of solutions to second order elliptic partial differential equations (PDE)
We denote by WMp,λ ≡ WMp,λ(Rn) the weak Morrey space of all functions f ∈ WLlpoc(Rn) for which f
Using Orlicz spaces, we can investigate the boundedness of the maximal operator near p = more precisely
Summary
As is well known Morrey [ ] introduced the classical Morrey spaces to investigate the local behavior of solutions to second order elliptic partial differential equations (PDE). Using Orlicz spaces, we can investigate the boundedness of the maximal operator near p = more precisely (see [ , ] and [ ]). It is well known that the maximal and singular integral operators play an important role in harmonic analysis (see [ , ]). We need to give a precise definition of Kf for the function f in Morrey spaces, for example, Kf (x) = K (f χ B) +. The main purpose of this paper is to find sufficient conditions on general Young function and functions φ , φ which ensure the boundedness of the sublinear operators generated by singular integral operators from vanishing generalized Orlicz-Morrey spaces VM ,φ (Rn) to another VM ,φ (Rn), from VM ,φ (Rn) to vanishing weak generalized. For the boundedness of the operators of harmonic analysis on Orlicz-Morrey spaces, see [ – ].
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