Maximal and singular integral operators and their commutators on generalized weighted Morrey spaces with variable exponent

Abstract

Abstract. We consider the generalized weighted Morrey spaces M p(·),ϕ ω (Ω) with variable exponent p(x) and a general function ϕ(x,r) defining the Morrey-type norm. In case of unbounded sets Ω ⊂ R n we prove the boundedness of the Hardy-Littlewood maximal operator and Calderón-Zygmund singular operators with standard kernel, in such spaces. We also prove the boundedness of the commutators of maximal operator and Calderón-Zygmund singular operators in the generalized weighted Morrey spaces with variable exponent Mathematics subject classification (2010): 42B20, 42B25, 42B35.

Extrapolation to weighted Morrey spaces with variable exponents and applications

“…In this paper, we further extend the studies in [20, 42] to establish the mapping properties of pseudo-differential operators and Fourier integral operators on weighted Morrey spaces with variable exponents. We obtain our main results by extending the well-known extrapolation theory introduced by Rubio de Francia [51–52] to the weighted Morrey spaces with variable exponents.…”

“…For the mapping properties of the singular integral operators and the Riesz potentials on the weighted Morrey spaces with variable exponents, the reader is referred to [20, 42].…”

“…The weighted Morrey spaces with variable exponents were introduced and studied in [20, 42]. Moreover, the mapping properties of singular integral operators and the Riesz potentials were obtained in [19] and [42], respectively.…”

Weighted local Morrey spaces

“…The two-weighted Morrey spaces of the type M p q (u, v) can be found in [24,43,49] including generalized Morrey spaces [5,9,12,13,20,21,25] and their closed subspaces [2].…”

Commutators of the B-maximal operator and B-maximal commutators

“…Alphonse [8] obtained weak type inequality for maximal commutators, and pointwise estimates of the maximal commutator and the commutator of the maximal function are proved by Agcayazı, Gogatishvili, Koca, Mustafayev [1]. Commutators have been research area many mathematicians such as Guliyev, Hasanov, Hu, Lin, Yang, Janson [18,20,21] and others.…”