Necessary and Sufficient Conditions for Boundedness of Commutators of the General Fractional Integral Operators on Weighted Morrey Spaces

Abstract

We prove that b is in Lip β (β) if and only if the commutator [b, L −α/2 ] of the multiplication operator by b and the general fractional integral operator L −α/2 is bounded from the weighed Morrey space L p,k (ω) to L q,kq/p (ω 1−(1−α/n)q , ω), where 0 < β < 1, 0 < α + β < n, 1 < p < n/(α + β), 1/q = 1/p − (α + β)/n, 0 ≤ k < p/q, ω q/p ∈ A1 and rω > 1−k p/q−k , and here rω denotes the critical index of ω for the reverse Hölder condition.2000 Mathematics Subject Classification. 42B20; 42B35.

Weighted local Morrey spaces

“…Many researchers investigated the boundedness properties of the linear operators acting on weighted Morrey spaces. such as sublinear operator [4,12,15,35], singular integral operators [14,35,63], commutators [17,12,35,59,61], pseudo-differential operators [26], the square functions [11], Toeplitz operators [56], the fractional integral operators [12,28,31,32] and fractional integrals associated to operators [51,54,55] including the related commutators. Applications to partial differential equations can be found in [8,19,50].…”