Commutator Theorems for Fractional Integral Operators on Weighted Morrey Spaces

Abstract

LetLbe the infinitesimal generator of an analytic semigroup onL2(Rn)with Gaussian kernel bounds, and letL-α/2be the fractional integrals ofLfor0<α<n. For any locally integrable functionb, the commutators associated withL-α/2are defined by[b,L-α/2](f)(x)=b(x)L-α/2(f)(x)-L-α/2(bf)(x). Whenb∈BMO(ω)(weightedBMOspace) orb∈BMO, the authors obtain the necessary and sufficient conditions for the boundedness of[b,L-α/2]on weighted Morrey spaces, respectively.

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

“…Our results not only extend the results of [12] from (−△) to a general operator L, but also characterize the (weighted) Lipschitz spaces by means of the boundedness of [b, L −α/2 ] on the weighted Morrey spaces, which extend the results of [12] and [13]. The basic tool is based on a modification of sharp maximal function M ♯ L introduced by [6].…”

“…Shirai [9] proved that b ∈ Lip β (R n ) if and only if the commutator [b, I α ] is bounded from the classical Morrey spaces L p,λ (R n ) to L q,λ (R n ) for 1 < p < q < ∞, 0 < α, 0 < β < 1 and 0 < α + β = (1/p − 1/q)(n − λ) < n or L p,λ (R n ) to L q,µ (R n ) for 1 < p < q < ∞, 0 < α, 0 < β < 1, 0 < α + β = (1/p − 1/q) < n, 0 < λ < n − (α + β)p and µ/q = λ/p. Wang [12] established some weighted boundedness of properties of commutator [b, I α ] on the weighted Morrey spaces L p,k under appropriated conditions on the weight ω, where the symbol b belongs to (weighted) Lipschitz spaces. The weighted Morrey space was first introduced by Komori and Shirai [5].…”

Estimates for Multilinear Commutators of Generalized Fractional Integral Operators on Weighted Morrey Spaces

“…Recently, Wang [5] obtained some estimates for the commutator [ , ] on weighted Morrey space (see Definitions 3 and 4), where ∈ BMO 1 ( ). Furthermore, Wang and Si [6] obtained the necessary and sufficient conditions for the boundedness of [ , − /2 ] on weighted Morrey spaces when ∈ BMO 1 ( ).…”

“…Motivated by [1,3,5,6], it is natural to raise the following question: how to establish corresponding boundedness of the multilinear commutator − /2 ⃗ on the weighted Morrey space, where ⃗ = ( 1 , . .…”

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].…”