Compactness properties of commutators of bilinear fractional integrals

Abstract

Commutators of a large class of bilinear operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be jointly compact. Under a similar commutation, fractional integral versions of the bilinear Hilbert transform yield separately compact operators.

Boundedness and Compactness for the Commutators of Bilinear Operators on Morrey Spaces

“…We would like to point out that the conclusion of Theorem 3.1 holds also for the iterated commutator [[I α , b 1 ] 1 , b 2 ] 2 of the bilinear fractional integral I α with the same indexes, which is defined by Recently, Bényi et al [8] proved that if 0 < α < 2n, 1 < p 1 , p 2 < ∞ with 1 p 1 + 1 p 2 > α n , 1 < q < ∞ with 1 q = 1 p 1 + 1 p 2 − α n , then for b ∈ CMO, [I α , b] i (i = 1, 2) are all compact operators from L p 1 (R n ) × L p 2 (R n ) to L q (R n ).…”

Endpoint estimates for commutators of sublinear operators in the Morrey-type spaces

“…Compactness results in the multilinear setting have just began to be studied. Bényi et al [3], [4] and [6] showed that symbols in CMO again produce compact commutators. Ding and Mei [20] consider the compactness of linear commutator of bilinear operators from product of Morrey spaces to Morrey spaces.…”

Characterization of Functions via Commutators of Bilinear Fractional Integrals on Morrey Spaces

“…Also, let CM O(R n ) be the closure in the BM O(R n ) norm of C ∞ c (R n ), which represents the space of infinitely differentiable functions with compact support. The boundedness and compactness of [b, I α ] i on variant function spaces have been the topic in many articles recently, see [1,2,3,5,6,12,13,17,18,19,26], among numerous references. One of the interesting questions on [b, I α ] i is whether it can be used to characterize BM O(R n ) by boundedness, or CM O(R n ) by compactness, as those in the linear setting (see [7,9,27] etc).…”