Compact bilinear commutators: The weighted case

Proceedings of the American Mathematical Society, 2013

A notion of compactness in the bilinear setting is explored. Moreover, commutators of bilinear Calderón-Zygmund operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be compact.

Proceedings of the American Mathematical Society, 2018

Potential Analysis, 2015

The compactness of the commutators of bilinear fractional integral operators and point-wise multiplication, acting on products of Lebesgue spaces, is characterized in terms of appropriate mean oscillation properties of their symbols. The compactness of the commutators when acting on product of weighted Lebesgue spaces is also studied.

Journal of Function Spaces, 2014

The boundedness of multilinear commutators of Calderón-Zygmund operatorTb→on generalized weighted Morrey spacesMp,φ(w)with the weight functionwbelonging to Muckenhoupt's classApis studied. When1<p<∞andb→=(b1,…,bm),bi∈BMO,i=1,…,m, the sufficient conditions on the pair(φ1,φ2)which ensure the boundedness of the operatorTb→fromMp,φ1(w)toMp,φ2(w)are found. In all cases the conditions for the boundedness ofTb→are given in terms of Zygmund-type integral inequalities on(φ1,φ2), which do not assume any assumption on monotonicity ofφ1(x,r), φ2(x,r)inr.

Archiv der Mathematik

Mathematische Zeitschrift, 2015

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.

Bulletin of Mathematical Sciences

In this paper, we obtain the necessary and sufficient conditions for the weak/strong boundedness of the Calderón–Zygmund operators in generalized weighted Orlicz–Morrey spaces. We also study the boundedness of the commutators of Calderón–Zygmund operators on these spaces. Moreover, the boundedness of Calderón–Zygmund operators in the vector-valued setting is given.

Czechoslovak Mathematical Journal, 2014

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Advances in Mathematics

In recent years, it has been well understood that a Calderón-Zygmund operator T is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse operators). We obtain a similar pointwise estimate for the commutator [b, T ] with a locally integrable function b. This result is applied into two directions. If b ∈ BM O, we improve several weighted weak type bounds for [b, T ]. If b belongs to the weighted BM O, we obtain a quantitative form of the two-weighted bound for [b, T ] due to Bloom-Holmes-Lacey-Wick.

… , Integration and Related …, 2010