Weighted estimates for commutators of linear operators

Bilinear decompositions and commutators of singular integral operators

“…Unlike the theory of Calderón-Zygmund operators, the proof of this result does not rely on a weak type (1, 1) estimate for [b, T ]. In fact, it was shown in [38] that, in general, the linear commutator fails to be of weak type (1,1), when b is in BMO(R n ). Instead, an endpoint theory was provided for this operator.…”

“…We also study the commutators [b, T ] for T in a class K of sublinear operators containing almost all important operators in harmonic analysis. When T is linear, we prove that there exists a bilinear operators R = R T mapping continuously H 1 (R n ) × BM O(R n ) into L 1 (R n ) such that for all (f, b) ∈ H 1 (R n ) × BM O(R n ), we have (1) […”

“…The class K. The purpose of this subsection is to give some examples of operators in the class K. More precisely, the class K contains almost all important operators in Harmonic analysis: Calderón-Zygmund type operators, strongly singular integral operators, multiplier operators, pseudo-differential operators with symbols in the Hörmander class S m ̺,δ with 0 < ̺ ≤ 1, 0 ≤ δ < 1, m ≤ −n((1 − ̺)/2 + max{0, (δ − ̺)/2}) (see [2,1]), maximal type operators, the area integral operator of Lusin, Littlewood-Paley type operators, Marcinkiewicz operators, maximal Bochner-Riesz operators T δ * with δ > (n − 1)/2 (cf. [24]), etc...…”

On the commutators of singular integrals related to block spaces

“…( [6,2,10]) Let Ω, a, k be as above and h ≡ 1, 1 < p < ∞. If Ω ∈ ∪ q>1 L q (S n−1 ), then T a,k is bounded on L p (R n ).…”

“…Afterwards, by a well-known result of Duoandikoetxea [6] and the boundedness criterion of Alvarez-Bagby-KurtzPérez for the commutators of linear operator (see [2]), we have obtained the following theorem (see also [10]):…”