Boundedness results for commutators with BMO functions via weighted estimates: a comprehensive approach

Abstract

We present a unified method to obtain unweighted and weighted estimates of linear and multilinear commutators with BMO functions, that is amenable to a plethora of operators and functional settings. Our approach elaborates on a commonly used Cauchy integral trick, recovering many known results but yielding also numerous new ones. In particular, we solve a problem about the boundedness of the commutators of the bilinear Hilbert transform with functions in BMO.

A revisit on the compactness of commutators

“…Moreover, let d 0 be as in Proposition 4.4 and d 1 ≥ d 0 . Since b satisfies (4.7), conditions (1) and 2, we may take a subsequence of {Q j } ∞ j=1 , also denoted by {Q j } ∞ j=1 , such that for each j, |Q j | ∼ 1 and…”

A Survey on Function Spaces of John–Nirenberg Type

“…As an application of (4), we find that for any given [42], p. 125, Corollary 6.12). Recently, Bényi et al [44] gave a comprehensive approach for the boundedness of weighted commutators via a new equivalent Orlicz-type norm…”

End-point estimates, extrapolation for multilinear Muckenhoupt classes, and applications

“…Finally, our method allows us to deal with commutators of the previous operators with BMO functions. In [27, Section 2.6], the method developed in [5], was further pushed to obtain [27,Theorem 2.22]. The latter in conjunction with Theorem 2.1 easily yields the following extension where both in the hypotheses and the conclusions one can include the end-points: Corollary 3.3.…”