The Unified Theory for the Necessity of Bounded Commutators and Applications

Abstract

The general methods which are powerful for the necessity of bounded commutators are given. As applications, some necessary conditions for bounded commutators are first obtained in certain endpoint cases, and several new characterizations of BM O spaces, Lipschitz spaces and their weighted versions via boundedness of commutators in various function spaces are deduced.2010 Mathematics Subject Classification. 42B20; 42B25.

Lower bound of Riesz transform kernels and commutator theorems on stratified nilpotent Lie groups

“…It is well-known that the commutator of the Hilbert transform has the following properties: a) [ For more details, we refer to the references listed above. We also point out that there are quite a number of recent results on the characterisations of commutators in the above forms for singular integrals in different settings, see for example [10,22,9,21,25,24,15,19,13,8,1]. Inspired by these classical results above, it is natural to ask whether these results hold on the Heisenberg group H n .…”

“…For the sufficiency, we point out that it follows directly from [24] with only minor changes since for this part we only need to use the upper bound of the Riesz transform, which satisfies the standard size and smoothness condition of Calderón-Zygmund type. For the necessary part, we use the idea of Uchiyama [27] and the technique that has been further explored and studied in [13] and [1]. To be more specific, we write |[b, R j ](f )(g)| ≥ |R j (bf )(g)| − |b(g)| |R j (f )(g)|, and then by choosing a suitable function f that is closely related to b and with cancellation condition and by making good use of the lower bound in Theorem 1.1, we show that |R j (bf )(g)| is the main term and |b(g)| |R j (f )(g)| acts as the "error" term due to the cancellation of f and the smoothness condition of the kernel of R j .…”

“…In [16] and [13], they considered the class of rough homogeneous singular integrals, which is of the form T (f )(x) = R n Ω(x − y)|x − y| −n f (y)dy, and obtained the characterisation of weighted BMO space via the two weight commutator of T , by assuming certain conditions on the homogeneous function Ω(x) of degree zero. The proof of the lower bound depends on the specific form of the kernel and the assumptions on Ω.…”

Commutators of singular integrals revisited

“…Very recently we have learned that after finishing this paper some results concerning the necessity of BM O for the endpoint estimate of commutators have been obtained. We remit the interested reader to [1,9] for those results. In an even more recent work [14], a more general version of the second part of Theorem 1.1 has been obtained, answering positively a question posed in Remark 4.1.…”

Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces

“…Moreover, Uchiyama showed that [ b , T ] is compact on for any p ∈ (1, ∞ ) if and only if , which is the BMO ‐closure of , the set of all infinitely differentiable functions on with compact supports. Since then, there have been a lot of articles concerning the boundedness and the compactness of commutators on function spaces as well as their applications in PDEs (see, for instance, other studies) and references therein). In particular, Di Fazio and Ragusa in 1991 gave a characterization of the boundedness of [ b , T ] on the Morrey space for any λ ∈ (0, n ) and p ∈ (1, ∞ ).…”