A characterization of BMO in terms of endpoint bounds for commutators of singular integrals

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

“…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].…”

“…We also point out that this endpoint characterisation in Theorem 1.3 above is sharp since following the method in [24] in the Euclidean setting and using the lower bound in Theorem 1.1, it is easy to construct a function b ∈ BMO(G) such that [b, R j ] fails to be weak type (1,1).…”

“…Calderón [4]). It is well-known that the commutator of the Hilbert transform has the following properties: , where A p denotes the Muckenhoupt weights ( [2], [15]); c) [b, H] is bounded from L log + L(R) to L 1,∞ (R) if and only if b ∈ BMO(R) ( [24], [13], [1]); d) [b, H] is bounded from H 1 (R) to L 1 (R) if and only if b equals a constant almost everywhere ( [14]); e) [b, H] is bounded from L ∞ c (R) to BMO(R) if and only if b equals a constant almost everywhere ( [14]); f) [b, H] is of weak type (1, 1) if and only if b is in L ∞ (R) ( [1]).…”

“…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…”

“…Third, besides the weak type (1,1), it is natural to study the endpoint estimates like H 1 (G) to L 1 (G) and the L ∞ (G) to BMO(G). We also have the following characterisations.…”