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

TL;DR

This paper characterizes BMO space through endpoint bounds of commutators of singular integrals, establishing equivalences between BMO norms and bounds of these commutators in various dimensions and for different operators.

Contribution

It provides new endpoint boundedness characterizations of BMO via commutators of singular integrals, extending to higher dimensions and more general operators.

Findings

BMO norm is equivalent to the best constant in endpoint estimates for commutators.

Characterizations extend to higher order commutators of Hilbert transform and Riesz transforms.

Results apply to a broad class of convolution-type singular integral operators.

Abstract

We provide a characterization of $\mathrm{BMO}$ in terms of endpoint boundedness of commutators of singular integrals. In particular, in one dimension, we show that $\|b\|_{\mathrm{BMO}}\eqsim B$, where $B$ is the best constant in the endpoint $L\log L$ modular estimate for the commutator $[H,b]$. We provide a similar characterization of the space $\mathrm{BMO}$ in terms of endpoint boundedness of higher order commutators of the Hilbert transform. In higher dimension we give the corresponding characterization of $\mathrm{BMO}$ in terms of the first order commutators of the Riesz transforms. We also show that these characterizations can be given in terms of commutators of more general singular integral operators of convolution type.