An extension to BMO functions of some product properties of Hilbert transforms (Q1096836)

Let \(b\in BMO\) and \(f\in L^ p\), \(1<p<\infty\). The symbol H denotes the Hilbert transform: \[ HF(x)=\lim_{\epsilon \to 0,r\to \infty}\int_{0<\epsilon <| x-t| <r}\frac{F(t)}{\pi (x-t)}dt. \] The dual Hilbert transform on BMO according to the H-BMO duality is denoted by H': \(<H'b,h>=-<b,Hh>,\forall b\in BMO\), \(\forall h\in H^ 1\). The paper verifies an identity \(H'b.Hf-bf=H(b.Hf)+H(f.H'b).\) The proof uses distributional convolutions with vp(1/x) and a result of \textit{R. R. Coifman}, \textit{R. Rochberg}, and \textit{G. Weiss}; [Ann. Math., II. Ser. 103, 611-635 (1976; Zbl 0326.32011)].