An extension to BMO functions of some product properties of Hilbert transforms

AbstractLet b ϵ BMO, ƒ ϵ Lp, 1 < p < ∞, H′b denote the dual Hilbert transform of b defined on BMO according to the H1-BMO duality and H be the Hilbert transform defined as the pointwise limit almost everywhere of appropriate truncated integrals. We prove that H′b.Hƒ − bƒ = H(bHƒ) + H(ƒH′b) a.e. The proof uses distributional convolutions with vp(1x) and a result of R. R. Coifman, R. Rochberg, and G. Weiss on the Lp-boundedness of some commutators