Products and commutators of martingales in H1 and BMO

Abstract

Let f:=(fn)n∈Z+ and g:=(gn)n∈Z+ be two martingales related to the probability space (Ω,F,P) equipped with the filtration (Fn)n∈Z+. Assume that f is in the martingale Hardy space H1 and g is in its dual space, namely the martingale BMO. Then the semi-martingale f⋅g:=(fngn)n∈Z+ may be written as the sumf⋅g=G(f,g)+L(f,g). Here L(f,g):=(L(f,g)n)n∈Z+ with L(f,g)n:=∑k=0n(fk−fk−1)(gk−gk−1) for any n∈Z+, where f−1:=0=:g−1. The authors prove that L(f,g) is a process with bounded variation and limit in L1, while G(f,g) belongs to the martingale Hardy-Orlicz space Hlog associated with the Orlicz functionΦ(t):=tlog⁡(e+t),∀t∈[0,∞). The above bilinear decomposition L1+Hlog is sharp in the sense that, for particular martingales, the space L1+Hlog cannot be replaced by a smaller space having a larger dual. As an application, the authors characterize the largest subspace of H1, denoted by H1b with b∈BMO, such that the commutators [T,b] with classical sublinear operators T are bounded from H1b to L1. This endpoint boundedness of commutators allows the authors to give more applications. On the one hand, in the martingale setting, the authors obtain the endpoint estimates of commutators for both martingale transforms and martingale fractional integrals. On the other hand, in harmonic analysis, the authors establish the endpoint estimates of commutators both for the dyadic Hilbert transform beyond doubling measures and for the maximal operator of Cesàro means of Walsh–Fourier series.