Products and commutators of martingales in H1 and BMO

Let f := (fn)n is an element of Z+ and g := (gn)n is an element of Z+ be two martingales related to the probability space (omega,7,P) equipped with the filtration (7n)n is an element of 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 center dot g := (fngn)n is an element of Z+ may be written as the sum f center dot g = G(f, g) +L(f, g). Here L(f, g) := (L(f,g)n)n is an element of Z+ with L(f, g)n := Enk=0(fk - fk-1)(gk - gk-1) for any n E 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 function t phi(t) := log(e + t), Vt E [0, 00).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 E 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 Cesaro means of Walsh-Fourier series.(c) 2023 Elsevier Masson SAS. All rights reserved.