A decomposition for Hardy martingales. II (Q2927880)

This article is a continuation of the author's earlier work [Indiana Univ. Math. J. 61, No. 5, 1801--1816 (2012; Zbl 1288.60053)] concerning Davis and Garsia inequalities (DGI) for Hardy martingales on the infinite dimensional torus \(\mathbb{T}^\mathbb{N}\) which are adapted to the filtration \((\mathcal{F}_k)\) generated by cylinder sets. In the above mentioned article it has been shown that every Hardy martingale \(F = (F_k)\) can be written as a sum of two Hardy martingales \(G = (G_k)\) and \(B = (B_k)\) so that the following DGI for the martingale differences \(\Delta H_k = H_k - H_{k-1}\) are satisfied: NEWLINE\[NEWLINE \mathbb{E} \Big( \sum_{k=1}^n \mathbb{E}_{\mathcal{F}_{k-1}} | \Delta G_k |^2 \Big)^{1/2} + \mathbb{E} \Big( \sum_{k=1}^n | \Delta B_k | \Big) \leq C \mathbb{E} \Big( \sum_{k=1}^n | \Delta F_k |^2 \Big)^{1/2} NEWLINE\]NEWLINE (or, more compactly, \(\| G \|_{\mathcal{P}} + \| B \|_{\mathcal{A}} \leq C \| F \|_{H^1}\)), and NEWLINE\[NEWLINE | \Delta G_k | \leq C |F_{k-1}| . NEWLINE\]NEWLINENEWLINENEWLINEThe main result of this article asserts that dyadic perturbations of Hardy martingales (which need not remain Hardy martingales) continue to satisfy DGI in the form of a geometric mean. More precisely, given a Hardy martingale \(F = (F_k)\) and a dyadic martingale \(D = (D_k)\), consider the associated martingale transform NEWLINE\[NEWLINE T(H) = \Im \Big[ \sum_{k=1}^n \frac{\overline{F_{k-1} - D_{k-1}}}{|F_{k-1} - D_{k-1}|} \Delta H_k \Big] . NEWLINE\]NEWLINE The stability of DGI under dyadic perturbations is formulated as follows: for every Hardy martingale \(F\) and dyadic martingale \(D\) there exist Hardy martingales \(G\) and \(B\) so that \(F = G+B\), and NEWLINE\[NEWLINE \| T(G-D) \|_\mathcal{P} \leq C \| F-D \|_{L^1}^{1/2} \| F-D \|_{H^1}^{1/2} , NEWLINE\]NEWLINE NEWLINE\[NEWLINE \| B \|_\mathcal{A} \leq C \| F-D \|_{L^1}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \| G \|_{\mathcal{P}} \leq C \| F \|_{L^1} + C \| D \|_{H^1} . NEWLINE\]NEWLINENEWLINENEWLINEThe main application of this result arises when \(D\) is the conditional expectation of \(F\) with respect to the dyadic \(\sigma\)-algebra. It concerns the embedding theorem of \textit{J. Bourgain} [Trans. Am. Math. Soc. 278, 689--702 (1983; Zbl 0517.46042)], which states that \(L^1(\mathbb{T})\) is isomorphic to a subspace of \(L^1(\mathbb{T})/H_0^1(\mathbb{T})\), where \(H^1_0(\mathbb{T})\) consists of integrable functions on \(\mathbb{T}\) with vanishing mean for which the harmonic extension to the unit disk is analytic.