--- title: Hardy Martingales url: https://www.emergentmind.com/topics/hardy-martingales type: topic --- # Hardy Martingales Hardy martingales are martingales on the product torus \( \mathbb{T}^{\mathbb{N}} \) whose increments are analytic in the last variable, equivalently martingales for which each conditioned difference belongs to \( H_0^1(\mathbb{T}) \). Introduced and developed by Garling, Maurey, and Bourgain, they occupy a central position at the interface of martingale theory, Hardy space methods, Banach space geometry, and harmonic analysis; the subject includes sharp square-function estimates, analyticity-preserving Davis-type decompositions, dyadic perturbation theory, product decompositions with BMO, and vector-valued as well as noncommutative extensions [1009.3629][1504.06513][2509.07616]. ## 1. Definition and analytic model In the standard model, one works on \( \mathbb{T}^{\mathbb{N}} \) with product Haar measure \( \mathbb{P} \) and the natural filtration \( \mathcal{F}_k \) generated by the first \( k \) coordinates. A martingale \( F=(F_k) \) is a Hardy martingale if, for each \( k \), the martingale difference \[ \Delta F_k = F_k - F_{k-1} \] defines, conditionally on \( \mathcal{F}_{k-1} \), a function in \( H_0^1(\mathbb{T}) \); in the formulation used for \( H^1(\mathbb{T}^{\mathbb{N}}) \), this means that \[ t \mapsto \Delta_n f(x_1,\ldots,x_{n-1},t) \] belongs to \( H^1(\mathbb{T}) \) for almost every fixed history \((x_1,\ldots,x_{n-1})\) [1009.3629][2509.07616]. The same analytic constraint can be expressed spectrally. For functions in the Hardy martingale space on \( \mathbb{T}^{\mathbb{N}} \), the Fourier transform is supported in the positive cone for the lexicographic order, corresponding to analytic directions in each variable. At the \(H^1\)-level, the norm is equivalent to the square-function norm \[ \|f\|_{H^1_{\mathrm{last}}} \simeq \left\| \left( \sum_k |\Delta_k f|^2 \right)^{1/2} \right\|_{L^1(\mathbb{T}^{\mathbb{N}})}, \] which already indicates that analyticity and martingale square functions are inseparable in the subject [2509.07616]. Classical \(L^1\)-bounds persist in this analytic setting. For complex-valued Hardy martingales one has \[ \| \sup_k |F_k| \|_1 \leq e \sup_k \|F_k\|_1, \] and the conditional square function \[ \Bigl( \sum_k \mathbb{E}_{k-1} |\Delta G_k|^2 \Bigr)^{1/2} \] defines the previsible norm \[ \|G\|_{\mathcal P} = \Big\| \Bigl( \sum_k \mathbb{E}_{k-1} |\Delta G_k|^2 \Bigr)^{1/2} \Big\|_{L^1}, \] which plays a structural role in transform estimates, decomposition theorems, and quotient-space arguments [1611.02653]. ## 2. Square functions, thin-thick splitting, and analytic Davis-Garsia theory A foundational theorem due to Garling states that for every Hardy martingale \(F\) there are constants \(c,C>0\) such that \[ c \,\mathbb{E}|F_n| \leq \mathbb{E}\left(\sum_{k=1}^n |\Delta F_k|^2\right)^{1/2} \leq C\,\mathbb{E}|F_n|. \] This square-function characterization already distinguishes Hardy martingales from general martingales by replacing a purely probabilistic control with one adapted to analytic increments [1009.3629]. A decisive refinement is the thin-thick decomposition. Every Hardy martingale \(F=(F_k)_{k=1}^n\) can be written as \[ F=G+B, \] where \(G\) and \(B\) are again Hardy martingales and satisfy \[ \mathbb{E} \left( \sum_{k = 1 }^n \mathbb{E}_{k-1} |\Delta G_k |^2 \right)^{1/2} + \mathbb{E} \left( \sum_{k = 1 }^n |\Delta B_k | \right) \leq C_1\mathbb{E} | F_n |, \] together with the previsible estimate \[ |\Delta G_k| \leq A_0 |F_{k-1}|, \qquad \forall k \leq n. \] The right-hand side involves only \( \mathbb{E}|F_n| \), not the square function, and the control of \( \Delta G_k \) depends only on the immediately previous value rather than on the full past maximum [1009.3629]. The analytic input for this splitting is Bourgain’s complex convexity inequality: \[ \int_{\mathbb{T}} \big(|z|^2 + \alpha_0^2 |h|^2\big)^{1/2} dm \leq \int_{\mathbb{T}} |z + h| dm \] for all \(z\in\mathbb C\) and \(h\in H_0^1(\mathbb T)\). Combined with Brownian motion and a stopping time \[ \rho = \inf\{t < \tau: |h(B_t)| > 2 \alpha_0^{-1}|z|\}, \] this yields a “pointwise” analytic splitting \(h=g+b\) with \(g\in H_0^\infty\), \(b\in H_0^1\), and estimates that globalize to the martingale decomposition \(F=G+B\) [1009.3629]. More recently, the Davis-Garsia decomposition has also been identified as an intrinsic decomposition inside the Hardy martingale space itself: \[ \mathbb{E} \left(\sum_{n=0}^\infty \left|\Delta_n f\right|^2\right)^\frac{1}{2} \simeq \inf_{f=g+h} \mathbb{E}\sum_{n=0}^\infty \left|\Delta_n g\right| + \mathbb{E}\left(\sum_{n=0}^\infty \mathbb{E}\left(\left|\Delta_n h\right|^2\mid \mathcal{F}_{n-1}\right)\right)^\frac{1}{2}, \] with \(g,h\) remaining in the Hardy martingale space. That analyticity-preserving form is the key input in the extension of Fefferman’s multiplier theorem from \(H^1(\mathbb T)\) to \(H^1(\mathbb T^{\mathbb N})\) [2509.07616]. ## 3. Vector-valued Hardy martingales and Banach-space geometry The vector-valued theory addresses a basic obstruction: the classical Davis decomposition does not respect the analytic structure when applied to Hardy martingales. For an \(X\)-valued Hardy martingale \(F=(F_k)_{k=1}^n\), where \(X\) is a complex Banach space, one has a Davis-type decomposition \[ F=G+B \] into \(X\)-valued Hardy martingales such that \[ \|\Delta G_k\|_X \le 2 \max_{m \le k-1} \|F_m\|_X \] and \[ \sum_{k=1}^n \mathbb{E}\|\Delta B_k\|_X \le C\,\mathbb{E}\|F\|_X. \] A stronger version sharpens the predictable part to \[ \|\Delta G_k\|_X \le C\|F_{k-1}\|_X, \qquad \sum_{k=1}^\infty \mathbb{E}\|\Delta B_k\|_X \le C\,\mathbb{E}\|F\|_X. \] These decompositions preserve the Hardy structure and mirror the scalar theory at the level of predictable control [1504.06513]. The proofs combine Brownian motion, stopping times, and Doob’s projection with nonlinear telescoping and maximal estimates; the refined decomposition also uses Havin’s Lemma, outer-function methods, and a vector-valued version of Bourgain’s functional inequalities for Hardy spaces. The analytic content is therefore not decorative: it is the mechanism that permits predictable control without leaving the class of Hardy martingales [1504.06513]. Under additional geometric assumptions on \(X\), the decomposition yields Davis-Garsia inequalities of \(L^q\)-type. If \(X\) satisfies hypothesis \((q)\), meaning that for any \(M\) there is \( \delta(M)>0 \) such that for \(\|x\|=1\) and \(g\in H_0^\infty(\mathbb T,X)\) with \(\|g\|_\infty\le M\), \[ \int_\mathbb T \| z + g \|_X dm \ge \left( 1 + \delta(M) \int_\mathbb T \|g\|_X^q dm \right)^{1/q}, \] then \[ \left( \sum_{k=1}^\infty \mathbb{E}_{k-1} \|\Delta G_k\|_X^q \right)^{1/q} + \sum_{k=1}^\infty \mathbb{E}\|\Delta B_k\|_X \le A_q\,\mathbb{E}\|F\|_X. \] For \(q=2\) and \(X=\mathbb C\), this recovers the classical Garsia/Davis inequalities for scalar Hardy martingales [1504.06513]. Two applications are especially prominent. First, if a martingale multiplier \(T_\varepsilon\) has an \(L^2\) bound on square-integrable Hardy martingales, the decomposition extrapolates this to \(L^1\): \[ \mathbb{E}\|T_\varepsilon F_k\|_X \le C \mathbb{E}\|F_k\|_X. \] Second, any \(X\)-valued Hardy martingale can be embedded as a subsequence into another Hardy martingale with small, predictable increments, providing an alternative route to extrapolation arguments in the style of Burkholder or Maurey [1504.06513]. ## 4. Sine-cosine decomposition, dyadic perturbation, and quotient embeddings For a martingale \(G=(G_k)\), the sine-cosine decomposition isolates the even and odd parts in the last variable. Writing \(x=(x_1,\ldots,x_{k-1})\) and \(y\in\mathbb T\), \[ U_k(x,y)=\frac{1}{2}\bigl(G_k(x,y)+G_k(x,\overline y)\bigr) \] defines the cosine part \(U=(U_k)\), while \(V_k=G_k-U_k\) defines the sine part. Then \(G=U+V\), \(U_k(x,y)=U_k(x,\overline y)\), and \(\Delta V_k(x,y)=-\Delta V_k(x,\overline y)\). In previsible norm, one has the transform identity \[ \|U\|_{\mathcal P}=\|T_W(G)\|_{\mathcal P}, \qquad T_W(G)=\Im \sum w_{k-1}\Delta_k G, \] for adapted unimodular \(W=(w_k)\) [1611.02653]. This identity remains stable under dyadic perturbation. If \( \mathcal D \) is the dyadic sigma-algebra generated by \( \sigma_k(x)=\operatorname{sign}(\Re x_k) \), then \[ \|U - \mathbb{E}(U|\mathcal D)\|_{\mathcal P} \leq C\, \|T_W(G - \mathbb{E}(G | \mathcal D))\|_{\mathcal P}^{1/2} \|G\|_{\mathcal P}^{1/2}. \] The estimate shows that the \( \mathcal P \)-norm relation between a Hardy martingale and its cosine part survives the subtraction of coarse dyadic information [1611.02653]. A related dyadic-perturbation theory for Davis-Garsia inequalities starts with a Hardy martingale \(F\) and a dyadic martingale \(D\). There exists a Hardy martingale \(G\) and a remainder \(B=F-G\) such that \[ |\Delta G_k| \leq C_0 |F_{k-1} - D_{k-1}|, \qquad \mathbb{E} \sum |\Delta B_k| \leq C \mathbb{E}|F_n - D_n|, \] and \[ \| T(G - D) \|_{\mathcal{P}} \leq C \|F - D\|_{L^1}^{1/2} \|F - D\|_{H^1}^{1/2}, \] where \[ T(H)=\Im \left[ \sum w_{k-1}\Delta H_k \right], \qquad w_{k-1}=\frac{\overline{F_{k-1}-D_{k-1}}}{|F_{k-1}-D_{k-1}|}. \] For \(D=\mathbb E_{\mathcal D}F\), this gives \[ \|B\|_{\mathcal A}\leq C\|F-\mathbb E_{\mathcal D}F\|_{L^1}. \] These are analyticity-preserving perturbative versions of the Davis-Garsia mechanism [1209.3964]. The same circle of ideas leads to geometric consequences in quotient spaces. For dyadic martingales \(D\), \[ \inf_{F\in \text{Hardy}} \|D-F\|_{L^1} \geq \delta \|D\|_{L^1}, \] and Bourgain’s embedding \(L^1\hookrightarrow L^1/H^1\) is revisited through this lower bound and the dyadic stability of transform estimates. Both the dyadic perturbation theory and the sine-cosine stability estimate are used in the proof that \(L^1\) embeds into \(L^1/H^1\) [1209.3964][1611.02653]. ## 5. Products with dual spaces, paraproducts, and endpoint commutators The product of a Hardy-space martingale with an element of its dual does not generally remain in the same Hardy class, and its correct range is described by paraproduct decompositions. For \(f\in H^1(\Omega)\) and \(g\in \mathrm{BMO}(\Omega)\), there exist continuous bilinear operators \[ \Pi_1: H^1(\Omega)\times \mathrm{BMO}(\Omega)\to L^1(\Omega),\qquad \Pi_2: H^1(\Omega)\times \mathrm{BMO}(\Omega)\to H^1(\Omega), \] \[ \Pi_3: H^1(\Omega)\times \mathrm{BMO}(\Omega)\to H_\Phi^1(\Omega), \qquad \Phi(t)=\frac{t}{\log(e+t)}, \] such that \[ f\cdot g=\Pi_1(f,g)+\Pi_2(f,g)+\Pi_3(f,g). \] For finite martingale expansions, \[ \Pi_1(f,g)=\sum_k d_kf\,d_kg,\qquad \Pi_2(f,g)=\sum_k f_{k-1}d_kg,\qquad \Pi_3(f,g)=\sum_k g_{k-1}d_kf, \] and analogous decompositions hold for \(0