Hardy Martingales

• Hardy martingales are martingales on the product torus with analytic increments, defined via their conditional differences lying in H₀¹ and supported in the positive Fourier cone.
• They utilize sharp square-function estimates, thin-thick and Davis-type decompositions, along with dyadic perturbation techniques to analyze martingale behavior.
• Extensions cover vector-valued cases, Banach space geometry, and endpoint commutator analyses, bridging classical harmonic analysis with noncommutative and dual space frameworks.

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 (Mueller, 2010, Müller, 2015, Rzeszut, 9 Sep 2025).

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})$0; in the formulation used for $H_0^1(\mathbb{T})$1, this means that

$H_0^1(\mathbb{T})$2

belongs to $H_0^1(\mathbb{T})$3 for almost every fixed history $H_0^1(\mathbb{T})$4 (Mueller, 2010, Rzeszut, 9 Sep 2025).

The same analytic constraint can be expressed spectrally. For functions in the Hardy martingale space on $H_0^1(\mathbb{T})$5, the Fourier transform is supported in the positive cone for the lexicographic order, corresponding to analytic directions in each variable. At the $H_0^1(\mathbb{T})$6-level, the norm is equivalent to the square-function norm

$H_0^1(\mathbb{T})$7

which already indicates that analyticity and martingale square functions are inseparable in the subject (Rzeszut, 9 Sep 2025).

Classical $H_0^1(\mathbb{T})$8-bounds persist in this analytic setting. For complex-valued Hardy martingales one has

$H_0^1(\mathbb{T})$9

and the conditional square function

$\mathbb{T}^{\mathbb{N}}$0

defines the previsible norm

$\mathbb{T}^{\mathbb{N}}$1

which plays a structural role in transform estimates, decomposition theorems, and quotient-space arguments (Müller, 2016).

2. Square functions, thin-thick splitting, and analytic Davis-Garsia theory

A foundational theorem due to Garling states that for every Hardy martingale $\mathbb{T}^{\mathbb{N}}$2 there are constants $\mathbb{T}^{\mathbb{N}}$3 such that

$\mathbb{T}^{\mathbb{N}}$4

This square-function characterization already distinguishes Hardy martingales from general martingales by replacing a purely probabilistic control with one adapted to analytic increments (Mueller, 2010).

A decisive refinement is the thin-thick decomposition. Every Hardy martingale $\mathbb{T}^{\mathbb{N}}$5 can be written as

$\mathbb{T}^{\mathbb{N}}$6

where $\mathbb{T}^{\mathbb{N}}$7 and $\mathbb{T}^{\mathbb{N}}$8 are again Hardy martingales and satisfy

$\mathbb{T}^{\mathbb{N}}$9

together with the previsible estimate

$\mathbb{P}$0

The right-hand side involves only $\mathbb{P}$1, not the square function, and the control of $\mathbb{P}$2 depends only on the immediately previous value rather than on the full past maximum (Mueller, 2010).

The analytic input for this splitting is Bourgain’s complex convexity inequality: $\mathbb{P}$3 for all $\mathbb{P}$4 and $\mathbb{P}$5. Combined with Brownian motion and a stopping time

$\mathbb{P}$6

this yields a “pointwise” analytic splitting $\mathbb{P}$7 with $\mathbb{P}$8, $\mathbb{P}$9, and estimates that globalize to the martingale decomposition $\mathcal{F}_k$0 (Mueller, 2010).

More recently, the Davis-Garsia decomposition has also been identified as an intrinsic decomposition inside the Hardy martingale space itself: $\mathcal{F}_k$1 with $\mathcal{F}_k$2 remaining in the Hardy martingale space. That analyticity-preserving form is the key input in the extension of Fefferman’s multiplier theorem from $\mathcal{F}_k$3 to $\mathcal{F}_k$4 (Rzeszut, 9 Sep 2025).

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 $\mathcal{F}_k$5-valued Hardy martingale $\mathcal{F}_k$6, where $\mathcal{F}_k$7 is a complex Banach space, one has a Davis-type decomposition

$\mathcal{F}_k$8

into $\mathcal{F}_k$9-valued Hardy martingales such that

$k$0

and

$k$1

A stronger version sharpens the predictable part to

$k$2

These decompositions preserve the Hardy structure and mirror the scalar theory at the level of predictable control (Müller, 2015).

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 (Müller, 2015).

Under additional geometric assumptions on $k$3, the decomposition yields Davis-Garsia inequalities of $k$4-type. If $k$5 satisfies hypothesis $k$6, meaning that for any $k$7 there is $k$8 such that for $k$9 and $F=(F_k)$0 with $F=(F_k)$1,

$F=(F_k)$2

then

$F=(F_k)$3

For $F=(F_k)$4 and $F=(F_k)$5, this recovers the classical Garsia/Davis inequalities for scalar Hardy martingales (Müller, 2015).

Two applications are especially prominent. First, if a martingale multiplier $F=(F_k)$6 has an $F=(F_k)$7 bound on square-integrable Hardy martingales, the decomposition extrapolates this to $F=(F_k)$8: $F=(F_k)$9 Second, any $k$0-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 (Müller, 2015).

4. Sine-cosine decomposition, dyadic perturbation, and quotient embeddings

For a martingale $k$1, the sine-cosine decomposition isolates the even and odd parts in the last variable. Writing $k$2 and $k$3,

$k$4

defines the cosine part $k$5, while $k$6 defines the sine part. Then $k$7, $k$8, and $k$9. In previsible norm, one has the transform identity

$\Delta F_k = F_k - F_{k-1}$0

for adapted unimodular $\Delta F_k = F_k - F_{k-1}$1 (Müller, 2016).

This identity remains stable under dyadic perturbation. If $\Delta F_k = F_k - F_{k-1}$2 is the dyadic sigma-algebra generated by $\Delta F_k = F_k - F_{k-1}$3, then

$\Delta F_k = F_k - F_{k-1}$4

The estimate shows that the $\Delta F_k = F_k - F_{k-1}$5-norm relation between a Hardy martingale and its cosine part survives the subtraction of coarse dyadic information (Müller, 2016).

A related dyadic-perturbation theory for Davis-Garsia inequalities starts with a Hardy martingale $\Delta F_k = F_k - F_{k-1}$6 and a dyadic martingale $\Delta F_k = F_k - F_{k-1}$7. There exists a Hardy martingale $\Delta F_k = F_k - F_{k-1}$8 and a remainder $\Delta F_k = F_k - F_{k-1}$9 such that

$\mathcal{F}_{k-1}$0

and

$\mathcal{F}_{k-1}$1

where

$\mathcal{F}_{k-1}$2

For $\mathcal{F}_{k-1}$3, this gives

$\mathcal{F}_{k-1}$4

These are analyticity-preserving perturbative versions of the Davis-Garsia mechanism (Müller, 2012).

The same circle of ideas leads to geometric consequences in quotient spaces. For dyadic martingales $\mathcal{F}_{k-1}$5,

$\mathcal{F}_{k-1}$6

and Bourgain’s embedding $\mathcal{F}_{k-1}$7 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 $\mathcal{F}_{k-1}$8 embeds into $\mathcal{F}_{k-1}$9 (Müller, 2012, Müller, 2016).

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 $H_0^1(\mathbb{T})$00 and $H_0^1(\mathbb{T})$01, there exist continuous bilinear operators

$H_0^1(\mathbb{T})$02

$H_0^1(\mathbb{T})$03

such that

$H_0^1(\mathbb{T})$04

For finite martingale expansions,

$H_0^1(\mathbb{T})$05

and analogous decompositions hold for $H_0^1(\mathbb{T})$06 with the martingale Lipschitz/Campanato dual space $H_0^1(\mathbb{T})$07, where $H_0^1(\mathbb{T})$08 (Bakas et al., 2023).

A sharper endpoint decomposition isolates bounded variation. For $H_0^1(\mathbb{T})$09 and $H_0^1(\mathbb{T})$10,

$H_0^1(\mathbb{T})$11

where

$H_0^1(\mathbb{T})$12

is a process with bounded variation and limit in $H_0^1(\mathbb{T})$13, while $H_0^1(\mathbb{T})$14 belongs to the martingale Hardy-Orlicz space $H_0^1(\mathbb{T})$15 associated with

$H_0^1(\mathbb{T})$16

The range $H_0^1(\mathbb{T})$17 is sharp: for particular martingales, it cannot be replaced by a smaller space having a larger dual (Bonami et al., 2023).

This endpoint structure controls commutators. For $H_0^1(\mathbb{T})$18, the largest subspace $H_0^1(\mathbb{T})$19 for which the commutators

$H_0^1(\mathbb{T})$20

are bounded from $H_0^1(\mathbb{T})$21 to $H_0^1(\mathbb{T})$22 is characterized by

$H_0^1(\mathbb{T})$23

and equivalently by

$H_0^1(\mathbb{T})$24

For operators in the class $H_0^1(\mathbb{T})$25, one has

$H_0^1(\mathbb{T})$26

The relevant operator class includes martingale transforms, martingale fractional integrals, the Doob maximal operator, the martingale square function, the dyadic Hilbert transform, and the maximal operator of Césaro means for Walsh–Fourier series (Bonami et al., 2023).

6. Extensions, related frameworks, and recent directions

Several adjacent theories extend the Hardy-martingale toolkit beyond the analytic torus model. Some of these concern martingale Hardy spaces rather than Hardy martingales in the strict sense, but they preserve the same structural themes: square functions, atomic decompositions, interpolation, perturbation stability, and endpoint duality.

Framework	Representative result	Source
Non-homogeneous martingale Hardy spaces	$H_0^1(\mathbb{T})$27, proved via the relation between Carleson sequences and balayage	(Lai, 2014)
Variable and mixed exponents	$H_0^1(\mathbb{T})$28 admits a $H_0^1(\mathbb{T})$29-atomic decomposition; five mixed spaces $H_0^1(\mathbb{T})$30 have atomic decompositions and satisfy a mixed Burkholder-Davis-Gundy inequality	(Jiao et al., 2014, Szarvas et al., 2019)
Noncommutative martingale Hardy spaces	Constructive atomic decomposition for $H_0^1(\mathbb{T})$31, $H_0^1(\mathbb{T})$32; atomic block decomposition for arbitrary filtrations; K-closedness and real interpolation for $H_0^1(\mathbb{T})$33; square-function inequalities for weakly differentially subordinate martingales	(Chen et al., 2020, Conde-Alonso et al., 2014, Moyart, 2024, Jiao et al., 2019)
Fractional integration and irregular filtrations	For arbitrary filtrations, $H_0^1(\mathbb{T})$34 is bounded when $H_0^1(\mathbb{T})$35, and $H_0^1(\mathbb{T})$36 shares the same $H_0^1(\mathbb{T})$37 boundedness	(Stolyarov et al., 2020)
Multipliers, interpolation, and partial sums	Fefferman’s multiplier theorem extends to $H_0^1(\mathbb{T})$38; for holomorphic random variables, new truncation formulae yield Marcinkiewicz decompositions and $H_0^1(\mathbb{T})$39; on Vilenkin systems, subsequences of partial sums on $H_0^1(\mathbb{T})$40, $H_0^1(\mathbb{T})$41, satisfy sharp boundedness and convergence criteria	(Rzeszut, 9 Sep 2025, Müller et al., 2016, Tephnadze, 2018)

Within noncommutative martingale Hardy spaces, several structural facts parallel the classical analytic theory. For $H_0^1(\mathbb{T})$42, $H_0^1(\mathbb{T})$43 admits an explicit constructive decomposition into algebraic atoms, and for $H_0^1(\mathbb{T})$44 the $H_0^1(\mathbb{T})$45-atomic spaces coincide for all $H_0^1(\mathbb{T})$46 (Chen et al., 2020). For arbitrary filtrations, $H_0^1(\mathbb{T})$47 and its noncommutative analogue admit atomic block decompositions,

$H_0^1(\mathbb{T})$48

which replace classical atomic decompositions that fail outside the regular setting (Conde-Alonso et al., 2014). Real interpolation also persists: for $H_0^1(\mathbb{T})$49, $H_0^1(\mathbb{T})$50 is K-complemented in $H_0^1(\mathbb{T})$51, and

$H_0^1(\mathbb{T})$52

with equivalent norms (Moyart, 2024).

This suggests that the defining analytic increment condition of Hardy martingales is compatible with a broad transfer apparatus: square-function equivalences, atomic and block decompositions, interpolation, multiplier theory, and endpoint product estimates all survive, but typically only after the theory is reformulated so that analyticity, predictability, or noncommutative support conditions are preserved at every stage.