Davis–Garsia Decomposition

• Davis–Garsia Decomposition is a principle that splits a Hardy martingale into a predictable analytic part and an L1-summable remainder.
• It employs complex-analytic truncation to preserve the analyticity of increments while ensuring controlled quadratic variation and summability.
• The method underpins key inequalities and extrapolation results in scalar, vector-valued, and noncommutative Hardy martingale theories.

Searching arXiv for relevant Davis–Garsia decomposition papers. arXiv search query: "Davis Garsia decomposition Hardy martingales noncommutative biparameter" The Davis–Garsia decomposition is a martingale splitting principle for Hardy martingales in which a given martingale $F$ is written as $F=G+B$, with $G$ carrying the analytically structured, predictably controlled part of the process and $B$ carrying an $L^1$-summable remainder. In the Hardy-martingale setting, the decomposition is distinguished from the general Davis decomposition by preserving analyticity of the increments, and it serves as the structural input for Davis–Garsia inequalities, extrapolation of martingale transforms, vector-valued extensions, dyadic perturbation estimates, noncommutative analogues, and more recent multiparameter variants (Müller, 2015).

1. Analytic setting and basic definitions

Let $\mathbb T=\{e^{i\theta}:\theta\in[0,2\pi)\}$ be the unit circle with normalized arc-length measure $m$. For a complex Banach space $X$ and $1\le p\le\infty$,

$$L^p_0(\mathbb T,X)=\Bigl\{f\in L^p(\mathbb T,X):\int_{\mathbb T}f(\zeta)\,dm(\zeta)=0\Bigr\},$$

and

$F=G+B$0

On $F=G+B$1 with product Haar measure, let $F=G+B$2. An $F=G+B$3-valued martingale $F=G+B$4 satisfies $F=G+B$5. It is a vector-valued Hardy martingale if, for each $F=G+B$6 and each $F=G+B$7-atom $F=G+B$8, the increment $F=G+B$9, viewed as a function of the $G$0th coordinate, belongs to $G$1. In the scalar case $G$2, one writes $G$3 with $G$4 (Müller, 2015).

For scalar martingales on $G$5, the norm structure used in the decomposition theory includes

$G$6

$G$7

In the same setting, a dyadic martingale is one whose coordinates are measurable with respect to the $G$8-algebra generated by the Rademacher functions $G$9 (Müller, 2012).

2. Scalar Davis–Garsia decomposition for Hardy martingales

For scalar Hardy martingales, the decomposition theorem yields Hardy martingales $B$0 and $B$1 such that

$B$2

with increment control

$B$3

and the conditional $B$4-estimate

$B$5

Consequently,

$B$6

This form makes explicit the two complementary roles of the summands: $B$7 has predictable quadratic control, while $B$8 has summable increments in the $B$9-norm (Müller, 2012).

The proof is based on complex-analytic truncation. After freezing the past $L^1$0, one regards $L^1$1 as an $L^1$2-function $L^1$3. One then stops complex Brownian motion on the unit disk when $L^1$4 exceeds $L^1$5, and defines a truncated increment

$L^1$6

This yields $L^1$7, $L^1$8, and the one-variable estimate

$L^1$9

After conditioning and iteration over $\mathbb T=\{e^{i\theta}:\theta\in[0,2\pi)\}$0, the increment bound for $\mathbb T=\{e^{i\theta}:\theta\in[0,2\pi)\}$1 and the telescoping estimate for $\mathbb T=\{e^{i\theta}:\theta\in[0,2\pi)\}$2 follow. The analyticity of the increments is preserved throughout because the truncation is carried out inside the Hardy class (Müller, 2012).

3. Vector-valued and strong forms

For general $\mathbb T=\{e^{i\theta}:\theta\in[0,2\pi)\}$3-valued martingales, the original Davis decomposition guarantees

$\mathbb T=\{e^{i\theta}:\theta\in[0,2\pi)\}$4

In the Hardy-martingale setting, that construction is insufficient because it does not preserve analyticity. The vector-valued Hardy-martingale version remedies this by producing Hardy martingales $\mathbb T=\{e^{i\theta}:\theta\in[0,2\pi)\}$5 and $\mathbb T=\{e^{i\theta}:\theta\in[0,2\pi)\}$6 such that

$\mathbb T=\{e^{i\theta}:\theta\in[0,2\pi)\}$7

and, for $\mathbb T=\{e^{i\theta}:\theta\in[0,2\pi)\}$8,

$\mathbb T=\{e^{i\theta}:\theta\in[0,2\pi)\}$9

The construction proceeds by conditioning on $m$0, freezing the first $m$1 coordinates, and splitting the Hardy increment $m$2 as $m$3, with $m$4 and $m$5 small; Doob’s projection in the last coordinate preserves analyticity and produces $m$6, $m$7 (Müller, 2015).

A stronger version replaces the running maximum by the immediately preceding value. Under the same hypotheses, one can split $m$8 into Hardy martingales such that

$m$9

where $X$0, equivalently the $X$1-norm of the terminal value. The key analytic input is a Havin-type splitting lemma: for every $X$2 and every $X$3, there exists $X$4 such that

$X$5

and

$X$6

Setting $X$7 and $X$8, the integral inequality telescopes in $X$9. This suggests that the strong decomposition is not merely a truncation statement but an analytic factorization principle tied to the geometry of $1\le p\le\infty$0 (Müller, 2015).

4. Passage to Davis–Garsia inequalities and extrapolation

Once a decomposition

$1\le p\le\infty$1

is available with

$1\le p\le\infty$2

the nonlinear telescoping device of Bourgain–Garsia–Wilson yields

$1\le p\le\infty$3

In the scalar case this is the Davis–Garsia inequality. In the vector-valued setting, under the additional geometric hypothesis on $1\le p\le\infty$4 called property $1\le p\le\infty$5, one obtains

$1\le p\le\infty$6

for $1\le p\le\infty$7 (Müller, 2015).

The same decomposition underlies extrapolation of martingale transforms. For fixed signs $1\le p\le\infty$8, define

$1\le p\le\infty$9

If square-integrable Hardy martingales $$L^p_0(\mathbb T,X)=\Bigl\{f\in L^p(\mathbb T,X):\int_{\mathbb T}f(\zeta)\,dm(\zeta)=0\Bigr\},$$0 satisfy an $$L^p_0(\mathbb T,X)=\Bigl\{f\in L^p(\mathbb T,X):\int_{\mathbb T}f(\zeta)\,dm(\zeta)=0\Bigr\},$$1-bound

$$L^p_0(\mathbb T,X)=\Bigl\{f\in L^p(\mathbb T,X):\int_{\mathbb T}f(\zeta)\,dm(\zeta)=0\Bigr\},$$2

then every integrable Hardy martingale $$L^p_0(\mathbb T,X)=\Bigl\{f\in L^p(\mathbb T,X):\int_{\mathbb T}f(\zeta)\,dm(\zeta)=0\Bigr\},$$3 satisfies

$$L^p_0(\mathbb T,X)=\Bigl\{f\in L^p(\mathbb T,X):\int_{\mathbb T}f(\zeta)\,dm(\zeta)=0\Bigr\},$$4

The decomposition is therefore the mechanism that transfers $$L^p_0(\mathbb T,X)=\Bigl\{f\in L^p(\mathbb T,X):\int_{\mathbb T}f(\zeta)\,dm(\zeta)=0\Bigr\},$$5 control of transforms to $$L^p_0(\mathbb T,X)=\Bigl\{f\in L^p(\mathbb T,X):\int_{\mathbb T}f(\zeta)\,dm(\zeta)=0\Bigr\},$$6 control in the Hardy category (Müller, 2015).

5. Dyadic perturbations, small-increment embeddings, and distance phenomena

The decomposition is stable under dyadic perturbations. If $$L^p_0(\mathbb T,X)=\Bigl\{f\in L^p(\mathbb T,X):\int_{\mathbb T}f(\zeta)\,dm(\zeta)=0\Bigr\},$$7 is a Hardy martingale and $$L^p_0(\mathbb T,X)=\Bigl\{f\in L^p(\mathbb T,X):\int_{\mathbb T}f(\zeta)\,dm(\zeta)=0\Bigr\},$$8 is a dyadic martingale, then there exists a Hardy martingale $$L^p_0(\mathbb T,X)=\Bigl\{f\in L^p(\mathbb T,X):\int_{\mathbb T}f(\zeta)\,dm(\zeta)=0\Bigr\},$$9, with $F=G+B$00, such that

$F=G+B$01

and

$F=G+B$02

Hence

$F=G+B$03

and in particular

$F=G+B$04

This perturbative form is the basis for quantitative separation results between dyadic martingales and Hardy martingales (Müller, 2012).

One application specializes to $F=G+B$05, the conditional expectation onto the dyadic $F=G+B$06-algebra. Then

$F=G+B$07

and hence

$F=G+B$08

As a consequence, for every dyadic martingale $F=G+B$09,

$F=G+B$10

so dyadic martingales are uniformly far in $F=G+B$11 from the Hardy class. A further application transfers the estimate back to $F=G+B$12, yielding an isomorphic embedding of $F=G+B$13 into the quotient $F=G+B$14, and hence a realization of $F=G+B$15 as a complemented subspace of $F=G+B$16 (Müller, 2012).

A different application uses Maurey’s small-increment embedding. Any $F=G+B$17 can be embedded into a Hardy martingale $F=G+B$18 on $F=G+B$19 whose increments satisfy

$F=G+B$20

By iterating that construction, an arbitrary Hardy martingale $F=G+B$21 on $F=G+B$22 can be realized as a subsequence of another Hardy martingale $F=G+B$23 whose increments are uniformly small and predictable. This gives an alternative route to extrapolation bounds without splitting $F=G+B$24 (Müller, 2015).

6. Noncommutative algebraic and asymmetric variants

In the noncommutative setting, one works with a von Neumann algebra $F=G+B$25 equipped with an increasing filtration of von Neumann subalgebras $F=G+B$26, together with the corresponding $F=G+B$27-preserving conditional expectations $F=G+B$28. For a martingale $F=G+B$29, with differences $F=G+B$30, the column and row Hardy-space quasi-norms are

$F=G+B$31

$F=G+B$32

For $F=G+B$33, the theory refines these spaces into conditioned and diagonal pieces, and the algebraic Davis decomposition gives complete quasi-isomorphisms

$F=G+B$34

and likewise

$F=G+B$35

Thus every $F=G+B$36 splits as $F=G+B$37, with the two parts admitting single-atom factorizations (Hong et al., 2015).

A principal consequence is the asymmetric Davis–Garsia decomposition. For $F=G+B$38 and any $F=G+B$39, there exist $F=G+B$40 and contractions $F=G+B$41 such that

$F=G+B$42

Moreover, $F=G+B$43 and $F=G+B$44 converge in the row and column Hardy spaces, respectively. At the endpoint $F=G+B$45, the results establish a noncommutative form of the comparison between martingale maximal and square functions in $F=G+B$46. The distinctive features here are the algebraic atomic descriptions, the asymmetric factorization of conditional expectations, and the use of Cuculescu projections together with weak-$F=G+B$47 spaces at the endpoint (Hong et al., 2015).

7. Biparameter analogue and the open reverse inequality

For a two-parameter filtration $F=G+B$48 satisfying the Cairoli–Walsh $F=G+B$49-condition, the martingale difference operator is

$F=G+B$50

so that

$F=G+B$51

The natural biparameter square function and maximal function are

$F=G+B$52

with Hardy-space quasi-norms $F=G+B$53 and $F=G+B$54 (Rzeszut, 27 Sep 2025).

In this setting, the one-parameter two-term splitting $F=G+B$55 is replaced by a four-part decomposition. If $F=G+B$56, there exist adapted fields $F=G+B$57 such that

$F=G+B$58

and

$F=G+B$59

Assembling the corresponding martingales $F=G+B$60, one obtains $F=G+B$61. The central quantitative estimate is

$F=G+B$62

$F=G+B$63

Each term is then handled by a different one-parameter argument, yielding

$F=G+B$64

Thus the $F=G+B$65 half of the two-parameter Davis inequality follows (Rzeszut, 27 Sep 2025).

The reverse inequality remains open in full generality for $F=G+B$66 filtrations. The stated obstruction is that no single bi-parameter stopping time is known that simultaneously controls the full square function and the two one-parameter square functions. A proposed route is variational: one studies an $F=G+B$67-square-function quotient and, via a gradient-form lemma, reduces the problem to a lower bound in $F=G+B$68. Two possible approaches are identified: finite-model approximation and variational-embedding techniques. This suggests that, in the biparameter theory, the Davis–Garsia paradigm survives only after substantial structural modification, and that the precise analogue of the one-parameter reverse inequality remains a central open question (Rzeszut, 27 Sep 2025).