1 Introduction

The Rademacher functions \(\left( r_i\right) _{i\in {\mathbb {N}}}\) generate a well studied subspace of \(L^p[0,1]\), which we identify with \(L^p\left( {\mathbb {Z}}_2^{\mathbb {N}}\right) \). In particular by Khintchine inequality

$$\begin{aligned} \left\| \sum _i c_i r_i\right\| _{L^p}\simeq _p \left( \sum _i \left| c_i\right| ^2\right) ^\frac{1}{2} \end{aligned}$$
(1.1)

for \(0<p<\infty \), and \(\overline{\mathrm {span}}\left( r_i:i\in {\mathbb {N}}\right) \) is complemented in \(L^p\) for \(1<p<\infty \) but not for \(p=1\). We will index the Walsh system by finite subsets of \({\mathbb {N}}\), i.e.

$$\begin{aligned} w_A=\prod _{i\in A}r_i. \end{aligned}$$
(1.2)

The number |A| is called the mutiplicity of \(w_A\). Analogous problems for Walsh functions of finite multiplicity have been resolved independently by Bonami [2] and Kiener [13]. Namely, the inequality

$$\begin{aligned} \left\| \sum _{|A|\le m} c_A w_A\right\| _{L^p}\simeq _{p,m} \left( \sum _{|A|\le m} \left| c_A\right| ^2\right) ^\frac{1}{2} \end{aligned}$$
(1.3)

holds true, and the orthogonal projection \(P_m\) onto \(\overline{\mathrm {span}}\left( w_A:|A|\le m\right) \) is bounded if and only if \(1<p<\infty \). Some lower estimates for \(P_m\) are also known, see [12] or [20] for a detailed discussion on this subject.

In [4], Bourgain generalized these results to a setting in which \(\left( {\mathbb {Z}}_2,\left\{ \emptyset ,\{0\},\{1\},\{0,1\}\right\} ,\frac{1}{2}\#\right) \) is replaced with an arbitrary probability space \(\left( \Omega ,{\mathcal {F}},\mu \right) \). To be more precise, let \(\left( \Omega ^\infty ,{\mathcal {F}}^{\otimes \infty },\mu ^{\otimes \infty }\right) \) be the infinite product space. Any \(f\in L^2\left( \Omega ^\infty ,{\mathcal {F}}^{\otimes \infty },\mu ^{\otimes \infty }\right) \) can be decomposed in a unique way into a series

$$\begin{aligned} f(x)=\sum _m \sum _{i_1<\ldots <i_m}f_{i_1,\ldots ,i_m}\left( x_{i_1},\ldots ,x_{i_m}\right) \end{aligned}$$
(1.4)

where \(f_{i_1,\ldots ,i_m}\in L^2\left( \Omega ^m\right) \) is mean zero in each of its m arguments. Thus, \(P_A\) and \(P_m\) defined by

$$\begin{aligned} P_{\left\{ i_1,\ldots ,i_m\right\} }f(x)= f_{i_1,\ldots ,i_m}\left( x_{i_1},\ldots ,x_{i_m}\right) ,\quad P_m=\sum _{|A|=m}P_A \end{aligned}$$
(1.5)

are mutually orthogonal orthogonal projections. In the case of \(\Omega ={\mathbb {Z}}_2\), the image of \(P_A\) is just the one-dimensional space spanned by \(w_A\), so the above definition of \(P_m\) coincides for with the projection onto Walsh functions of multiplicity m. In [4] Bourgain proved that for \(1\le p<\infty \),

$$\begin{aligned} \left\| \sum _{|A|\le m}P_A f\right\| _{L^p}\simeq _{p,m}\left\| \left( \sum _{|A|\le m}\left| P_A f\right| ^2\right) ^\frac{1}{2}\right\| _{L^p}, \end{aligned}$$
(1.6)

which is a direct generalization of (1.3). Moreover, he proved that \(P_m\) is bounded on \(L^p\) if and only if \(1<p<\infty \), with norm smaller than \(c_p^m\) where \(c_p\lesssim \frac{ {\hat{p}}^\frac{5}{2}}{\log {\hat{p}}}\) and \({\hat{p}}=p\vee \frac{p}{p-1}\).

It turns out that the projections \(P_m\) have a well established probabilistic interpretation. In [14], Kwapień connected them to the notion of Hoeffding decomposition, which originated from Hoeffding’s work [11]. More precisely, elements of the image of \(P_m\) are what is called generalized canonical U-statistics and the decomposition \(f=\sum _m P_m f\) plays a crucial role in the proofs of many theorems concerning U-statistics. For more information, we refer the reader to [16]. Kwapień provided a shorter proof of Bourgain’s result about boundedness of \(P_m\), with a better constant \(c_p\lesssim \frac{{\hat{p}}}{\log {\hat{p}}}\).

Let us decribe the main results of this paper, which give certain endpoint estimates for \(P_m\). One of them (Theorem 4.5 in the text) is obtained by restricting the domain of \(P_m\). For exact definition of \(H^1_{\mathrm {all}}\), see Sect. 2.

Theorem A

\(P_m\) is bounded on the subspace \(H^1_{\mathrm {all}}\left( {\mathbb {T}}^\infty \right) \) of \(L^1\left( {\mathbb {T}}^\infty \right) \) consisting of functions analytic in each variable.

We also find a norm stronger than \(L^1\) and weaker than \(L^p\) (\(p>1\)), in which \(P_m\) is bounded. The detailed construction is described in Sect. 5.2.

Theorem B

For any \(m\in {\mathbb {N}}\), there is a partition of the family of finite subsets of \({\mathbb {N}}\) into \({\dot{\bigcup }}_{i\in {\mathcal {I}}}{\mathcal {A}}_i\) such that the norm

$$\begin{aligned} \Vert f\Vert := {\mathbb {E}}\left( \sum _i \left| \sum _{A\in {\mathcal {A}}_i}P_A f\right| ^2\right) ^\frac{1}{2} \end{aligned}$$
(1.7)

is between \(L^1\) and all \(L^p\) (\(p>1\)) and \(P_m\) is bounded in this norm.

It is worth noting that Theorem A translates directly to the space \({\mathcal {H}}^1\) of Dirichlet series, i.e. the closure of polynomials of the form \(\sum _{n=1}^N b_n n^{-s}\) in the norm

$$\begin{aligned} \left\| \sum _{n=1}^N b_n n^{-s}\right\| _{{\mathcal {H}}^1}:= \lim _{T\rightarrow \infty }\frac{1}{2T} \int _{-T}^T \left| \sum _{n=1}^N b_n n^{-it}\right| \mathrm {d}t. \end{aligned}$$
(1.8)

The Bohr lift, dating back to [1], is the map

$$\begin{aligned} H^1_{\mathrm {all}}\left( {\mathbb {T}}^\infty \right) \ni \sum _{k\in {\mathbb {N}}^{\oplus {\mathbb {N}}}}a_k \mathrm {e}^{i \langle k,t\rangle }\mapsto \sum _{n\in {\mathbb {N}}} b_n n^{-s}\in {\mathcal {H}}^1 \end{aligned}$$
(1.9)

where \(a_k=b_n\) for n having the prime number factorization \(n=\prod _{j}p_j^{k_j}\). It is an isometry between \(H^1_{\mathrm {all}}\left( {\mathbb {T}}^\infty \right) \) and the space \({\mathcal {H}}^1\) of Dirichlet series. Thus, our result is equivalent to the fact that the projection from \({\mathcal {H}}^1\) onto

$$\begin{aligned} \overline{\mathrm {span}}\left\{ n^{-s}:n\text { has at most }m\text { prime factors}\right\} \subset {\mathcal {H}}^1 \end{aligned}$$
(1.10)

is bounded. For a more detailed exposition of Dirichlet series and their relation to polydisc Hardy spaces, see [21].

The paper is organized as follows. In Sect. 2 we introduce necessary notation and definitions. In Sect. 3, we provide a new simple proof of the historic \(L^p\) boundedness result. The proof of the estimate \(\left\| P_m\right\| \le \left( \mathrm {e}\left\| P_1\right\| \right) ^m\) is done by means of a combinatorial identity expressing \(P_m\) in terms of tensor products of \(P_1\). In Sect. 4, we show that the same argument carries over with little modification showing boundedness of \(P_m\) on \(H^1_{\mathrm {all}}\left( {\mathbb {T}}^\infty \right) \). In Sect. 5.1, we define, purely in terms of square functions and not referring to analyticity, a multiple indexed martingale Hardy space \(H^1\left[ {\mathcal {T}}_m\right] \) of functions on \(\Omega ^\infty \) that admits a bounded action of \(P_m\). It turns out that if \(\Omega ={\mathbb {T}}\), there is a subspace \(H^1_{m\text { last}}\left( {\mathbb {T}}^{\mathbb {N}}\right) \) of \(L^1\left( {\mathbb {T}}^\infty \right) \), much bigger than \(H^1_{\mathrm {all}}\left( {\mathbb {T}}^\infty \right) \), on which the \(L^1\) norm is equivalent to \(H^1\left[ {\mathcal {T}}_m\right] \) norm. The arguments rely heavily on \(L^1\) square function theorem for Hardy martingales and decoupling inequality of Zinn. We present two proofs of the latter in Sect. 1.

2 Preliminaries

Probability spaces and conditional expectations  In all of the text, \(\left( \Omega ,{\mathcal {F}},\mu \right) \) will be a probability space. We will equip sets of the form \(\Omega ^I\), where I is an at most countable index set, with the product measure \(\mu ^{\otimes I}\) defined on \({\mathcal {F}}^{\otimes I}\). In case we are only concerned with the cardinality of I, we will write \(\Omega ^n\), where n is a natural number or \(\infty \). By the natural filtration on \(\Omega ^{\mathbb {N}}\) we mean the filtration \(\left( {\mathcal {F}}_n:n=0,1,\ldots \right) \), where \({\mathcal {F}}_k\) is generated by the coordinate projection \(\omega \mapsto \left( \omega _1,\ldots ,\omega _k\right) \) and denote \({\mathbb {E}}_k={\mathbb {E}}\left( \cdot \mid {\mathcal {F}}_k\right) \). In general, for a subset A of the index set, \({\mathcal {F}}_A\) will be the sigma algebra generated by the coordinate projection \(\omega \mapsto \left( \omega _i\right) _{i\in A}\) and \({\mathbb {E}}_A={\mathbb {E}}\left( \cdot \mid {\mathcal {F}}_A\right) \). In more explicit terms, measurability with respect to \({\mathcal {F}}_A\) is equivalent to being dependent only on variables with indices belonging to A and the conditional expectation operator \({\mathbb {E}}_A\) integrates away the dependence on all other variables, so that the formulas

$$\begin{aligned}&{\mathbb {E}}_k f\left( x\right) = \int _{\Omega ^{[k+1,\infty )}} f\left( x_1,\ldots ,x_k,y_{k+1},y_{k+2},\ldots \right) \mathrm {d}\mu ^{\otimes [k+1,\infty )}(y), \end{aligned}$$
(2.1)
$$\begin{aligned}&{\mathbb {E}}_A f(x)= \int _{\Omega ^{{\mathbb {N}}{\setminus } A}}f\left( x_A,y_{{\mathbb {N}}{\setminus } A}\right) \mathrm {d}\mu ^{\otimes {\mathbb {N}}{\setminus } A}(y) \end{aligned}$$
(2.2)

are satisfied (with the convention that sequences indexed by A and \({\mathbb {N}}{\setminus } A\) are merged in a natural way into a sequence indexed by \({\mathbb {N}}\)). It will often be convenient to identify a function f defined on \(\Omega ^A\) with an \({\mathcal {F}}_A\)-measurable function \(\Omega ^I\ni \omega \mapsto f\left( \left( \omega _i\right) _{i\in A}\right) \). In order to save space, we will often write \(\mathrm {d}x\) instead of \(\mathrm {d}\mu (x)\) whenever the measure is implied by context.

Tensor products  Let \(1\le p<\infty \). For \(f_k\in L^p\left( \Omega _k\right) \), we will denote by \(\bigotimes _{k=1}^n f_k\) the function on \(\prod _k \Omega _k\) satisfying

$$\begin{aligned} \left( \bigotimes _k f_k\right) (x)=\prod _k f_k\left( x_k\right) . \end{aligned}$$
(2.3)

Because of separation of variables, we have \(\left\| \bigotimes _k f_k\right\| _{L^p\left( \prod _k \Omega _k\right) }= \prod _k \left\| f_k\right\| _{L^p\left( \Omega _k\right) }\). This way we actually define an injection of the algebraic tensor product \(\bigotimes _k L^p\left( \Omega _k\right) \) into \(L^p\left( \prod _k \Omega _k\right) \), the image of which is dense.

Let \(X_k\) be subspaces (by a subspace we always mean a closed linear subspace) of \(L^p\left( \Omega _k\right) \). By \(\bigotimes _k X_k\) we will denote the subspace of \(L^p\left( \prod _k \Omega _k\right) \) spanned by functions of the form \(\bigotimes _k f_k\), where \(f_k\in X_k\), and the norm is inherited from \(L^p\left( \prod _k \Omega _k\right) \) (care has to be taken, as \(\bigotimes _k X_k\) is not determined solely by \(X_k\) as Banach spaces, but rather by the particular way they are embedded in \(L^p\left( \Omega _k\right) \)). If \(T_k: X_k\rightarrow L^p\left( \Omega _k\right) \) are bounded operators, then we can define an operator \(\bigotimes _k T_k: \bigotimes _k X_k\rightarrow L^p\left( \prod _k \Omega _k\right) \) by the formula

$$\begin{aligned} \left( \bigotimes _k T_k\right) \left( \bigotimes _k f_k\right) = \bigotimes _k T_kf_k, \end{aligned}$$
(2.4)

and easily check that the property

$$\begin{aligned} \left\| \bigotimes _k T_k:\bigotimes _k X_k\rightarrow L^p\left( \prod _k \Omega _k\right) \right\| \le \prod _k \left\| T_k:X_k\rightarrow L^p\left( \Omega _k\right) \right\| \end{aligned}$$
(2.5)

is satisfied. Indeed, \(\bigotimes _k T_k= \prod _k \mathrm {id}_{L^p\left( \prod _{j\ne k}\Omega _j\right) }\otimes T_k\), and any operator of the form \(\mathrm {id}\otimes T\) has norm bounded by \(\Vert T\Vert \), because \(\left( \mathrm {id}\otimes T\right) f\left( \omega _1,\omega _2\right) = T\left( f\left( \omega _1,\cdot \right) \right) \left( \omega _2\right) \).

Fourier transform  Let \({\mathbb {T}}\) be the interval \([0,2\pi )\) equipped with addition modulo \(2\pi \) and normalized Lebesgue measure \(\mathrm {d}\mu = \frac{\mathrm {d}x}{2\pi }\). We will be exclusively dealing with Fourier transforms of functions on \({\mathbb {T}}\) or some power of \({\mathbb {T}}\). Since the group dual to \({\mathbb {T}}\) is \({\mathbb {Z}}\), the dual group to the product \({\mathbb {T}}^{\mathbb {N}}\) is the direct sum \({\mathbb {Z}}^{\oplus {\mathbb {N}}}\) (i.e., integer-valued sequences that are eventually 0), on which we define the Fourier transform by

$$\begin{aligned} {\widehat{f}}(n)=\int _{{\mathbb {T}}^{\mathbb {N}}} f\left( x\right) \mathrm {e}^{-i\sum _{k\in {\mathbb {N}}} n_k x_k}\mathrm {d}\mu ^{\otimes {\mathbb {N}}}(x). \end{aligned}$$
(2.6)

Hardy spaces of martingales and analytic functions  By \({\mathbb {D}}\) we denote the unit disk in the complex plane. We can identify \({\mathbb {T}}\) with the unit circle by the map \(t\mapsto \mathrm {e}^{it}\). For \(N\in {\mathbb {N}}\), the space \(H^1({\mathbb {D}}^N)\) is defined as the space of functions analytic in the polydisc \({\mathbb {D}}^N\) such that the norm

$$\begin{aligned} \Vert F\Vert _{H^1\left( {\mathbb {D}}^N\right) }= \sup _{0<r_1,\ldots ,r_n<1} \int _{{\mathbb {T}}^N} \left| F\left( r_1 \mathrm {e}^{it_1},\ldots ,r_n \mathrm {e}^{i t_N}\right) \right| \frac{\mathrm {d}t}{\left( 2\pi \right) ^N} \end{aligned}$$
(2.7)

is finite. It is well-known [10] that such a function has an a.e. radial limit \(f\left( t_1,\ldots ,t_n\right) =\lim _{r\rightarrow 1} F\left( r\mathrm {e}^{it_1},\ldots ,r\mathrm {e}^{it_n}\right) \) on the distinguished boundary \({\mathbb {T}}^N\) and F can be recovered from f by convolution with a Poisson kernel. This sets a one-to-one correspondence between \(H^1\left( {\mathbb {D}}^N\right) \) and the space

$$\begin{aligned} H^1_{\mathrm {all}}\left( {\mathbb {T}}^N\right) = \overline{\mathrm {span}}\,\left\{ \mathrm {e}^{i \langle n,t\rangle }: n_1,\ldots ,n_N\ge 0\right\} \subset L^1\left( {\mathbb {T}}^N\right) . \end{aligned}$$
(2.8)

We also can define \(H^1_{\mathrm {all}}\left( {\mathbb {T}}^{\mathbb {N}}\right) \) in the same manner as in (2.8), but care has to be taken, since these functions are can no longer be extended analytically to \({\mathbb {D}}^{\mathbb {N}}\) in general (hence the shorthand \(H^1\left( {\mathbb {D}}^{\mathbb {N}}\right) \), which we will sometimes use, is an abuse of notation). Later we will use two more \(H^1\) spaces, namely \(H^1_{\mathrm {last}}\left( {\mathbb {T}}^{\mathbb {N}}\right) \) (also called Hardy martingales) and \(H^1_{m\,\mathrm {last}}\left( {\mathbb {T}}^{\mathbb {N}}\right) \), which we will define as follows.

$$\begin{aligned}&H^1_{\mathrm {last}}\left( {\mathbb {T}}^{\mathbb {N}}\right) = H^1_{1\,\mathrm {last}}\left( {\mathbb {T}}^{\mathbb {N}}\right) = \overline{\mathrm {span}}\,\left\{ \mathrm {e}^{i \langle n,t\rangle }: n_{i_0}>0 \text { for }i_0=\max \left\{ i:n_i\ne 0\right\} \right\} \subset L^1\left( {\mathbb {T}}^{\mathbb {N}}\right) ,\nonumber \\ \end{aligned}$$
(2.9)
$$\begin{aligned}&H^1_{m\,\mathrm {last}}\left( {\mathbb {T}}^{\mathbb {N}}\right) = \overline{\mathrm {span}}\,\left\{ \mathrm {e}^{i \langle n,t\rangle }:m\text { last of nonzero }n_i\text {'s are }>0\right\} \subset L^1\left( {\mathbb {T}}^{\mathbb {N}}\right) . \end{aligned}$$
(2.10)

In the space \(H^1_{m\,\mathrm {last}}\left( {\mathbb {T}}^{\mathbb {N}}\right) \) we allow characters of the form \(\mathrm {e}^{i\langle n,t\rangle }\), where \(\left| \mathrm {supp}\,n\right| <m\) and \(n_j\ge 0\) for all j.

Now we recall the definition of a martingale Hardy space and some related inequalities. A standard reference in this matter is [9]. Let \(\left( {\mathcal {F}}_n\right) _{n=0}^\infty \) be an arbitrary filtration on a probability space \(\left( \Omega ,{\mathcal {F}},\mu \right) \), where \({\mathcal {F}}\) is generated by \(\bigcup {\mathcal {F}}_n\). We denote \({\mathbb {E}}_k={\mathbb {E}}\left( \cdot \mid {\mathcal {F}}_k\right) \), \(\Delta _0={\mathbb {E}}_0\), \(\Delta _{k}={\mathbb {E}}_k-{\mathbb {E}}_{k-1}\) for \(k\ge 1\), and define the square function and maximal function of f respectively by

$$\begin{aligned} Sf= \left( \sum _{n=0}^\infty \left| \Delta _n f\right| ^2\right) ^\frac{1}{2}, \quad f^*= \sup _{n}\left| {\mathbb {E}}_n f\right| . \end{aligned}$$
(2.11)

This allows us to define the martingale Hardy space.

Definition 2.1

The space \(H^1\left[ \left( {\mathcal {F}}_n\right) _{n=1}^\infty \right] \) is a function space on \(\Omega \) with the norm

$$\begin{aligned} \left\| f\right\| _{H^1\left[ \left( {\mathcal {F}}_n\right) _{n=1}^\infty \right] }={\mathbb {E}}Sf. \end{aligned}$$
(2.12)

We will make use of three following classical martingale inequalities.

Theorem 2.2

(Burkholder, Gundy [7] for  \(1<p<\infty \); Davis [8] for  \(p=1\)) For \(1\le p<\infty \),

$$\begin{aligned} \left\| Sf\right\| _{L^p}\simeq _p \left\| f^*\right\| _{L^p}. \end{aligned}$$
(2.13)

Theorem 2.3

(Burkholder [6])  For \(1<p<\infty \),

$$\begin{aligned} \left\| f\right\| _{L^p}\simeq _p \left\| Sf\right\| _{L^p}. \end{aligned}$$
(2.14)

Theorem 2.4

(Stein [24]) For \(1<p<\infty \) and an arbitrary sequence \(\left( f_n\right) _{n=0}^\infty \),

$$\begin{aligned} \left\| \left( \sum _{n=0}^\infty \left| {\mathbb {E}}_n f_n\right| ^2 \right) ^\frac{1}{2}\right\| _{L^p}\lesssim _p \left\| \left( \sum _{n=0}^\infty \left| f_n\right| ^2 \right) ^\frac{1}{2}\right\| _{L^p}. \end{aligned}$$
(2.15)

Definition 2.5

A martingale atom is a function of the form

$$\begin{aligned} a=u-{\mathbb {E}}_{j-1}u, \end{aligned}$$
(2.16)

where

$$\begin{aligned} A\in {\mathcal {F}}_j,\quad \mathrm {supp}\,u\subset A,\quad \left\| u\right\| _{L^2}\le |A|^{-\frac{1}{2}}. \end{aligned}$$
(2.17)

Theorem 2.6

Let \(f \in H^1\left[ \left( {\mathcal {F}}_n\right) _{n=1}^\infty \right] \) be of mean 0. Then there are atoms \(a_1,a_2,\ldots \) and scalars \(c_1,c_2,\ldots \) such that

$$\begin{aligned} f=\sum _{n=1}^\infty c_n a_n \end{aligned}$$
(2.18)

and

$$\begin{aligned} \sum _{n=1}^\infty \left| c_n\right| \lesssim \Vert f\Vert _{H^1\left[ \left( {\mathcal {F}}_n\right) _{n=1}^\infty \right] }. \end{aligned}$$
(2.19)

Theorem 2.7

[Fefferman] The dual space to \(H^1\left[ \left( {\mathcal {F}}_n\right) _{n=1}^\infty \right] \) is \(BMO\left[ \left( {\mathcal {F}}_n\right) _{n=1}^\infty \right] \), where

$$\begin{aligned} \Vert g\Vert _{BMO\left[ \left( {\mathcal {F}}_n\right) _{n=1}^\infty \right] }\simeq \sup _k \left\| \left( {\mathbb {E}}_k \sum _{n\ge k}\left| \Delta _n g\right| ^2 \right) ^\frac{1}{2}\right\| _{L^\infty }, \end{aligned}$$
(2.20)

where the duality is given by \(\langle f,g\rangle =\lim _{n\rightarrow \infty }{\mathbb {E}}\left( {\mathbb {E}}_n f{\mathbb {E}}_n g\right) \).

Vector-valued inequalities  For a Banach space B, by \(L^p\left( S,B\right) \) we denote the Bochner space of strongly measurable B-valued random variables equipped with the norm

$$\begin{aligned} \Vert f\Vert _{L^p(S,B)}=\left( \int _{s}\Vert f(x)\Vert _B^p\mathrm {d}\mu (s)\right) ^\frac{1}{p} \end{aligned}$$
(2.21)

(or, equivalently, the closed span of functions of the form \((f\otimes v)(x)= f(x)v\), where \(f\in L^p(S)\) and \(v\in B\), in the \(L^p(S,B)\) norm). For an operator T between subspaces of \(L^p\left( S_1\right) \) and \(L^p\left( S_2\right) \) and a linear operator \(F:B_1\rightarrow B_2\) we can define \(T\otimes F\) and the algebraic tensor product by \(\left( T\otimes F\right) \left( f\otimes v\right) = T(f)\otimes F(v)\), but this construction does not necessarlily produce a bounded operator on the closure. The main tool for obtaining vector-vlaued extensions of inequalities will be the following lemma, which for \(I_1,I_2\) being singletons is due to Marcinkiewicz and Zygmund [18] (in this case \(\lesssim \Vert T\Vert \) can be replaced with \(\le \Vert T\Vert \)).

Lemma 2.8

Let \(X_i\subset L^1\left( S_i,\ell ^2\left( I_i\right) \right) \) for \(i=1,2\), B be a Hilbert space and \(T:X_1\rightarrow X_2\) be bounded. Then \(T\otimes \mathrm {id}_B: X_1\otimes B\rightarrow X_2\otimes B\), where \(X_i\otimes B\) is treated as a subspace of \(L^1\left( \Omega _i,\ell ^2\left( I_i,B\right) \right) \), is bounded with norm \(\lesssim \Vert T\Vert \).

Proof

Without loss of generality, B is finite-dimensional, say \(B=\ell ^2\left( J\right) \) for some finite J. Let \(X_1\otimes \ell ^2\left( J\right) \ni f=\left( f_j\right) _{j\in J}\), so that \(f_j\in X_1\). Let also \(r_j\) for \(j\in J\) be Rademacher variables. Then, applying \(\ell ^2\left( I_2\right) \)-valued Khintchine inequality,

$$\begin{aligned}&\left\| \left( T\otimes \mathrm {id}\right) f\right\| _{L^1\left( S_2,\ell ^2\left( I_2\times J\right) \right) }\nonumber \\&\quad =\int _{S_2}\left( \sum _{j\in J}\left\| Tf_j\left( s\right) \right\| _{\ell ^2\left( I_2\right) }^2 \right) ^\frac{1}{2}\mathrm {d}\mu _2(s) \end{aligned}$$
(2.22)
$$\begin{aligned}&\quad \simeq \int _{S_2}{\mathbb {E}}\left\| \sum _j r_j Tf_j(s)\right\| _{\ell ^2\left( I_2\right) }\mathrm {d}\mu _2(s) \end{aligned}$$
(2.23)
$$\begin{aligned}&\quad = {\mathbb {E}}\int _{S_2}\left\| T\left( \sum _j r_j f_j\right) (s)\right\| _{\ell ^2\left( I_2\right) }\mathrm {d}\mu _2(s) \end{aligned}$$
(2.24)
$$\begin{aligned}&\quad \le \Vert T\Vert {\mathbb {E}}\int _{S_1}\left\| \sum _j r_j f_j(s)\right\| _{\ell ^2\left( I_1\right) }\mathrm {d}\mu _1(s) \end{aligned}$$
(2.25)
$$\begin{aligned}&\quad \le \Vert T\Vert \int _{S_1}\left( \sum _j \left\| f_j(s)\right\| _{\ell ^2\left( I_1\right) }^2 \right) ^\frac{1}{2}\mathrm {d}\mu _1(s) \end{aligned}$$
(2.26)
$$\begin{aligned}&\quad =\Vert T\Vert \Vert f\Vert _{L^1\left( S_1,\ell ^2\left( I_1\times J\right) \right) }. \end{aligned}$$
(2.27)

\(\square \)

Hoeffding decomposition  Now we define the main object of our interest. In order to avoid technicalities with convergence in strong operator topology, we will work in a finite product of \(\Omega \) (all the results extend automatically to \(\Omega ^\infty \) by density). We will see in a moment that any function \(f\in L^1\left( \Omega ^n\right) \) can be decomposed in a unique way as

$$\begin{aligned} f=\sum _{m=0}^n \sum _{1\le i_1<\ldots <i_m\le n}P_{i_1,\ldots ,i_m}f, \end{aligned}$$

where \(P_{i_1,\ldots ,i_m}f\left( x_1,\ldots ,x_n\right) \) depends only on \(x_{i_1},\ldots ,x_{i_m}\) and is of mean 0 with respect to each of \(x_{i_1},\ldots ,x_{i_m}\) (equivalently, \(P_A f\) is \({\mathcal {F}}_A\)-measurable and is orthogonal to all \({\mathcal {F}}_B\)-measurable functions for \(B\subsetneq A\)). This decomposition has been studied in [4, 14]. In particular, \(P_{i_1,\ldots ,i_m}\) are pairwise orthogonal orthogonal projections. Let

$$\begin{aligned} P_m= \sum _{1\le i_1<\ldots <i_m\le n}P_{i_1,\ldots ,i_m} \end{aligned}$$

and \(U_m\) be the range of \(P_m\). It is known [4, 14] that \(P_m\) is bounded on \(L^p\left( \Omega ^n\right) \), \(1<p<\infty \), with norm independent on n, but this is not true for \(L^1\left( \Omega ^n\right) \).

One of the possible ways to prove the existence of the above decomposition in \(L^2\left( \Omega ^n\right) \) is as follows. First we define the subspace

$$\begin{aligned} U_{\le m}=\overline{\mathrm {span}}\,\bigcup _{|A|\le m} \left\{ f\in L^2\left( \Omega ^n\right) : f\text { is }{\mathcal {F}}_A\text {-measurable}\right\} \subset L^2\left( \Omega ^n\right) \end{aligned}$$
(2.28)

for each \(m\ge 0\). The sequence of subspaces \(U_{\le 0},U_{\le 1},\ldots ,U_{\le n}\) is increasing, so by putting

$$\begin{aligned} U_0=U_{\le 0}, \quad U_m=U_{\le m}\cap U_{\le m-1}^{\perp } \end{aligned}$$
(2.29)

we obtain a decomposition

$$\begin{aligned} L^2\left( \Omega ^n\right) = \bigoplus _{m=0}^{n} U_m \end{aligned}$$
(2.30)

into an orthogonal direct sum of \(U_m\). We will denote the orthogonal projection onto \(U_m\) by \(P_m\).

A more explicit formula for \(P_m\) can be obtained. For \(A\subset [1,n]\), let

$$\begin{aligned} P_A= \left( \mathrm {id}-{\mathbb {E}}\right) ^{\otimes A}\otimes {\mathbb {E}}^{\otimes [1,n]{\setminus } A}, \end{aligned}$$
(2.31)

where \(\mathrm {id}\) and \({\mathbb {E}}\) are understood to act on \(L^2(\Omega )\), and let \(U_A\) be the range of the projection \(P_A\). It is easy to see that

$$\begin{aligned} {\mathbb {E}}_A= (\mathrm {id}- {\mathbb {E}}+{\mathbb {E}})^{\otimes A}\otimes {\mathbb {E}}^{\otimes [1,n]{\setminus } A}= \sum _{B\subset A}\left( \mathrm {id}-{\mathbb {E}}\right) ^{\otimes B}\otimes {\mathbb {E}}^{\otimes [1,n]{\setminus } B} \end{aligned}$$
(2.32)

and, since the subspaces \(U_B\) are mutually orthogonal,

$$\begin{aligned} L^2\left( \Omega ^n,{\mathcal {F}}_A\right) = \bigoplus _{B\subset A} U_B. \end{aligned}$$
(2.33)

Moreover

$$\begin{aligned} U_{\le m}= & {} \overline{\mathrm {span}}\,\bigcup _{|A|\le m}L^2\left( \Omega ^n,{\mathcal {F}}_A\right) \end{aligned}$$
(2.34)
$$\begin{aligned}= & {} \overline{\mathrm {span}}\,\bigcup _{|A|\le m} \bigoplus _{B\subset A} U_B \end{aligned}$$
(2.35)
$$\begin{aligned}= & {} \bigoplus _{|B|\le m} U_B \end{aligned}$$
(2.36)

and consequently

$$\begin{aligned} U_m= \bigoplus _{|B|=m} U_B, \quad P_m=\sum _{|B|=m}P_B. \end{aligned}$$
(2.37)

Decoupling inequalities  We are going to present a special case of a theorem of J. Zinn [26], which will be one of the most important tools.

Theorem 2.9

For \(k=1,\ldots ,N\), let \(f_k\) be a function on \(\Omega ^k\). Then

$$\begin{aligned} \int _{\Omega ^N}\left( \sum _{k=1}^N \left| f_k\left( x_1,\ldots ,x_k\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}x \simeq \int _{\Omega ^N}\int _{\Omega ^N}\left( \sum _{k=1}^N \left| f_k\left( x_1,\ldots ,x_{k-1},y_k\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}x\mathrm {d}y.\nonumber \\ \end{aligned}$$
(2.38)

We will provide two new proofs of the above in Sect. 1. Below, we state two corollaries obtained by iterating Zinn’s inequality.

Corollary 2.10

For \(1\le a< b\le N\), let \(f_{a,b}\in L^1\left( \Omega ^N, {\mathcal {F}}_{[a,b]}\right) \). Denote \(\left( x_a,\ldots ,x_b\right) \) by \(x_{[a,b]}\). Then

$$\begin{aligned} \int _{\Omega ^N}\left( \sum _{a<b} \left| f_{a,b}\left( x_{[a,b]}\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}x \simeq \int _{\left( \Omega ^N\right) ^3}\left( \sum _{a<b} \left| f_{a,b}\left( z_a,x_{[a+1,b-1]},y_b\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}x \mathrm {d}y\mathrm {d}z\nonumber \\ \end{aligned}$$
(2.39)

Proof

Let \(F_b\in L^1\left( \Omega ^N, {\mathcal {F}}_b,\ell ^2\right) \) be defined by \(\left( F_b\right) _a= f_{a,b}\) for \(a<b\) and 0 otherwise. Then, by Theorem 2.9 applied for functions \(\left\| F_b\right\| _{\ell ^2}\),

$$\begin{aligned}&\int _{\Omega ^N}\left( \sum _{a< b} \left| f_{a,b}\left( x_{[a,b]}\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}x\nonumber \\&\quad = \int _{\Omega ^N}\left( \sum _{b} \left\| F_{b}\left( x_{\le b}\right) \right\| ^2_{\ell ^2} \right) ^\frac{1}{2}\mathrm {d}x \end{aligned}$$
(2.40)
$$\begin{aligned}&\quad \simeq \int _{\left( \Omega ^N\right) ^2}\left( \sum _{b} \left\| F_{b}\left( x_{\le b-1},y_b\right) \right\| ^2_{\ell ^2} \right) ^\frac{1}{2}\mathrm {d}x\mathrm {d}y \end{aligned}$$
(2.41)
$$\begin{aligned}&\quad = \int _{\left( \Omega ^N\right) ^2}\left( \sum _{a< b} \left| f_{a,b}\left( x_{[a,b-1]},y_b\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}x\mathrm {d}y. \end{aligned}$$
(2.42)

Analogously, by setting y as fixed, and applying Theorem 2.9 with reversed order of variables (which we can do, because we are dealing with finite sums),

$$\begin{aligned} \int _{\left( \Omega ^N\right) ^2}\left( \sum _{a\le b} \left| f_{a,b}\left( x_{[a,b-1]},y_b\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}x\mathrm {d}y\simeq \int _{\left( \Omega ^N\right) ^3}\left( \sum _{a\le b} \left| f_{a,b}\left( z_a,x_{[a+1,b-1]},y_b\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}x\mathrm {d}y\mathrm {d}z\nonumber \\ \end{aligned}$$
(2.43)

as desired. \(\square \)

Corollary 2.11

For all \(i=\left( i_1,\ldots ,i_m\right) \) such that \(i_1<\ldots <i_m\), let \(f_i\) be an \({\mathcal {F}}_{\left[ 1,i_1-1\right] \cup \left\{ i_1,\ldots ,i_m\right\} }\)-measurable function on \(\Omega ^{\mathbb {N}}\). Then, treating each \(f_i\) as a function on \(\Omega ^{\left[ 1,i_1-1\right] }\times \Omega ^{\left\{ i_1,\ldots ,i_m\right\} }\),

$$\begin{aligned}&\int _{\Omega ^{\mathbb {N}}}\left( \sum _i \left| f_i\left( x_{<i_1},x_{i_1},\ldots ,x_{i_m}\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}x \nonumber \\&\quad \simeq _m \int _{\Omega ^{\mathbb {N}}}\int _{\left( \Omega ^{\mathbb {N}}\right) ^m}\left( \sum _i \left| f_i\left( x_{<i_1},y^{(1)}_{i_1},\ldots ,y^{(m)}_{i_m}\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}y^{(1,\ldots ,m)}\mathrm {d}x, \end{aligned}$$
(2.44)

where \(y^{(1)},\ldots ,y^{(m)}\) are variables in \(\Omega ^{\mathbb {N}}\).

Proof

Let us fix \(k\in \{1,\ldots ,m\}\) and for each \(j\in {\mathbb {N}}\) define a function \(\varphi _j\) on \(\Omega ^{\left[ 1,j\right] }\times \left( \Omega ^{\mathbb {N}}\right) ^{m-k}\) by the formula

$$\begin{aligned}&\varphi _j \left( x_{\le j},y^{(k+1)},\ldots ,y^{(m)}\right) \nonumber \\&\quad =\left( \sum _{\begin{array}{c} i_1<\ldots<i_{k-1}< j<i_{k+1}<\ldots<i_{m} \end{array}} \left| f_{i_1,\ldots ,i_{k-1},j,i_{k+1},\ldots ,i_{m}}\left( x_{<i_1},x_{i_1},\ldots ,x_{i_{k-1}},x_j,y^{(k+1)}_{i_{k+1}},\ldots ,y^{(m)}_{i_m}\right) \right| ^2 \right) ^\frac{1}{2}. \end{aligned}$$
(2.45)

Then, for fixed \(y^{(> k)}=\left( y^{(k+1)},\ldots ,y^{(m)}\right) \in \left( \Omega ^{\mathbb {N}}\right) ^{m-k}\),

$$\begin{aligned}&\int _{\Omega ^{\mathbb {N}}}\left( \sum _{i_1<\ldots<i_m} \left| f_i\left( x_{<i_1},x_{i_1},\ldots ,x_{i_{k}},y^{(k+1)}_{i_{k+1}},\ldots ,y^{(m)}_{i_m}\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}x \end{aligned}$$
(2.46)
$$\begin{aligned}&\quad = \int _{\Omega ^{\mathbb {N}}}\left( \sum _{j\in {\mathbb {N}}} \left| \varphi _j \left( x_{\le j},y^{(k+1)},\ldots ,y^{(m)}\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}x \end{aligned}$$
(2.47)
$$\begin{aligned}&\quad \simeq \int _{\Omega ^{\mathbb {N}}}\int _{\Omega ^{\mathbb {N}}}\left( \sum _{j\in {\mathbb {N}}} \left| \varphi _j \left( x_{<j},y^{(k)}_{j},y^{(k+1)},\ldots ,y^{(m)}\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}x\mathrm {d}y^{(k)} \end{aligned}$$
(2.48)
$$\begin{aligned}&\quad = \int _{\Omega ^{\mathbb {N}}}\int _{\Omega ^{\mathbb {N}}}\left( \sum _{i_1<\ldots<i_m} \left| f_i\left( x_{<i_1},x_{i_1},\ldots ,x_{i_{k-1}},y^{(k)}_{i_k},y^{(k+1)}_{i_{k+1}},\ldots ,y^{(m)}_{i_m}\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}x\mathrm {d}y^{(k)}. \end{aligned}$$
(2.49)

Here, \(i_k\) plays the role of j and (2.48) is an application of Theorem 2.9 to functions \(\left| \varphi _j\right| ^2\). Integrating the resulting inequality with respect to \(y^{(>k)}\), we get

$$\begin{aligned}&\int _{\left( \Omega ^{\mathbb {N}}\right) ^{m-k}}\int _{\Omega ^{\mathbb {N}}}\left( \sum _{i_1<\ldots<i_m} \left| f_i\left( x_{<i_1},x_{i_1},\ldots ,x_{i_{k}},y^{(k+1)}_{i_{k+1}},\ldots ,y^{(m)}_{i_m}\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}x\mathrm {d}y^{(\ge k+1)}\nonumber \\&\quad \simeq \int _{\left( \Omega ^{\mathbb {N}}\right) ^{m-k+1}}\int _{\Omega ^{\mathbb {N}}}\left( \sum _{i_1<\ldots<i_m} \left| f_i\left( x_{<i_1},x_{i_1},\ldots ,x_{i_{k-1}},y^{(k)}_{i_k},y^{(k+1)}_{i_{k+1}},\ldots ,y^{(m)}_{i_m}\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}x\mathrm {d}y^{(\ge k)}, \end{aligned}$$
(2.50)

which by induction from \(k=m\) to \(k=1\) proves (2.44). \(\square \)

3 Boundedness of \(P_m\) on \(L^p\left( \Omega ^{\mathbb {N}}\right) \)

The main motivation for this part is the following theorem, proved by Bourgain with \(c_p\lesssim \frac{{\hat{p}}^{\frac{5}{2}}}{\ln {\hat{p}}}\) and by Kwapień with \(c_p\lesssim \frac{{\hat{p}}}{\ln {\hat{p}}}\), where \({\hat{p}}=\max \left( p,\frac{p}{p-1}\right) \).

Theorem 3.1

[4, 14] \(P_m\) is bounded on \(L^p\left( \Omega ^{\mathbb {N}}\right) \) for \(1<p<\infty \), with norm \(\lesssim c_p^m\).

We will present a proof that yields and .

Proof

Without loss of generality, we may assume that we are working in \(L^p\left( \Omega ^{[1,N]},{\mathcal {F}}^{\otimes N}\right) \). Indeed, by (2.33) and (2.37), \(P_m\) preserves \(L^2\left( \Omega ^{\mathbb {N}},{\mathcal {F}}_{[1,N]}\right) \), which can be canonically identified with \(L^2\left( \Omega ^{[1,N]},{\mathcal {F}}^{\otimes N}\right) \). Since the sequence \(\left( L^2\left( \Omega ^{\mathbb {N}},{\mathcal {F}}_{[1,N]}\right) :N\in {\mathbb {N}}\right) \) is increasing and its sum is dense in \(L^p\left( \Omega ^{\mathbb {N}},{\mathcal {F}}^{\otimes {\mathbb {N}}}\right) \), all we need to prove is

(3.1)

\(P_0={\mathbb {E}}\) is bounded. The \(L^p\) boundedness of \(P_1\) is essentially a known result [5], but we provide a proof for the sake of completeness. Let \(\left( {\mathcal {F}}_k:k\in [0,N]\right) \) be the natural filtration and \(\left( {\mathcal {F}}^*_k\right) _{k=0}^N\) be the natural reversed filtration, i.e. \({\mathcal {F}}^*_k={\mathcal {F}}_{[k,N]}\). By (2.32) and (2.37) we see that

$$\begin{aligned} P_1=\sum _{k=1}^N P_{\{k\}},\quad \Delta _k=\sum _{\max A=k}P_A,\quad {\mathbb {E}}^*_k= \sum _{A\subset [k,N]}P_A. \end{aligned}$$
(3.2)

By mutual orthogonality of \(P_A\)’s

$$\begin{aligned} \Delta _k P_1= P_{\{k\}}= {\mathbb {E}}_k^*\Delta _k. \end{aligned}$$
(3.3)

Applying Theorem 2.3, (3.3) and Theorem 2.4, we obtain

$$\begin{aligned} \left\| P_1f\right\| _{L^p}&\simeq _p&\left\| \left( \sum _{k=0}^N \left| \Delta _k P_1 f\right| ^2 \right) ^\frac{1}{2}\right\| _{L^p} \end{aligned}$$
(3.4)
$$\begin{aligned}= & {} \left\| \left( \sum _{k=0}^N \left| {\mathbb {E}}^*_k \Delta _k f\right| ^2 \right) ^\frac{1}{2}\right\| _{L^p} \end{aligned}$$
(3.5)
$$\begin{aligned}&\lesssim _p&\left\| \left( \sum _{k=0}^N \left| \Delta _k f\right| ^2 \right) ^\frac{1}{2}\right\| _{L^p} \end{aligned}$$
(3.6)
$$\begin{aligned}&\simeq _p&\left\| f\right\| _{L^p}. \end{aligned}$$
(3.7)

We will now proceed by induction. Suppose that (3.1) is satisfied with \(m-1\) in the place of m. Let \(N=mn\) and define an operator \(Q_m\) acting on \(L^p\left( \Omega ^{[1,N]}\right) \) by

(3.8)

Utilising (2.37) we get

(3.9)
(3.10)

By (2.31),

(3.11)

Putting the last four equations together, we get

(3.12)
(3.13)
(3.14)
$$\begin{aligned}= & {} m\frac{\left( {\begin{array}{c}N-m\\ n-1\end{array}}\right) }{\left( {\begin{array}{c}N\\ n\end{array}}\right) }P_m. \end{aligned}$$
(3.15)

However, by (3.8) and the induction hypothesis,

(3.16)
(3.17)
$$\begin{aligned} = \frac{c_p^m}{\mathrm {e}^2}. \end{aligned}$$
(3.18)

Let \(a_n\approx b_n\) denote \(\lim _{n\rightarrow \infty }\frac{a_n}{b_n}=1\). By the Stirling formula,

$$\begin{aligned} \left( {\begin{array}{c}nm\\ n\end{array}}\right)\approx & {} \frac{ \left( 2\pi nm \right) ^\frac{1}{2} \left( \frac{nm}{\mathrm {e}}\right) ^{nm}}{\left( 2\pi n \right) ^\frac{1}{2}\left( \frac{n}{\mathrm {e}}\right) ^n\left( 2\pi n(m-1) \right) ^\frac{1}{2}\left( \frac{n(m-1)}{\mathrm {e}}\right) ^{n(m-1)}} \end{aligned}$$
(3.19)
$$\begin{aligned}= & {} \left( \frac{m}{2\pi n(m-1)} \right) ^\frac{1}{2} \frac{n^{nm} m^{nm}}{n^n n^{n(m-1)} (m-1)^{n(m-1)}} \end{aligned}$$
(3.20)
$$\begin{aligned}= & {} \left( \frac{m}{2\pi n(m-1)} \right) ^\frac{1}{2} \left( \frac{m^m}{(m-1)^{m-1}}\right) ^n. \end{aligned}$$
(3.21)

Thus

$$\begin{aligned} \frac{\left( {\begin{array}{c}nm\\ n\end{array}}\right) }{m\left( {\begin{array}{c}(n-1)m\\ n-1\end{array}}\right) }\approx & {} \frac{1}{m}\left( \frac{n+1}{n} \right) ^\frac{1}{2}\frac{m^m}{(m-1)^{(m-1)}} \end{aligned}$$
(3.22)
$$\begin{aligned}\approx & {} \left( \frac{m}{m-1}\right) ^{m-1}. \end{aligned}$$
(3.23)

Finally, by (3.15), (3.18) and (3.23),

(3.24)
$$\begin{aligned}= & {} \lim _{n\rightarrow \infty } \frac{\left( {\begin{array}{c}nm\\ n\end{array}}\right) }{m\left( {\begin{array}{c}nm-m\\ n-1\end{array}}\right) }\left\| Q_m\right\| \end{aligned}$$
(3.25)
$$\begin{aligned}= & {} \left( 1+\frac{1}{m-1}\right) ^{m-1} \left\| Q_m\right\| \end{aligned}$$
(3.26)
$$\begin{aligned}\le & {} \frac{c_p^m}{\mathrm {e}}. \end{aligned}$$
(3.27)

\(\square \)

We provide a short proof of a fact taken from [5] that Theorem 3.1 can not be, extended to \(p=1\) or \(\infty \), which motivates the next section.

Proposition 3.2

If \(\Omega \) is not a single atom, then \(P_m\) for \(m\ge 1\) is not bounded on \(L^1\left( \Omega ^\infty \right) \) or \(L^\infty \left( \Omega ^\infty \right) \).

Proof

It is enough to consider \(L^1\left( \Omega ^\infty \right) \), because \(P_m\)’s are self-adjoint. Let \(f\in L^2\left( \Omega \right) \) be such that \({\mathbb {E}}f=1\), \(f\ge 0\) and \(\mu \left( \mathrm {supp}\,f\right) <1\). Then \({\mathbb {E}}|f-1|^2>0\). For \(F_n=f^{\otimes n}\in L^2\left( \Omega ^n\right) \) we have

$$\begin{aligned} \left\| P_1 F_n\right\| _{L^1\left( \Omega ^n\right) }=&\int _{\Omega ^n}\left| \sum _i \left( f\left( x_i\right) -1\right) \right| \mathrm {d}x \end{aligned}$$
(3.28)
$$\begin{aligned} \simeq&\int _{\Omega ^n}\left( \sum _i \left| f\left( x_i\right) -1\right| ^2 \right) ^\frac{1}{2}\mathrm {d}x \end{aligned}$$
(3.29)
$$\begin{aligned} \ge&\left( n {\mathbb {E}}|f-1|^2 \right) ^\frac{1}{2} \end{aligned}$$
(3.30)

which is not dominated by \(\left\| F_n\right\| _{L^1\left( \Omega ^n\right) }= \left\| f\right\| _{L^1\left( \Omega \right) }^n=1\). To prove the unboundedness of \(P_m\) for \(m>1\), we simply notice that

$$\begin{aligned} P_m\left( (f-1)^{\otimes (m-1)}\otimes F_n\right) = (f-1)^{\otimes (m-1)}\otimes P_1 F_n. \end{aligned}$$
(3.31)

\(\square \)

4 Boundedness of \(P_m\) on \(H^1 \left( {\mathbb {D}}^{\mathbb {N}}\right) \)

The projection \(P_m\) can be described even more explicitly in the case \(\Omega ={\mathbb {T}}\). Indeed, if \(n\in {\mathbb {Z}}^{\oplus {\mathbb {N}}}\) is supported on the set A, then

$$\begin{aligned} P_A\mathrm {e}^{i\langle n,t\rangle }= \bigotimes _{j\in A}\left( \mathrm {id}-{\mathbb {E}}\right) \mathrm {e}^{i n_j t_j}= \prod _{j\in A}\mathrm {e}^{i n_j t_j}= \mathrm {e}^{i\langle n,t\rangle }. \end{aligned}$$
(4.1)

Thus

$$\begin{aligned} \mathrm {e}^{i \langle n,t\rangle }\in U_{\mathrm {supp}\,n} \end{aligned}$$
(4.2)

and

$$\begin{aligned} U_m=\overline{\mathrm {span}}\,\left\{ \mathrm {e}^{i \langle n,t\rangle }:|\mathrm {supp}\,n|=m\right\} . \end{aligned}$$
(4.3)

In particular, \(P_m\) preserves the space \(H^2\left( {\mathbb {D}}^{\mathbb {N}}\right) = L^2\left( {\mathbb {T}}^{\mathbb {N}}\right) \cap H^1\left( {\mathbb {D}}^{\mathbb {N}}\right) \).

In order to adapt the proof of Theorem 3.1 to the \(H^1\left( {\mathbb {D}}^{\mathbb {N}}\right) \) case, we will need a replacement for the argument proving that \(P_1\) is bounded. The role of the combination of Burkholder–Gundy and Doob inequalities will be played by the following theorem, which can be found in [3].

Theorem 4.1

(Bourgain) For \(f\in H^1_{\mathrm {last}}\left( {\mathbb {T}}^{\mathbb {N}}\right) \), there is an equivalence of norms

$$\begin{aligned} \Vert f\Vert _{L^1\left( {\mathbb {T}}^{\mathbb {N}}\right) }\simeq \Vert f\Vert _{H^1\left[ \left( {\mathcal {F}}_n\right) _{n=0}^\infty \right] }, \end{aligned}$$
(4.4)

where \(\left( {\mathcal {F}}_n\right) _{n=0}^\infty \) is the natural filtration on \({\mathbb {T}}^{\mathbb {N}}\).

For later use, we note the Hilbert space valued extension.

Corollary 4.2

Let B be a Hilbert space. For \(f\in H^1_{\mathrm {last}}\left( {\mathbb {T}}^{\mathbb {N}},B\right) \), there is an equivalence of norms

$$\begin{aligned} \Vert f\Vert _{L^1\left( {\mathbb {T}}^{\mathbb {N}},B\right) }\simeq \Vert f\Vert _{H^1\left[ \left( {\mathcal {F}}_n\right) _{n=0}^\infty ,B\right] }=\int _{{\mathbb {T}}^{\mathbb {N}}} \left( \sum _{k=0}^\infty \left\| \Delta _k f(t)\right\| _{B}^2 \right) ^\frac{1}{2}\mathrm {d}t ,\end{aligned}$$
(4.5)

where \(\left( {\mathcal {F}}_n\right) _{n=0}^\infty \) is the natural filtration on \({\mathbb {T}}^{\mathbb {N}}\).

Proof

Theorem 4.1 gives a map

$$\begin{aligned} T: H^{1}_{\mathrm {last}}\left( {\mathbb {T}}^{\mathbb {N}}\right) \rightarrow L^1\left( {\mathbb {T}}^{\mathbb {N}},\ell ^2\right) , \end{aligned}$$
(4.6)

which is an isomorphism onto the subspace of \(L^1\left( {\mathbb {T}}^{\mathbb {N}},\ell ^2\right) \) consisting of functions f such that \(f_k\) is a k-th martingale difference and is analytic in the k-th variable, defined by

$$\begin{aligned} Tf=\left( \Delta _k f\right) _{k=0}^\infty . \end{aligned}$$
(4.7)

Thus, applying Lemma 2.8 with \(I_1\) being a singleton, \(I_2={\mathbb {N}}\), T as above (and then the same for \(T^{-1}\)) we get

$$\begin{aligned} \Vert f\Vert _{H^1_{\mathrm {last}}\left( {\mathbb {T}}^{\mathbb {N}},B\right) }\simeq \Vert \left( T\otimes \mathrm {id}_{B}\right) f\Vert _{L^1\left( {\mathbb {T}}^{\mathbb {N}}, B\right) } =\int _{{\mathbb {T}}^{\mathbb {N}}} \left( \sum _k \left\| \Delta _k f(t)\right\| _{B}^2 \right) ^\frac{1}{2}\mathrm {d}t. \end{aligned}$$
(4.8)

\(\square \)

The role of the Stein martingale inequality will be played by the following simple observation.

Corollary 4.3

For any sequence \(\left( f_n:n\in {\mathbb {N}}\right) \) adapted to the natural filtration on \(\Omega ^{\mathbb {N}}\),

$$\begin{aligned} {\mathbb {E}}\left( \sum _{n=1}^\infty \left| f_n\right| ^2 \right) ^\frac{1}{2} > rsim {\mathbb {E}}\left( \sum _{n=1}^\infty \left| {\mathbb {E}}_{\{n\}}f_n\right| ^2 \right) ^\frac{1}{2}. \end{aligned}$$
(4.9)

Proof

Let \({\tilde{f}}_n\) be a sequence of functions on \(\Omega ^{\mathbb {N}}\times \Omega ^{\mathbb {N}}\) defined by

$$\begin{aligned} {\tilde{f}}_n\left( x,y\right) =f_n\left( x_1,\ldots ,x_{n-1},y_n\right) . \end{aligned}$$
(4.10)

Applying Theorem 2.9 and conditional expectation with respect to the second of two sets of variables,

$$\begin{aligned} {\mathbb {E}}\left( \sum _{n=1}^\infty \left| f_n\right| ^2 \right) ^\frac{1}{2} > rsim & {} {\mathbb {E}}\left( \sum _{n=1}^\infty \left| {\tilde{f}}_n\right| ^2 \right) ^\frac{1}{2} \end{aligned}$$
(4.11)
$$\begin{aligned}\ge & {} {\mathbb {E}}\left( \sum _{n=1}^\infty \left| \left( {\mathbb {E}}\otimes \mathrm {id}\right) {\tilde{f}}_n\right| ^2 \right) ^\frac{1}{2} \end{aligned}$$
(4.12)
$$\begin{aligned}= & {} {\mathbb {E}}\left( \sum _{n=1}^\infty \left| 1\otimes {\mathbb {E}}_{\{n\}} f_n\right| ^2 \right) ^\frac{1}{2} \end{aligned}$$
(4.13)
$$\begin{aligned}= & {} {\mathbb {E}}\left( \sum _{n=1}^\infty \left| {\mathbb {E}}_{\{n\}} f_n\right| ^2 \right) ^\frac{1}{2}. \end{aligned}$$
(4.14)

\(\square \)

By conditioning with respect to the first set of variables, we obtain the inequality

$$\begin{aligned} {\mathbb {E}}\left( \sum \left| f_n\right| ^2\right) ^\frac{1}{2} > rsim {\mathbb {E}}\left( \sum \left| {\mathbb {E}}_{n-1}f_n\right| ^2\right) ^\frac{1}{2} \end{aligned}$$
(4.15)

due to Lepingle [17].

Theorem 4.4

For any \(\Omega \), \(P_1\) is bounded on \(H^1\left[ \left( {\mathcal {F}}_n\right) _{n=0}^\infty \right] \).

Proof

We proceed as in the proof of Theorem 3.1. First, we reduce the problem to the \(\Omega ^{[1,N]}\) realm. Then we notice that

$$\begin{aligned} \Delta _k P_1= P_{\{k\}}= {\mathbb {E}}_{\{k\}}\Delta _k, \end{aligned}$$
(4.16)

which by Corollary 4.3 yields

$$\begin{aligned} \left\| P_1f\right\| _{H^1}= & {} \left\| \left( \sum _{k=0}^N \left| \Delta _k P_1 f\right| ^2 \right) ^\frac{1}{2}\right\| _{L^1} \end{aligned}$$
(4.17)
$$\begin{aligned}= & {} \left\| \left( \sum _{k=0}^N \left| {\mathbb {E}}_{\{k\}} \Delta _k f\right| ^2 \right) ^\frac{1}{2}\right\| _{L^1} \end{aligned}$$
(4.18)
$$\begin{aligned}\lesssim & {} \left\| \left( \sum _{k=0}^N \left| \Delta _k f\right| ^2 \right) ^\frac{1}{2}\right\| _{L^1} \end{aligned}$$
(4.19)
$$\begin{aligned}= & {} \left\| f\right\| _{H^1}. \end{aligned}$$
(4.20)

\(\square \)

Theorem 4.5

\(P_m\) is bounded on \(H^1\left( {\mathbb {D}}^{\mathbb {N}}\right) \) with norm \(\le \frac{1}{\mathrm {e}}c_1^m\), where

(4.21)

Proof

The case \(m=0\) is trivial, \(m=1\) follows directly from Theorems 4.1 and 4.4. The induction step is identical to the proof of Theorem 3.1, up to changing \(L^p\left( \Omega ^I\right) \) to \(H^1\left( {\mathbb {D}}^I\right) \). Alternatively, we can prove the same in a single step. Set

(4.22)

It is easily seen that for each set B of cardinality m, \(P_B\) appears \(m!\left( {\begin{array}{c}(n-1)m\\ n-1,\ldots ,n-1\end{array}}\right) \) times in the sum. Therefore

$$\begin{aligned} \left\| P_1:H^1\left( {\mathbb {D}}^\infty \right) \right\| ^m \ge \left\| Q_{m,n}\right\| \end{aligned}$$
(4.23)
(4.24)

and since

$$\begin{aligned} \lim _{n\rightarrow \infty } \frac{\left( {\begin{array}{c}nm\\ n,\ldots ,n\end{array}}\right) }{m!\left( {\begin{array}{c}(n-1)m\\ n-1,\ldots ,n-1\end{array}}\right) }= & {} \lim _{n\rightarrow \infty } \frac{\frac{(nm)!}{n!^m}}{m!\frac{(nm-m)!}{(n-1)!^m}} \end{aligned}$$
(4.25)
$$\begin{aligned}= & {} \lim _{n\rightarrow \infty } \frac{nm}{m}n^{-m} \end{aligned}$$
(4.26)
$$\begin{aligned}= & {} \frac{m^m}{m!} \end{aligned}$$
(4.27)

we get

(4.28)

\(\square \)

It has to be noted that our proofs of Theorems 3.1 and 4.5 extend naturally to a vector valued case, respectively UMD and AUMD valued. Indeed, Bourgain’s proof of Theorem 2.4, as presented in [20], extends to the UMD valued version, while Theorem 4.1 is just the statement that a one-dimensional space has the AUMD property. In both cases, the induction follows without change. There is also a second direction in which we can generalize. Namely, by looking carefully at the proof of Theorem 4.1, one can see that the only place in which analyticity plays a role is the \(H^1=H^2\cdot H^2\) theorem, which is true for \(H^1\) on any compact and connected group with ordered dual [22], which means that we can replace \({\mathbb {T}}\) with any such group.

Given that Kwapień’s constant \(c_p\) in Theorem 3.1 has the best known asymptotics as a function of p for \(m=1\), one can ask about the dependence of and on m.

Proposition 4.6

The inequalities

(4.29)

for nontrivial \(\Omega \) and

(4.30)

where \(L^p_0\) and \(H^1_0\) stand for functions of mean 0, are true. Also,

(4.31)

Proof

Let \(f\in L^p\left( \Omega ^n\right) \) be of mean 0. Then \(f^{\otimes m}\in L^p\left( \Omega ^{mn}\right) \) and

(4.32)

Indeed, we have \(f=\sum _{|A|\ge 1}P_A f\) because \(P_{\emptyset }f={\mathbb {E}}f=0\), hence

$$\begin{aligned} f^{\otimes m}= \sum _{\begin{array}{c} A_i\subset [n(i-1)+1,ni]\\ \left| A_i\right| \ge 1\text { for }i=1,\ldots ,m \end{array}} \bigotimes _{i=1}^m P_{A_i}f \end{aligned}$$
(4.33)

The only way to get a summand in \(U_m\) is to have \(\left| A_i\right| =1\) for all i and the sum of such summands is the right hand side of (4.32). Taking an f which is close to attaining the norm of \(P_1\) on a respective space proves (4.29) and (4.30).

In order to see (4.31), assume for the sake of contradiction that \(P_1\) is a contraction on \(H^1_0\left( {\mathbb {D}}^2\right) \). We will test it on functions of the form \({\mathbb {T}}^2 \ni (w,z)\mapsto F(z)+w+azw \in {\mathbb {C}}\), where \(F\in H^1_0\left( {\mathbb {D}}\right) \) and a is a scalar. It is easy to see that

$$\begin{aligned} {\mathbb {E}}\left| \alpha + \beta w\right| ={\mathbb {E}}\left| |\alpha |+|\beta |w\right| \end{aligned}$$
(4.34)

for \(\alpha , \beta \in {\mathbb {C}}\). Hence, from the inequality

$$\begin{aligned} {\mathbb {E}}\left| F(z)+w(1+az)\right| \ge {\mathbb {E}}\left| F(z)+w\right| \end{aligned}$$
(4.35)

we get

$$\begin{aligned} {\mathbb {E}}\left| \left| F(z)\right| +w(1+az)\right| \ge {\mathbb {E}}\left| \left| F(z)\right| +w\right| . \end{aligned}$$
(4.36)

Since any nonnegative function can be approximated by the modulus of an \(H^1_0\left( {\mathbb {D}}\right) \) function, (4.35) is true for any nonnegative F. In particular, the left hand side attains a local minimum at \(a=0\), so by \(|u+v|=|u|+\mathrm {Re}\,\frac{u{\overline{v}}}{|u|}+o(v)\) we infer that

$$\begin{aligned} \mathrm {Re}\,{\mathbb {E}}\frac{\left( F(z)+w\right) {\overline{wz}}}{|F(z)+w|}=0. \end{aligned}$$
(4.37)

Now let

$$\begin{aligned} \phi (r)= {\mathbb {E}}\frac{1+r{\overline{w}}}{|1+r{\overline{w}}|} \end{aligned}$$
(4.38)

for \(r\ge 0\). This is a continuous function, whose values lie on some curve \(\gamma \) connecting 0 and 1 (because \(\phi (0)=1\) and \(\lim _{r\rightarrow \infty }\phi (r)=0\)). The condition (4.37) can be rewritten as

$$\begin{aligned} \mathrm {Re}\,{\mathbb {E}}{\overline{z}}\phi (F(z))=0. \end{aligned}$$
(4.39)

Since F was allowed to be any positive function, \(\phi (F)\) can be any function with values in \(\gamma \), making (4.39) obviously false. \(\square \)

5 Martingale Hardy spaces

5.1 Double indexed martingales

Above we noticed that the boundedness of \(P_1\) on \(H^1\left( {\mathbb {D}}^{\mathbb {N}}\right) \) follows from the boundedness of \(P_1\) on a bigger space \(H^1\left[ \left( {\mathcal {F}}_n\right) \right] \). It is tempting to find an abstract martingale inequality responsible for the boundedness of \(P_m\) on \(H^1\left( {\mathbb {D}}^{\mathbb {N}}\right) \). We can do this for \(m=2\).

By the natural double-indexed filtration on \(\Omega ^{\mathbb {N}}\) we will mean the family \(\left( {\mathcal {F}}_{[a,b]}:a\le b\right) \) (note that the inclusion order in the first index is reversed). Let \(\Delta _n={\mathbb {E}}_n-{\mathbb {E}}_{n-1}\) be the martinagle differences with respect to \(\left( {\mathcal {F}}_n\right) \) and \(\Delta ^*_n={\mathbb {E}}^*_n-{\mathbb {E}}^*_{n+1}\) be the martingale differences with repsect to \(\left( {\mathcal {F}}^*_n\right) \), where \({\mathcal {F}}^*_n={\mathcal {F}}_{[n,\infty )}\). We define the martingale differences with respect to \(\left( {\mathcal {F}}_{[a,b]}\right) \) by

$$\begin{aligned} \Delta _{[a,b]}= \Delta _{a}^*\Delta _b = {\mathbb {E}}_{[a+1,b-1]}+{\mathbb {E}}_{[a,b]}- {\mathbb {E}}_{[a+1,b]}-{\mathbb {E}}_{[a,b-1]} \end{aligned}$$
(5.1)

and an \(H^1\) norm for this filtration by

$$\begin{aligned} \Vert f\Vert _{H^1\left[ \left( {\mathcal {F}}_{[a,b]}\right) \right] }= {\mathbb {E}}\left( \left| {\mathbb {E}}f\right| ^2+\sum _{1\le a\le b} \left| \Delta _{[a,b]}f\right| ^2 \right) ^\frac{1}{2}. \end{aligned}$$
(5.2)

The definition of double martingale differences coincides with what is considered in [25].

Corollary 5.1

For \(f\in H^1\left( {\mathbb {D}}^{\mathbb {N}}\right) \), there is an equivalence of norms

$$\begin{aligned} \Vert f\Vert _{H^1\left( {\mathbb {D}}^{\mathbb {N}}\right) }\simeq \Vert f\Vert _{H^1\left[ \left( {\mathcal {F}}_{[a,b]}\right) _{a\le b}\right] }, \end{aligned}$$
(5.3)

where \(\left( {\mathcal {F}}_{[a,b]}\right) _{a\le b}\) is the natural double-indexed filtration on \({\mathbb {T}}^{\mathbb {N}}\).

Proof

For any \(\pm 1\)-valued sequence \(\left( \varepsilon _n:n\in {\mathbb {N}}\right) \), we define operators \(S_\varepsilon \) and \(S^*_\varepsilon \) by

$$\begin{aligned} S_\varepsilon f={\mathbb {E}}f+ \sum _{n=1}^\infty \varepsilon _n \Delta _n f, \quad S^*_\varepsilon f={\mathbb {E}}f+ \sum _{n=1}^\infty \varepsilon _n \Delta ^*_n f. \end{aligned}$$
(5.4)

By Theorem 4.1, \(S_\varepsilon \) is an isomorphism from \(H^1\left( {\mathbb {D}}^{\mathbb {N}}\right) \) to itself, uniformly in \(\varepsilon \). By reversing the order of variables, the same can be said about \(S^*_\varepsilon \). Thus for any \(\varepsilon , \varepsilon '\),

$$\begin{aligned} \Vert f\Vert _{H^1\left( {\mathbb {D}}^{\mathbb {N}}\right) }\simeq & {} \left\| S_\varepsilon S^*_{\varepsilon '}f\right\| _{H^1\left( {\mathbb {D}}^{\mathbb {N}}\right) } \end{aligned}$$
(5.5)
$$\begin{aligned}= & {} \left\| {\mathbb {E}}f+\sum _{a\le b}\varepsilon _a\varepsilon '_b \Delta ^*_a\Delta _b f\right\| _{H^1\left( {\mathbb {D}}^{\mathbb {N}}\right) } \end{aligned}$$
(5.6)
$$\begin{aligned}= & {} {\mathbb {E}}\left| {\mathbb {E}}f+\sum _{a\le b}\varepsilon _a\varepsilon '_b \Delta _{[a,b]}f\right| . \end{aligned}$$
(5.7)

By averaging the last quantity over all choices of \(\varepsilon , \varepsilon '\) and applying the Khintchine-Kahane inequality twice, we get the desired inequalities. \(\square \)

Theorem 5.2

\(P_2\) is bounded on \(H^1\left[ \left( {\mathcal {F}}_{a,b}\right) _{a\le b}\right] \), for any \(\Omega \).

Proof

As usual, we reduce the problem to the \(\Omega ^{[1,N]}\) version. By (3.2),

$$\begin{aligned} \Delta _{[a,b]}=\sum _{\begin{array}{c} \min A=a\\ \max A=b \end{array}}P_{A}. \end{aligned}$$
(5.8)

Thus

$$\begin{aligned} \Delta _{[a,b]}P_2= P_{\{a,b\}}= {\mathbb {E}}_{\{a,b\}}\Delta _{[a,b]} \end{aligned}$$
(5.9)

for \(a<b\). We can assume that \(P_0 f=P_1 f=0\) (i.e. \({\mathbb {E}}f=0\) and \(\Delta _{[a,a]}f=0\) for all a), because \(U_{\le 1}\), being the image of \({\mathbb {E}}+ \sum _a \Delta _{[a,a]}\) is trivially complemented in the underlying norm and \(P_2\) is 0 on \(U_{\le 1}\). By applying Corollary 2.10,

$$\begin{aligned} \Vert f\Vert _{H^1}= & {} {\mathbb {E}}\left( \sum _{a< b}\left| \Delta _{[a,b]}f\right| ^2 \right) ^\frac{1}{2} \end{aligned}$$
(5.10)
$$\begin{aligned}= & {} \int _{\Omega ^N} \left( \sum _{a<b} \left| \Delta _{[a,b]}f\left( x_{[a,b]}\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}x \end{aligned}$$
(5.11)
$$\begin{aligned}\simeq & {} \int _{\left( \Omega ^N\right) ^3} \left( \sum _{a<b} \left| \Delta _{[a,b]}f\left( z_a,x_{[a+1,b-1]},y_b\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}x \mathrm {d}y \mathrm {d}z \end{aligned}$$
(5.12)
$$\begin{aligned}\ge & {} \int _{\left( \Omega ^N\right) ^2} \left( \sum _{a<b} \left| \int _{\Omega ^N}\Delta _{[a,b]}f\left( z_a,x_{[a+1,b-1]},y_b\right) \mathrm {d}x\right| ^2 \right) ^\frac{1}{2}\mathrm {d}y \mathrm {d}z \end{aligned}$$
(5.13)
$$\begin{aligned}\simeq & {} {\mathbb {E}}\left( \sum _{a<b}\left| {\mathbb {E}}_{\{a,b\}}\Delta _{[a,b]}f\right| ^2 \right) ^\frac{1}{2} \end{aligned}$$
(5.14)
$$\begin{aligned}= & {} {\mathbb {E}}\left( \sum _{a<b}\left| \Delta _{[a,b]}P_2 f\right| ^2 \right) ^\frac{1}{2} \end{aligned}$$
(5.15)
$$\begin{aligned}= & {} \left\| P_2f\right\| _{H^1} \end{aligned}$$
(5.16)

as desired.\(\square \)

5.2 Multiple indexed martingales

We will make an attempt at generalizing the above for multiple indexed martingales. Suppose there is a family \(\left( T_i,\partial T_i\right) _{i\in {\mathcal {I}}}\) of pairs of finite subsets of some set X (finite or not) indexed by some set \({\mathcal {I}}\), such that \(\partial T_i\subseteq T_i\) (\(\partial T_i\) is not a boundary in a topological sense - we use this notation for resemblance with the case where \(T_i\) are intervals and \(\partial T_i\) are their endpoints). We would like to define operators \(\Delta _i\) on \(L^2\left( \Omega ^X\right) \) by the formula

$$\begin{aligned} \Delta _i= \left( \mathrm {id}-{\mathbb {E}}\right) ^{\otimes \partial T_i}\otimes \mathrm {id}^{\otimes T_i{\setminus } \partial T_i}\otimes {\mathbb {E}}^{\otimes T_i'}, \end{aligned}$$
(5.17)

where \(T_i'\) stands for the complement of \(T_i\) in X. This is supposed to mimic the standard martingale differences when \(X={\mathbb {N}}\), \({\mathcal {I}}={\mathbb {N}}\), \(T_i=[0,i]\), \(\partial T_i=\{i\}\) and double martingale differences when \({\mathcal {I}}=\left\{ \left( a,b\right) :a\le b\right\} \), \(T_{a,b}=[a,b]\), \(\partial T_{a,b}=\{a,b\}\). The natural condition

$$\begin{aligned} \sum _i \Delta _{i} f= f \end{aligned}$$
(5.18)

is guaranteed by

$$\begin{aligned} \text {for any finite }A\subset X, \text {there exists unique }i\in {\mathcal {I}}\text { such that } \partial T_i\subseteq A\subseteq T_i. \end{aligned}$$
(5.19)

Indeed,

$$\begin{aligned} \Delta _i= \left( \mathrm {id}-{\mathbb {E}}\right) ^{\otimes \partial T_i}\otimes \mathrm {id}^{\otimes T_i{\setminus } \partial T_i}\otimes {\mathbb {E}}^{\otimes T_i'} \end{aligned}$$
(5.20)
(5.21)
$$\begin{aligned}= & {} \left( \mathrm {id}-{\mathbb {E}}\right) ^{\otimes \partial T_i} \otimes {\mathbb {E}}^{\otimes T_i'}\otimes \sum _{B\subseteq T_i{\setminus } \partial T_i} \left( \mathrm {id}-{\mathbb {E}}\right) ^{\otimes B}\otimes {\mathbb {E}}^{\otimes T_i{\setminus }\left( \partial T_i\cup B\right) } \end{aligned}$$
(5.22)
$$\begin{aligned}= & {} \sum _{B\subseteq T_i{\setminus } \partial T_i}P_{\partial T_i\cup B}. \end{aligned}$$
(5.23)

Hence

$$\begin{aligned} \sum _{i\in {\mathcal {I}}} \Delta _i= \sum _{i\in {\mathcal {I}}} \sum _{B\subseteq T_i{\setminus } \partial T_i}P_{\partial T_i\cup B} \end{aligned}$$
(5.24)

and each \(P_A\) appears in the above sum exactly once if and only if the condition (5.19) is satisfied. For a family \(\left( T_i,\partial T_i\right) _{i\in {\mathcal {I}}}\) we may define a norm by the formula

$$\begin{aligned} \Vert f\Vert _{H^1\left[ \left( T_i,\partial T_i\right) _{i\in {\mathcal {I}}}\right] }= {\mathbb {E}}\left( \sum _{i\in {\mathcal {I}}} \left| \Delta _i f\right| ^2 \right) ^\frac{1}{2} \end{aligned}$$
(5.25)

and ask the following:

  • Is it true that

    $$\begin{aligned} \Vert f\Vert _{H^1\left[ \left( T_i,\partial T_i\right) _{i\in {\mathcal {I}}}\right] }\simeq \Vert f\Vert _{H^1\left( {\mathbb {D}}^{\mathbb {N}}\right) } \end{aligned}$$
    (5.26)

    for \(f\in \Vert f\Vert _{H^1\left( {\mathbb {D}}^{\mathbb {N}}\right) }\)?

  • If yes, is there any interesting example of a set \({\mathbb {N}}^{\oplus \infty }\subset \Gamma \subset {\mathbb {Z}}^{\oplus \infty }\) such that (5.26) is true for \(f\in L^1\left( {\mathbb {T}}^\infty \right) \) with \(\mathrm {supp}\,{\widehat{f}}\subset \Gamma \)?

  • For which, if any, m is \(P_m\) bounded on \(H^1\left[ \left( T_i,\partial T_i\right) _{i\in {\mathcal {I}}}\right] \)?

We are able to answer them in the case when

$$\begin{aligned}&{\mathcal {I}}=\left\{ A\subset {\mathbb {N}}: |A|\le m\right\} \end{aligned}$$
(5.27)
$$\begin{aligned}&\partial T_A=A,\quad T_A= \left\{ \begin{array}{lcr} A&{}\text { if }&{}|A|<m,\\ \left( 0,\min A\right) \cup A&{}\text { if }&{} |A|=m. \end{array}\right. \end{aligned}$$
(5.28)

For a finite set \(B\subset {\mathbb {N}}\), the unique \(A\in {\mathcal {I}}\) such that \(\partial T_A\subseteq B\subseteq T_A\), which we will denote by \(\partial B\), is

$$\begin{aligned} \partial B=\left\{ \begin{array}{lcr} B&{}\text { if }&{}|B|<m,\\ m\text { last elements of }B&{} \text { if }&{} |B|\ge m. \end{array}\right. \end{aligned}$$
(5.29)

Theorem 5.3

Let \(m\ge 1\) be fixed and \(\left( T_i,\partial T_i\right) _{i\in {\mathcal {I}}}\) be defined by (5.27), (5.28). Then

$$\begin{aligned} \Vert f\Vert _{H^1_{m\,\mathrm {last}}\left( {\mathbb {T}}^{\mathbb {N}}\right) }\simeq \Vert f\Vert _{H^1\left[ \left( T_i,\partial T_i\right) _{i\in {\mathcal {I}}}\right] } \end{aligned}$$
(5.30)

for \(f\in H^1_{m\,\mathrm {last}}\left( {\mathbb {T}}^{\mathbb {N}}\right) \), where \({\mathbb {T}}\) is used as \(\Omega \). Moreover, for \(m'\in {\mathbb {N}}\) and nontrivial \(\Omega \), the following are equivalent.

  1. (i)

    \(m'\le m\)

  2. (ii)

    \(P_{m'}\) is bounded on \(H^1\left[ \left( T_i,\partial T_i\right) _{i\in {\mathcal {I}}}\right] \)

  3. (iii)

    \(P_{m'}\) is bounded on \(H^1_{m\,\mathrm {last}}\left( {\mathbb {T}}^{\mathbb {N}}\right) \).

Proof

For \(A,B\subset {\mathbb {N}}\), we will write \(T^{(m)}_A\), \(\partial T^{(m)}_A\), \(\Delta ^{(m)}_A\), \(\partial ^{(m)}B\) to indicate the value of m we are currently using. For brevity we will denote \(\left( T^{(m)}_A,\partial T^{(m)}_A\right) _{|A|\le m}\) by \({\mathcal {T}}_m\). For \(|A|<m\), we have \(\Delta ^{(m)}_A=\left( \mathrm {id}-{\mathbb {E}}\right) ^{\otimes A}\otimes {\mathbb {E}}^{{\mathbb {N}}{\setminus } A}= P_A\). In particular, \(\Delta ^{(m)}_{\emptyset }= {\mathbb {E}}\). Therefore, by definition of the \(H^1\left[ {\mathcal {T}}_m\right] \) norm and Corollary 2.11,

$$\begin{aligned} \Vert f\Vert _{H^1\left[ {\mathcal {T}}_m\right] }&\simeq \int _{\Omega ^{\mathbb {N}}}\int _{\left( \Omega ^{\mathbb {N}}\right) ^m} \left( \sum _{i_1<\ldots<i_m}\left| \Delta ^{(m)}_i f\left( x_{<i_1},y^{(1)}_{i_1},\ldots ,y^{(m)}_{i_m}\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}y^{(1,\ldots ,m)}\mathrm {d}x\nonumber \\&\quad +\sum _{0<s<m} \int _{\left( \Omega ^{\mathbb {N}}\right) ^s}\left( \sum _{i_1<\ldots <i_s}\left| P_i f\left( y^{(1)}_{i_1},\ldots ,y^{(s)}_{i_s}\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}y^{(1,\ldots ,s)}+ \left| {\mathbb {E}}f\right| . \end{aligned}$$
(5.31)

Here, we identify an increasing sequence with the set of its elements, write \(\mathrm {d}y^{(1,\ldots ,m)}\) to denote \(\mathrm {d}y^{(1)}\ldots \mathrm {d}y^{(m)}\) and treat \(\Delta ^{(m)}_if\) as a function on \(\Omega ^{\left[ 1,i_1-1\right] }\times \Omega ^{\left\{ i_1,\ldots ,i_m\right\} }\). From this expression, we immediately see the implication \((i)\implies (ii)\). Indeed, for \(m'<m\),

$$\begin{aligned} \Delta ^{(m)}_A P_{m'}=\left\{ \begin{matrix}\Delta ^{(m)}_A&{}\text { if }&{}|A|=m'\\ 0&{}\text { if }&{}|A|\ne m',\end{matrix}\right. \end{aligned}$$
(5.32)

which trivializes the inequality \(\Vert f\Vert _{H^1\left[ {\mathcal {T}}_m\right] } > rsim \Vert P_{m'}f\Vert _{H^1\left[ {\mathcal {T}}_m\right] }\). For \(m'=m\), we notice that

$$\begin{aligned} \Delta ^{(m)}_A P_{m}=\left\{ \begin{array}{lll}{\mathbb {E}}^{\otimes \left[ 1,\min A-1\right] }\Delta ^{(m)}_A&{}\text { if }&{}|A|=m\\ 0&{}\text { if }&{}|A|\ne m,\end{array}\right. \end{aligned}$$
(5.33)

and the desired inequality follows from

$$\begin{aligned} \Vert f\Vert _{H^1\left[ {\mathcal {T}}_m\right] }\ge&\int _{\Omega ^{\mathbb {N}}}\int _{\left( \Omega ^{\mathbb {N}}\right) ^m} \left( \sum _{i_1<\ldots<i_m}\left| \Delta ^{(m)}_i f\left( x_{<i_1},y^{(1)}_{i_1},\ldots ,y^{(m)}_{i_m}\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}y^{(1,\ldots ,m)}\mathrm {d}x \end{aligned}$$
(5.34)
$$\begin{aligned} \ge&\int _{\left( \Omega ^{\mathbb {N}}\right) ^m} \left( \sum _{i_1<\ldots<i_m}\left| \int _{\Omega ^{\mathbb {N}}}\Delta ^{(m)}_i f\left( x_{<i_1},y^{(1)}_{i_1},\ldots ,y^{(m)}_{i_m}\right) \mathrm {d}x\right| ^2 \right) ^\frac{1}{2}\mathrm {d}y^{(1,\ldots ,m)} \end{aligned}$$
(5.35)
$$\begin{aligned} =&\int _{\left( \Omega ^{\mathbb {N}}\right) ^m} \left( \sum _{i_1<\ldots<i_m}\left| \Delta ^{(m)}_i P_m f\left( x_{<i_1},y^{(1)}_{i_1},\ldots ,y^{(m)}_{i_m}\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}y^{(1,\ldots ,m)} \end{aligned}$$
(5.36)
$$\begin{aligned} =&\Vert P_m f\Vert _{H^1\left[ {\mathcal {T}}_m\right] }. \end{aligned}$$
(5.37)

The implication \((ii)\implies (iii)\) follows from (5.30), which we will prove by induction with respect to m. For \(m=1\) this is just Theorem 4.1. Suppose it is true for some m and let \(f\in H^1_{m+1\,\mathrm {last}}\left( {\mathbb {T}}^{\mathbb {N}}\right) \). In particular, \(f\in H^1_{m\,\mathrm {last}}\left( {\mathbb {T}}^{\mathbb {N}}\right) \). By (5.30), which is now the induction hypothesis, and (5.31),

$$\begin{aligned} \Vert f\Vert _{L^1\left( {\mathbb {T}}^{\mathbb {N}}\right) }&\simeq \int _{{\mathbb {T}}^{\mathbb {N}}}\int _{\left( {\mathbb {T}}^{\mathbb {N}}\right) ^m} \left( \sum _{i_1<\ldots<i_m}\left| \Delta ^{(m)}_i f\left( x_{<i_1},y^{(1)}_{i_1},\ldots ,y^{(m)}_{i_m}\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}y^{(1,\ldots ,m)}\mathrm {d}x\nonumber \\&\quad +\sum _{0<s<m} \int _{\left( {\mathbb {T}}^{\mathbb {N}}\right) ^s}\left( \sum _{i_1<\ldots <i_s}\left| P_i f\left( y^{(1)}_{i_1},\ldots ,y^{(s)}_{i_s}\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}y^{(1,\ldots ,s)}+ \left| {\mathbb {E}}f\right| . \end{aligned}$$
(5.38)

The last two summands are as they are in the desired expression for \(m+1\) instead of m and we only have to deal with the first. For any \(i_1<\cdots <i_m\) and \(t\in {\mathbb {T}}^{\mathbb {N}}\) we have

$$\begin{aligned} \Delta ^{(m)}_{i_1,\ldots ,i_m}f(t)= \sum _{\partial ^{(m)}\mathrm {supp}\,n=i }{\widehat{f}}(n)\mathrm {e}^{i\langle n,t\rangle }. \end{aligned}$$
(5.39)

Thus, treating \(\Delta ^{(m)}_i f\) as a function on \({\mathbb {T}}^{\left[ 1,i_1-1\right] }\times {\mathbb {T}}^{\left\{ i_1,\ldots ,i_m\right\} }\),

$$\begin{aligned} \Delta ^{(m)}_{i_1,\ldots ,i_m}f\left( x_{<i_1},y_1,\ldots ,y_m\right) = \sum _{\partial ^{(m)}\mathrm {supp}\,n=i }{\widehat{f}}(n)\mathrm {e}^{i\sum _{j<i_1}n_j x_j+i\sum _{1\le j\le m}n_{i_j}y_j }. \end{aligned}$$
(5.40)

Let y be fixed and \(\Delta ^{(1)}_k\), where \(k\in {\mathbb {N}}\cup \{\emptyset \}\), act with respect to the variable \(x\in {\mathbb {T}}^{\mathbb {N}}\) (so, technically, \(\Delta ^{(1)}_k\) stands for \(\Delta ^{(1)}_k\otimes \mathrm {id}\)). Then

$$\begin{aligned} \Delta ^{(1)}_\emptyset \Delta ^{(m)}_{i_1,\ldots ,i_m}f\left( x_{<i_1},y_1,\ldots ,y_m\right) = P_i f\left( y_1,\ldots ,y_m\right) \end{aligned}$$
(5.41)

and

$$\begin{aligned} \Delta ^{(1)}_k \Delta ^{(m)}_{i_1,\ldots ,i_m}f\left( x_{<i_1},y_1,\ldots ,y_m\right) =0\quad \text { for }k\ge i_1. \end{aligned}$$
(5.42)

For \(k<i_1\),

$$\begin{aligned}&\Delta ^{(1)}_k\Delta ^{(m)}_{i_1,\ldots ,i_m}f\left( x_{<i_1},y_1,\ldots ,y_m\right) \end{aligned}$$
(5.43)
$$\begin{aligned} =&\sum _{\partial ^{(m)}\mathrm {supp}\,n=i }{\widehat{f}}(n)\mathrm {e}^{i\sum _{1\le j\le m}n_{i_j}y_j }\Delta ^{(1)}_k\mathrm {e}^{i\sum _{j<i_1}n_j \cdot _j}\left( x\right) \end{aligned}$$
(5.44)
$$\begin{aligned} =&\sum _{\begin{array}{c} \partial ^{(m)}\mathrm {supp}\,n=i\\ \max \left( \mathrm {supp}\,n{\setminus } i\right) =k \end{array}}{\widehat{f}}(n)\mathrm {e}^{i\sum _{1\le j\le m}n_{i_j}y_j +i\sum _{j\le k}n_j x_j} \end{aligned}$$
(5.45)
$$\begin{aligned} =&\sum _{\partial ^{(m+1)}\mathrm {supp}\,n=\left\{ k,i_1,\ldots ,i_m\right\} }{\widehat{f}}(n) \mathrm {e}^{i\sum _{j<k}n_j x_j + in_k x_k +i \sum _{1\le j\le m}n_{i_j}y_j} \end{aligned}$$
(5.46)
$$\begin{aligned} =&\Delta ^{(m+1)}_{k,i_1,\ldots ,i_m}f\left( x_{<k},x_k,y_1,\ldots ,y_m\right) . \end{aligned}$$
(5.47)

By (5.40), \(\Delta ^{(m)}_{i_1,\ldots ,i_m}f\left( x_{<i_1},y_1,\ldots ,y_m\right) \) is in \(H^1_{\mathrm {last}}\left( {\mathbb {T}}^{\mathbb {N}}\right) \) with respect to x. Therefore, applying Corollary 4.2 to the vector valued function \(x\mapsto \left( \Delta ^{(m)}_{i}f\left( x_{<i_1},y^{(1)}_{i_1},\ldots ,y^{(m)}_{i_m}\right) \right) _{i_1<\ldots <i_m}\) with fixed \(y^{(1,\ldots ,m)}\), plugging in (5.41), (5.42), (5.47) and using Corollary 2.11, we get

$$\begin{aligned}&\int _{{\mathbb {T}}^{\mathbb {N}}} \left( \sum _{i_1<\ldots<i_m}\left| \Delta ^{(m)}_i f\left( x_{<i_1},y^{(1)}_{i_1},\ldots ,y^{(m)}_{i_m}\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}x \end{aligned}$$
(5.48)
$$\begin{aligned}&\quad = \int _{{\mathbb {T}}^{\mathbb {N}}} \left\| \left( \Delta ^{(m)}_i f\left( x_{<i_1},y^{(1)}_{i_1},\ldots ,y^{(m)}_{i_m}\right) \right) _{i_1<\ldots <i_m}\right\| _{\ell ^2}\mathrm {d}x \end{aligned}$$
(5.49)
$$\begin{aligned}&\quad \simeq \left\| \left( \Delta ^{(1)}_\emptyset \Delta ^{(m)}_i f\left( x_{<i_1},y^{(1)}_{i_1},\ldots ,y^{(m)}_{i_m}\right) \right) _{i_1<\ldots <i_m}\right\| _{\ell ^2} \end{aligned}$$
(5.50)
$$\begin{aligned}&\qquad + \int _{{\mathbb {T}}^{\mathbb {N}}}\left( \sum _k \left\| \left( \Delta ^{(1)}_k\Delta ^{(m)}_i f\left( x_{<i_1},y^{(1)}_{i_1},\ldots ,y^{(m)}_{i_m}\right) \right) _{i_1<\ldots <i_m}\right\| _{\ell ^2}^2 \right) ^\frac{1}{2}\mathrm {d}x \end{aligned}$$
(5.51)
$$\begin{aligned}&\quad =\left\| \left( P_{i_1,\ldots ,i_m} f\left( y^{(1)}_{i_1},\ldots ,y^{(m)}_{i_m}\right) \right) _{i_1<\ldots <i_m}\right\| _{\ell ^2} \end{aligned}$$
(5.52)
$$\begin{aligned}&\qquad + \int _{{\mathbb {T}}^{\mathbb {N}}} \left( \sum _{k<i_1<\ldots<i_m}\left| \Delta ^{(m+1)}_{k,i_1,\ldots ,i_m} f\left( x_{<k}, x_{k},y^{(1)}_{i_1},\ldots ,y^{(m)}_{i_m}\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}x \end{aligned}$$
(5.53)
$$\begin{aligned}&\quad = \left( \sum _{i_1<\ldots <i_m}\left| P_{i_1,\ldots ,i_m}f \left( y^{(1)}_{i_1},\ldots ,y^{(m)}_{i_m}\right) \right| ^2 \right) ^\frac{1}{2} \end{aligned}$$
(5.54)
$$\begin{aligned}&\qquad + \int _{{\mathbb {T}}^{\mathbb {N}}} \int _{{\mathbb {T}}^{\mathbb {N}}}\left( \sum _{k<i_1<\ldots<i_m}\left| \Delta ^{(m+1)}_{k,i_1,\ldots ,i_m} f\left( x_{<k}, y^{(0)}_{k},y^{(1)}_{i_1},\ldots ,y^{(m)}_{i_m}\right) \right| ^2 \right) ^\frac{1}{2}\mathrm {d}x\mathrm {d}y^{(0)}. \end{aligned}$$
(5.55)

Integrating the resulting equivalence with respect to \(y^{(1,\ldots ,m)}\) and plugging into (5.38), we verify that \(\Vert f\Vert _{L^1\left( {\mathbb {T}}^{\mathbb {N}}\right) }\simeq H^1\left[ {\mathcal {T}}_{m+1}\right] \), which finishes the proof of (5.30).

In order to see that \((iii)\implies (i)\), let us take \(m'>m\). For any \(g\in L^1\left( {\mathbb {T}}^n\right) \), the function \(G\in L^1\left( {\mathbb {T}}^{\mathbb {N}}\right) \) defined by

$$\begin{aligned} G\left( t\right) =g\left( t_1,\ldots ,t_n\right) e^{i\sum _{j=n+1}^{n+m}t_j} \end{aligned}$$
(5.56)

is in \(H^1_{m\,\mathrm {last}}\left( {\mathbb {T}}^{\mathbb {N}}\right) \). But

$$\begin{aligned} \left( P_{m'}G\right) (t)=\left( P_{m'-m}g\right) \left( t_1,\ldots ,t_n\right) e^{i\sum _{j=n+1}^{n+m}t_j}, \end{aligned}$$
(5.57)

so

(5.58)
$$\begin{aligned} \quad = \frac{\int _{{\mathbb {T}}^{\mathbb {N}}}\left| \left( P_{m'-m}g\right) \left( t_1,\ldots ,t_n\right) \right| \mathrm {d}t}{\int _{{\mathbb {T}}^{\mathbb {N}}}\left| g\left( t_1,\ldots ,t_n\right) \right| \mathrm {d}t}, \end{aligned}$$
(5.59)

which by Proposition 3.2 can be arbitrarily big.\(\square \)

It is worth noting that by repeating the above proof of the equivalence between \(H^1_{m\text { last}}\) norm and \(H^1\left[ {\mathcal {T}}_m\right] \), one can obtain

$$\begin{aligned} \left\| f\right\| _{H^p\left[ {\mathcal {T}}_m\right] }\simeq _p \Vert f\Vert _{L^p} \end{aligned}$$
(5.60)

where \(H^p\left[ {\mathcal {T}}_m\right] \) is defined in a natural way. Moreover, by iterating the \(\Vert f\Vert _{H^1}\ge \Vert f\Vert _{L^1}\) inequality for linearly ordered martingales,

$$\begin{aligned} \Vert f\Vert _{H^1\left[ {\mathcal {T}}_m\right] } > rsim \Vert f\Vert _{L^1}. \end{aligned}$$
(5.61)