Weighted and multivariate Johnson--Schechtman inequalities with application to interpolation theory

TL;DR

This paper extends classical Johnson--Schechtman inequalities to weighted and multivariate cases, deriving new decomposition theorems for moments of generalized U-statistics and exploring their implications for interpolation theory.

Contribution

It introduces weighted and multivariate versions of Johnson--Schechtman inequalities, leading to new decomposition results for U-statistics and insights into interpolation properties of function spaces.

Findings

Weighted Johnson--Schechtman inequality established

Decomposition theorem for p-th moments of U-statistics derived

Interpolation properties of certain function spaces analyzed

Abstract

We prove a weighted version of a classical inequality of Johnson and Schechtman from which we derive a decomposition theorem for $p$-th moments ($0<p\leq 1$) of nonnegative generalized $U$-statistics with constant not dependent on $p$. In particular, for $1\leq p\leq 2$, the norm in the subspace $U^p_{\leq m}\left(\Omega^\infty\right)$ of $L^p\left(\Omega^\infty\right)$ spanned by functions dependent on at most $m$ variables is equivalent to the norm in a suitable interpolation sum of $L^p\left(L^2\right)$ spaces. As a consequence, we obtain some interpolation properties of $U^1_m\left(\Omega^\infty,\ell^p\right)$ that are known to imply cotype 2 of $L^1/U_{\leq m}^1\left(\Omega^\infty\right)$.

Equations

$${\Vert \sum _i{X}_i\Vert }_{L^p(\Omega )}{\simeq }_p\max \left(\sum _i{\Vert X_i\Vert }_{L^1},{\left(\sum _i{\Vert X_i\Vert }_{L^p}^p\right)}^{\frac{1}{p}}\right)$$

$${\Vert \sum _i{X}_i\Vert }_{L^p}{\simeq }_p\underset{X_i=Y_i+Z_i}{\inf }\sum _i{\Vert Y_i\Vert }_{L^1}+{\left(\sum _i{\Vert Z_i\Vert }_{L^p}^p\right)}^{\frac{1}{p}},$$

$${\Vert \sum _i{X}_i\Vert }_{L^p}{\simeq }_p\{\begin{array}{ll}{\Vert {⨆}_i{X}_i\Vert }_{L^1\cap L^p\left(\overline{\Omega }\right)}& \text{ for }1\le p<\infty \\ {\Vert {⨆}_i{X}_i\Vert }_{L^1+L^p\left(\overline{\Omega }\right)}& \text{ for }0<p\le 1\end{array}{\simeq }_p{\Vert \underset{i}{⨆}{X}_i\Vert }_{L^{{\varphi }_p}\left(\overline{\Omega }\right)}$$

$${\Vert \sum _{i_1<\dots <i_m}{f}_{i_1,\dots ,i_m}\left(X_{i_1}^{(1)},\dots ,X_{i_m}^{(m)}\right)\Vert }_{L^p}$$

$${\Vert \sum _{i_1<\dots <i_m}{f}_{i_1,\dots ,i_m}\left(X_{i_1},\dots ,X_{i_m}\right)\Vert }_{L^p}.$$

$${\Vert \sum X_i\Vert }_{L^p}{\simeq }_p{\Vert \sum {\left|X_i\right|}^2\Vert }_{L^{\frac{p}{2}}}^{\frac{1}{2}}$$

$$\sum _{i_1<\dots <i_m}{f}_{i_1,\dots ,i_m}\left(X_{i_1},\dots ,X_{i_m}\right)$$

$${\left(\sum _{i_1<\dots <i_m}{\left|f_{i_1,\dots ,i_m}\left(X_{i_1},\dots ,X_{i_m}\right)\right|}^2\right)}^{\frac{1}{2}}.$$

$$\sum _{i_1<\dots <i_m}{f}_{i_1,\dots ,i_m}\left(X_{i_1},\dots ,X_{i_m}\right)$$

$$\left(X_1,X_2\right)\text{ is }K\text{-closed in }\left(L^1,L^2\right).$$

$$\left(X\left({\ell }^1\right),X\left({\ell }^2\right)\right)\text{ is }K\text{-closed in }\left(L^1\left({\ell }^1\right),L^2\left({\ell }^2\right)\right),$$

$$\mathbb{E}{\left(\sum _i{\left|W_i{X}_i\right|}^r\right)}^{\frac{1}{r}}\ge \frac{1}{2}{\kappa }^r\underset{X_i=Y_i+Z_i}{\inf }{\left(\sum \mathbb{E}{Y}_i^r\right)}^{\frac{1}{r}}+\sum \mathbb{E}{Z}_i.$$

$${\int }_{{\Omega }^n}{\int }_{{\Omega }^n}{\left(\sum _{i,j}{\left|f_{i,j}\left(x_i,y_j\right)\right|}^p\right)}^{\frac{1}{p}}\mathrm{d}x\mathrm{d}y\simeq \underset{{⨆}_{i,j}{f}_{i,j}=a+b+c+d}{\inf }$$

$$+$$

$${\int }_{{\Omega }^n}{\int }_{{\Omega }^n}{\left(\sum _{i,j}{\left|f_{i,j}\left(x_i,y_j\right)\right|}^p\right)}^{\frac{1}{p}}\mathrm{d}x\mathrm{d}y\le 1,$$

$$f_{i,j}=a_{i,j}+b_{i,j}+c_{i,j}+d_{i,j}$$

$$\sum _{i,j}{\int }_{\Omega }{\int }_{\Omega }\left|a_{i,j}\left(\xi ,\upsilon \right)\right|\mathrm{d}\xi \mathrm{d}\upsilon \lesssim 1,$$

$${\left(\sum _{i,j}{\int }_{\Omega }{\int }_{\Omega }{\left|b_{i,j}\left(\xi ,\upsilon \right)\right|}^p\mathrm{d}\xi \mathrm{d}\upsilon \right)}^{\frac{1}{p}}\lesssim 1,$$

$$\sum _i{\int }_{\Omega }{\left(\sum _j{\int }_{\Omega }{\left|c_{i,j}\left(\xi ,\upsilon \right)\right|}^p\mathrm{d}\upsilon \right)}^{\frac{1}{p}}\mathrm{d}\xi \lesssim 1,$$

$$\sum _j{\int }_{\Omega }{\left(\sum _i{\int }_{\Omega }{\left|d_{i,j}\left(\xi ,\upsilon \right)\right|}^p\mathrm{d}\xi \right)}^{\frac{1}{p}}\mathrm{d}\upsilon \lesssim 1.$$

$${\Vert f\Vert }_{L^p\left({\Omega }^n\right)}\simeq \underset{f_{i,j}=a_{i,j}+b_{i,j}+c_{i,j}+d_{i,j}}{\inf }$$

$$+$$

$${\mathbb{E}}_{k}f\left(x\right)={\int }_{{\Omega }^{[k+1,\infty )}}f\left(x_1,\dots ,x_k,y_{k+1},y_{k+2},\dots \right)\mathrm{d}{\mu }^{\otimes [k+1,\infty )}(y),$$

$${\mathbb{E}}_{A}f(x)={\int }_{{\Omega }^{\mathbb{N}\setminus A}}f\left(x_A,y_{\mathbb{N}\setminus A}\right)\mathrm{d}{\mu }^{\otimes \mathbb{N}\setminus A}(y)$$

$$\left(\underset{k}{⨂}{f}_k\right)(x)=\prod _k{f}_k\left(x_k\right).$$

$$\left(\underset{k}{⨂}{T}_k\right)\left(\underset{k}{⨂}{f}_k\right)=\underset{k}{⨂}{T}_k{f}_k,$$

$$\Vert \underset{k}{⨂}{T}_k:\underset{k}{⨂}{X}_k\to L^p\left(\prod _k{\Omega }_k\right)\Vert \le \prod _k\Vert T_k:X_k\to L^p\left({\Omega }_k\right)\Vert$$

$${\left(\mathbb{E}{\Vert \sum r_i{X}_i\Vert }_B^p\right)}^{\frac{1}{p}}{\simeq }_{p,q}{\left(\mathbb{E}{\Vert \sum r_i{X}_i\Vert }_B^q\right)}^{\frac{1}{q}}.$$

$$\mathbb{E}{\Vert \sum X_i\Vert }_{L^p(S)}{\simeq }_p\mathbb{E}{\Vert {\left(\sum {\left|X_i\right|}^2\right)}^{\frac{1}{2}}\Vert }_{L^p(S)}.$$

Full text

1. 1 Introduction

2. 2 Overview of basic notions and facts

3. 3 Weighted inequalities

4. 4 Decomposition theorem for $U^{p}_{m}\left(\Omega^{\infty}\right)$

5. 4.1 The case $m=2$

6. 4.2 The general case

7. 5 Interpolation of $U^{1}_{m}\left(\Omega^{\infty},L^{p}(\mathbb{R})\right)$

8. 5.1 The case $m=2$

9. 5.2 The general case

10. 6 Interpolation of $U^{p}_{m}\left(\Omega^{\infty},L^{p}(\mathbb{R})\right)$

1. Introduction

The well known inequality due to Rosenthal [28] states that for $1\leq p<\infty$,

[TABLE]

where $X_{i}$ are nonnegative independent random variables on $\left(\Omega,\mathcal{F},\mu\right)$. Originally, it was proved for the purpose of Banach space geometry. The precise growth of the constant as a function of $p$ was found in [19]. In the case of $0<p<1$, it appears that there is no known expression for $\left\|\sum X_{i}\right\|_{L^{p}}$ that would be as explicit as the right hand side of (1.1). Theorems providing two sided bounds for this quantity, valid for all $0<p<\infty$, were proved by Johnson and Schechtman [18], Klass and Nowicki [21] and Latała [24]; see also [9]. All of them contain an Orlicz norm in some form. The most important for us is a special case of the main theorem from [18], namely the inequality

[TABLE]

valid for $0<p\leq 1$. It is a natural counterpart to (1.1) in the following sense. Suppose that $\left(\mathcal{F}_{i}\right)_{i=1}^{\infty}$ are independent and $X_{i}$ is $\mathcal{F}_{i}$-measurable. Since the sequence $\left(X_{i}\right)_{i=1}^{\infty}$ carries the same information as a function $\bigsqcup_{i}X_{i}$ on the disjoint union $\overline{\Omega}=\bigsqcup_{i}\left(\Omega,\mathcal{F}_{i},\mu\right)$ (which is now a sigma-finite measure space), the last two inequalities can be conveniently written as

[TABLE]

where $+$ denotes the interpolation sum and $\phi_{p}$ is an Orlicz function such that $\phi_{p}(t)\simeq t$ for $0\leq t\leq 1$ and $\phi_{p}(t)\simeq t^{p}$ for $t\geq 1$. For more information about Orlicz norms in this context, we refer the reader to [10].

It is a common practice to search for analogues of classical theorems concerning independent random variables in the setting of $U$-statistics, introduced by Hoeffding in [14]. This has been done for CLT (see e.g. [8], [16]), LIL (see e.g. [11], [2], [3]), SLLN (see e.g. [15]) just to name a few. A natural multivariate counterpart to $\left\|\sum X_{i}\right\|_{L^{p}}$ for nonnegative and independent $X_{i}$ is the $p$-th moment of a nonnegative generalized decoupled $U$-statistic, i.e. the quantity

[TABLE]

where $X^{(j)}_{i_{j}}$ are independent random variables and $f_{i_{1},\ldots,i_{m}}$ are nonnegative functions on $\mathbb{R}^{m}$. By virtue of a decoupling inequality due to Zinn [31], if the distribution of $X^{(j)}_{i}$ is the same for all $i,j$, then (1.4) is equivalent to its undecoupled version

[TABLE]

For $p\geq 1$, two-sided bounds for (1.4) in terms of mixed $L^{1}\left(L^{p}\right)$ norms were developed in [12]. They were generalized to Banach space valued $U$-statistics in [1], extending the inequalities of Rosenthal and of Klass and Nowicki. However, the authors indicated the lack of a satisfactory counterpart to these results for $0<p<1$. For more information about $U$-statistics and decoupling we refer to [13].

Let us shift our attention to the mean zero setting. Assuming that $\mathbb{E}X_{i}=0$, for $p\geq 1$, by Marcinkiewicz and Zygmund inequality [25], we have

[TABLE]

which allows to directly translate (1.3) to the mean zero case. A usual multivariate counterpart of independent mean zero variables are generalized canonical $U$-statistics, i.e. sums of the form

[TABLE]

where $X_{i}$ are independent and identically distributed, while $f_{i_{1},\ldots,i_{m}}$ are mean zero in each variable with respect to the law of $X_{i}$. By an inequality due to Bourgain [7, Proposition 7], we get an analogous equivalence of $p$-th moment of (1.7) to the $p$-th moment of a square function

[TABLE]

Bourgain and Kwapień in [7] and [22] considered subspaces $U^{p}_{m}\left(\Omega\right)$ of $L^{p}\left(\Omega\right)$ spanned by random variables of the form

[TABLE]

for all $f_{i_{1},\ldots,i_{m}}\in L^{1}\left(\mathbb{R}^{m}\right)$ mean zero in each argument with respect to the law of $X$. The subspaces $U^{2}_{m}\left(\Omega\right)$ for $m=0,1,2,\ldots$ form an orthogonal decomposition of $L^{2}\left(\Omega\right)$ and it turned out that $U^{p}_{m}\left(\Omega\right)$ is complemented in $L^{p}\left(\Omega\right)$ for $1<p<\infty$, but not for $p\in\{1,\infty\}$. Moreover, by (1.6) and (1.3), $U^{p}_{1}\left(\Omega\right)$ is isomorphic to $L^{2}\cap L^{p}\left(\overline{\Omega}\right)$ or $L^{2}+L^{p}\left(\overline{\Omega}\right)$ when $2\leq p<\infty$ or $1\leq p\leq 2$, respectively. If $\Omega=[0,1]$, then $U^{p}_{1}\left(\Omega\right)$ is isomorphic to $L^{p}\left(\mathbb{R}\right)$ for $1<p<\infty$, but not for $p=1$, see [17]. This makes the case $p=1$ the most interesting to study.

Let us briefly introduce some aspects of inteprolation theory that will be of some importance to us. Let $X_{1}\subset L^{1}$ and $X_{2}=\left(X_{1}\cap L^{2},\|\cdot\|_{L^{2}}\right)$. A desirable property of such a pair is $K$-closedness in $\left(L^{1},L^{2}\right)$, from which one can derive real interpolation spaces between $X_{1}$ and $X_{2}$, see Section 2 for details. It is trivially satisfied if the orthogonal projection on $X_{2}$ is bounded in $L^{1}$. Bourgain proved in [6] that if this projection is a Calderón–Zygmund operator, then

[TABLE]

A little is known about possible weaker assumptions on the projection onto $X_{2}\subset L^{2}$ that would imply (1.10). It has been proved in [30] for an $m$-fold tensor of Riesz projection. It has also been shown in [20] for a tensor of a Riesz projection and a Calderón–Zygmund projection for the usually more difficult $(2,\infty)$ side of the interpolation scale.

Another interesting inteprolation property of subspaces of $L^{1}$ is connected to work of Bourgain [5], Pisier [27] and Xu [30], resulting in a theorem that if $X\subset L^{1}$ is such that

[TABLE]

then $L^{1}/X$ is of cotype 2 and every operator $L^{1}/X\to\ell^{2}$ is 1-summing. In fact, it was originally motivated by the question of cotype of $L^{1}/H^{1}$ answered by Bourgain [5]. Later, it was extended to $X=H^{1}\left(\mathbb{D}^{m}\right)$ in [30] and for $X=H^{1}\left(\mathbb{B}_{m}\right)$ in [6].

Let us turn to a detailed description of the main results of the paper. We are going to provide a weighted version of (1.2), which in particular shows that in this inequality the constant is independent of $p$. Let us state a simplified version of Theorem 3.5.

Theorem A**.**

If $1\leq r<\infty$, and $X_{i}$ are independent and $W_{i}$ are $[0,1]$-valued weigths satisfying $\mathbb{E}\left(W_{i}\mid X_{i}\right)\geq\kappa$ for some constant $\kappa>0$ and all $i$, then

[TABLE]

From this, we derive the following theorem (see Theorem 4.3).

Theorem B**.**

Let $f_{i,j}\in L^{1}\left(\Omega^{2}\right)$ for $i,j=1,\ldots,n$ and $1\leq p<\infty$. Then

[TABLE]

with a constant not dependent on $p$. In more explicit terms, the inequality ‘$\gtrsim$’ means that if

[TABLE]

then there is a decomposition

[TABLE]

such that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Moreover, it can be chosen in such a way that for each $i,j$, supports of $a_{i,j},b_{i,j},c_{i,j},d_{i,j}$ are disjoint.

This is easily seen to be equivalent to the following, see Corollary 4.4.

Theorem C**.**

For $1\leq p\leq 2$ and $f\in U^{p}_{2}\left(\Omega^{n}\right)$ such that $f(x)=\sum_{i<j}f_{i,j}\left(x_{i},x_{j}\right)$,

[TABLE]

Moreover, the decomposition can be chosen such that $a_{i,j},b_{i,j},c_{i,j},d_{i,j}$ are mean zero in each variable.

Both Theorems B and C have natural $m$-variable extensions (see Theorems 4.5 and 4.6 for details). Finally, Theorems A and C are later utilized by us to prove that Hoeffding subspaces enjoy the mentioned interpolation properties (1.10) and (1.11), compare Theorems 5.5 and 6.1.

Theorem D**.**

The couple $\left(U^{1}_{m}\left(\ell^{1}\right),U^{1}_{m}\left(\ell^{2}\right)\right)$ is $K$-closed in $\left(L^{1}\left(\ell^{1}\right),L^{1}\left(\ell^{2}\right)\right)$.

Theorem E**.**

The couple $\left(U^{1}_{m},U^{2}_{m}\right)$ is $K$-closed in $\left(L^{1},L^{2}\right)$.

The paper is organized as follows. In Section 2, we recall some definitions and tools to be used later. In Section 3, we prove a weighted generalization of the classical Johnson–Schechtman inequality, as a byproduct getting a new proof of the historic result with an absolute constant. This weighted inequality will be applied in Section 4 to remove the obstacles that arise while iterating one-variable results leading to a Johnson–Schechtman type decomposition for low moments of nonnegative $U$-statistics. In Section 5, we apply the resulting decomposition theorems to obtain results about real interpolation of $U^{p}_{m}\left(\Omega^{\infty}\right)$ and $U^{1}_{m}\left(\Omega^{\infty},\ell^{p}\right)$ spaces between $p=1$ and $p=2$.

Acknowledgements

Most of results of this paper are taken from my doctoral thesis [29]. I am grateful to my advisors: Fedor Nazarov and Michał Wojciechowski for their mentorship and support, especially their unending willingness to discuss my research.

2. Overview of basic notions and facts

Notation. $\gtrsim,\lesssim,\simeq$ denote $\geq,\leq,=$ respectively up to a consatnt. The expression $\|f\|_{L^{p}(X)}$ is defined as $\left(\int\|f\|_{X}^{p}\right)^{\frac{1}{p}}$ for all $0<p<\infty$, which makes it a norm for $p\geq 1$ and a quasinorm for $p\in(0,1)$.

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

[TABLE]

[TABLE]

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 $1\leq p<\infty$. 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

[TABLE]

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 $T_{k}:X_{k}\to L^{p}\left(\Omega_{k}\right)$ are bounded operators, then we can define an operator $\bigotimes_{k}T_{k}:\bigotimes_{k}X_{k}\to L^{p}\left(\prod_{k}\Omega_{k}\right)$ by the formula

[TABLE]

and easily check that the property

[TABLE]

is satisfied. Indeed, $\bigotimes_{k}T_{k}=\prod_{k}\mathrm{id}_{L^{p}\left(\prod_{j\neq k}\Omega_{j}\right)}\otimes T_{k}$, and any operator of the form $\mathrm{id}\otimes T$ has norm bounded by $\|T\|$, because $\left(\mathrm{id}\otimes T\right)f\left(\omega_{1},\omega_{2}\right)=T\left(f\left(\omega_{1},\cdot\right)\right)\left(\omega_{2}\right)$.

Khintchine’s and related inequalities. First, we recall the classical Khintchine-Kahane inequality.

Theorem 2.1** (Khintchine for $B=\mathbb{C}$, Kahane for $B$ Banach).**

Let $X_{1},\ldots,X_{n}$ be vectors in a Banach space $B$ and $r_{1},\ldots,r_{n}$ be Rademacher variables (i.e. independent random variables, each of them attaining $\pm 1$ with probability $\frac{1}{2}$). Then, for $1\leq p,q<\infty$,

[TABLE]

In particular, $\mathbb{E}\left|\sum r_{i}z_{i}\right|\simeq\left(\sum\left|z_{i}\right|^{2}\right)^{\frac{1}{2}}$ for $z_{i}\in\mathbb{C}$. As a consequence we obtain the Marcinkiewicz-Zygmund moment inequality.

Lemma 2.2**.**

Let $S$ be a (not necessarily probability) measure space. If $X_{1},\ldots,X_{n}$ are independent $L^{p}(S)$-valued mean [math] random variables ($1\leq p<\infty$), then

[TABLE]

Proof. If $X,Y$ are mean [math] and independent, then $\mathbb{E}\|X+Y\|_{L^{p}}\geq\mathbb{E}\|X\|_{L^{p}}$ by taking the conditional expectation with respect to (the sigma-algebra generated by) $X$ and $\mathbb{E}\|X+Y\|_{L^{p}}\leq\mathbb{E}\|X\|_{L^{p}}+\mathbb{E}\|Y\|_{L^{p}}$ because of triangle inequality. Thus $\mathbb{E}\|X+Y\|_{L^{p}}\simeq\mathbb{E}\|X\|_{L^{p}}+\mathbb{E}\|Y\|_{L^{p}}$ and in particular $\mathbb{E}\|X+Y\|_{L^{p}}\simeq\mathbb{E}\|X-Y\|_{L^{p}}$. Hence, there is an equivalence $\mathbb{E}\left\|\sum X_{i}\right\|_{L^{p}}\simeq\mathbb{E}\left\|\sum r_{i}X_{i}\right\|_{L^{p}}$, where $r_{i}$ are Rademacher variables independent of $X_{i}$’s, which is true a.e. on the space where $r_{i}$’s are defined. Let us distinguish the expectations on spaces on which $r_{i}$’s and $X_{i}$’s are defined by denoting them respectively by $\mathbb{E}^{(r)}$ and $\mathbb{E}^{(X)}$. Then

[TABLE]

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

[TABLE]

(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 $F:B_{1}\to B_{2}$ we can define $T\otimes F$ on 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 [25] (in this case $\lesssim\|T\|$ can be replaced with $\leq\|T\|$).

Lemma 2.3**.**

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 $T:X_{1}\to X_{2}$ be bounded. Then $T\otimes\mathrm{id}_{B}:X_{1}\otimes B\to X_{2}\otimes B$, 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 $\lesssim\|T\|$.

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,

[TABLE]

∎

Interpolation. Let us recall basic information about the real interpolation method. The standard reference is [4]. A couple $\left(X_{0},X_{1}\right)$ of Banach spaces is called compatible if $X_{0}$ and $X_{1}$ are embedded in a linear topological space. For a compatible couple $\left(X_{0},X_{1}\right)$ we define the $K$-functional by the formula

[TABLE]

for $f\in X_{0}+X_{1}$. For $Y_{0}\subset X_{0}$, $Y_{1}\subset X_{1}$, we will say that the couple $\left(Y_{0},Y_{1}\right)$ is $K$-closed in $\left(X_{0},X_{1}\right)$ if

[TABLE]

for any $f$ in the algebraic sum $Y_{0}+Y_{1}$ (the reverse inequality holds trivially). This is equivalent to the following property: for any $f\in Y_{0}+Y_{1}$ and any decomposition $f=f_{0}+f_{1}$, where $f_{0}\in X_{0}$, $f_{1}\in X_{1}$, there exists a decomposition $f=\tilde{f_{0}}+\tilde{f_{1}}$ such that $\tilde{f_{0}}\in Y_{0}$, $\tilde{f_{1}}\in Y_{1}$, $\|\tilde{f_{0}}\|_{Y_{0}}\lesssim\left\|f_{0}\right\|_{X_{0}}$, $\|\tilde{f_{1}}\|_{Y_{1}}\lesssim\left\|f_{1}\right\|_{X_{1}}$.

The $K$-functional plays a crucial role in the real interpolation method. The norm in the real interpolation space $\left(X_{0},X_{1}\right)_{\theta,q}$, where $0<\theta<1$ and $1\leq q<\infty$ is defined by

[TABLE]

Operators bounded simultaneously on $X_{0}$ and $X_{1}$ are also bounded on $\left(X_{0},X_{1}\right)_{\theta,q}$. The canonical example is $\left(L^{p_{0}},L^{p_{1}}\right)_{\theta,p_{\theta}}=L^{p_{\theta}}$, where $\frac{1}{p_{\theta}}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$ and, more generally, $\left(L^{p_{0}}\left(\ell^{q_{0}}\right),L^{p_{1}}\left(\ell^{q_{1}}\right)\right)_{\theta,p_{\theta}}=L^{p_{\theta}}\left(\ell^{q_{\theta},p_{\theta}}\right)$. If $\left(Y_{0},Y_{1}\right)$ is $K$-closed in $\left(X_{1},X_{2}\right)$, then it is easily seen that $\left(Y_{0},Y_{1}\right)_{\theta,q}=\left(Y_{0}+Y_{1}\right)\cap\left(X_{0},X_{1}\right)_{\theta,q}$, which is particularly useful in case of couples $K$-closed in Lebesgue spaces.

The algebraic sum $X_{0}+X_{1}$ becomes a Banach space when equipped with the norm

[TABLE]

The intersection $X_{0}\cap X_{1}$ will be equipped with the norm

[TABLE]

It is easily checked that the dual space $\left(X_{0}+X_{1}\right)^{*}$ can be isometrically identified with $X_{0}^{*}\cap X_{1}^{*}$.

Hoeffding decomposition. 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

[TABLE]

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 [math] 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 [7], [22]. In particular, $P_{i_{1},\ldots,i_{m}}$ are pairwise orthogonal orthogonal projections. Let

[TABLE]

and $U_{m}$ be the range of $P_{m}$. It is known [7], [22] 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

[TABLE]

for each $m\geq 0$. The sequence of subspaces $U_{\leq 0}^{2},U_{\leq 1}^{2},\ldots,U_{\leq n}^{2}$ is increasing, so by putting

[TABLE]

we obtain a decomposition

[TABLE]

into an orthogonal direct sum of $U_{m}^{2}$. We will denote the orthogonal projection onto $U_{m}^{2}$ by $P_{m}$ and the closure of $U^{2}_{m}$ equipped with $L^{p}$ norm by $U^{p}_{m}$.

A more explicit formula for $P_{m}$ can be obtained. For $A\subset[1,n]$, let

[TABLE]

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

[TABLE]

and, since the subspaces $U_{B}^{2}$ are mutually orthogonal,

[TABLE]

Moreover

[TABLE]

and consequently

[TABLE]

Decoupling inequalities. We are going to present a special case of a theorem of J. Zinn [31], which will be one of the most important tools.

Theorem 2.4** (Zinn).**

Let $X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{n}$ be independent random variables such that $X_{k}$ and $Y_{k}$ have the same distribution for any $k$. Let also $\varphi_{k}$ be a nonnegative Borel function on $\mathbb{R}^{k}$ and $0<q\leq 1$. Then

[TABLE]

Corollary 2.5**.**

For all $i=\left(i_{1},\ldots,i_{m}\right)$ such that $i_{1}<\ldots<i_{m}$, let $f_{i}$ be an $\mathcal{F}_{\left\{i_{1},\ldots,i_{m}\right\}}$-measurable nonnegative function on $\Omega^{\mathbb{N}}$. Then, treating each $f_{i}$ as a function on $\Omega^{\left\{i_{1},\ldots,i_{m}\right\}}$,

[TABLE]

where $y^{(1)},\ldots,y^{(m)}$ are variables in $\Omega^{\mathbb{N}}$ and $0<q\leq 1$.

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

[TABLE]

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

[TABLE]

Here, $i_{k}$ plays the role of $j$ and (2.32) is an application of Theorem 2.4 to functions $\varphi_{j}$. Integrating the resulting inequality with respect to $y^{(>k)}$, we get

[TABLE]

which by induction from $k=m$ to $k=1$ proves (2.28). ∎

3. Weighted inequalities

If $\left(\Omega_{i},\mu_{i},\mathcal{F}_{i}\right)$ are probability spaces, we can form a (no longer probability) measure space $\left(\bigsqcup_{i}\Omega_{i},\bigsqcup_{i}\mu_{i},\bigsqcup_{i}\mathcal{F}_{i}\right)$ by considering the disjoint union of sets $\Omega_{i}$ with a measure $\bigsqcup_{i}\mu_{i}\left(\bigcup A_{i}\right)=\sum_{i}\mu_{i}\left(A_{i}\right)$ for $A_{i}\in\mathcal{F}_{i}$. We will often write $\mathrm{d}x$ instead of $\mathrm{d}\mu(x)$ if the choice of measure is clear.

Our main motivation is the following special case of a theorem due to Johnson and Schechtman.

Theorem 3.1** (Johnson, Schechtman [18]).**

Let $f_{i}\in L^{1}(\Omega)$ for $i=1,\ldots,n$ and $1\leq p<\infty$. Then

[TABLE]

In more explicit terms, the inequality ‘$\gtrsim$’ means that if

[TABLE]

then there exists a decomposition $f_{i}=g_{i}+h_{i}$ such that

[TABLE]

Moreover, the decomposition can be chosen so that for any $i$ the supports of $g_{i}$ and $h_{i}$ are disjoint.

This can be use to expresses $L^{p}$ norm, $1\leq p\leq 2$ (the case $\infty>p\geq 2$ is handled by Rosenthal inequality), on $U_{1}^{p}$ as a rearrangement invariant norm on the disjoint union $\bigsqcup_{i}\Omega$ in the following way. For $f\in U_{1}^{p}$, we have

[TABLE]

for some $f_{i}$ of mean [math] and consequently

[TABLE]

by Marcinkiewicz-Zygmund inequality. Applying Theorem 3.1 with expontent $\frac{2}{p}$ to $\left|f_{i}\right|^{p}$, we get an equivalent form of Theorem 3.1.

Corollary 3.2**.**

For $1\leq p\leq 2$ and $f_{i}\in L^{p}(\Omega)$ of mean zero,

[TABLE]

The building blocks $g_{i}$, $h_{i}$ of $g,h$ in the above can be also chosen to be of mean [math], because

[TABLE]

In this section we will develop a weighted version of Theorem 3.1 (in particular producing a new proof with constant independent on $p$), which will be Theorem 3.5, which for $J$ a singleton, $w\equiv 1$, $\kappa=1$ and $\varepsilon=0$ gives a constant $\frac{1}{2}$.

We start with a calculation lemma.

Lemma 3.3**.**

Let $\phi:\mathbb{R}^{\mathbb{N}}\to\mathbb{R}$ be a function differentiable outside of [math] and satisfying $\phi(cx)=c\phi(x)$ for $c>0$. Then

[TABLE]

Proof.

Let $|\cdot|$ denote the Euclidean norm. By direct calculation we verify that

[TABLE]

Therefore

[TABLE]

Thus

[TABLE]

∎

The following inequality is useful for obtaining lower bounds for subspaces of vector valued $L^{1}$.

Lemma 3.4**.**

*Let $N\in\mathbb{N}$, $\left(\Omega,\mathcal{F},\mu\right)$ be a finite probability space, $V_{k}$ be subspaces of $L^{2}\left(\Omega\right)$, $V$ be the subspace of $L^{2}\left(\Omega,\ell^{2}_{N}\right)$ consisting of sequences $\left(f_{n}\right)_{n=1}^{N}$ of functions such that $f_{n}\in V_{n}$, $P_{V_{k}}$, $P_{V}$ be orthogonal projections onto $V_{k}$, $V$ respectively, $\|\cdot\|_{X}$ be a random norm on $\mathbb{R}^{N}$ differentiable outside of [math], $\|\cdot\|_{Y}$ be a norm on $V$, $Y^{*}$ be the dual norm on $V$ in the sense of the usual pairing. Then for any constants $C>0$, $q>1$, the following are equivalent:

(i) for any $\left(f_{1},\ldots,f_{N}\right)\in V$,*

[TABLE]

(ii) for any $\left(f_{1},\ldots,f_{N}\right)\in V$ not identically zero,

[TABLE]

where $\nabla\|\cdot\|_{X}$ is extended to be equal to [math] in [math]. Moreover, if for any $\left(f_{1},\ldots,f_{N}\right)\in V$ not identically zero,

[TABLE]

where $F=\left\{\varphi\in V:\mathrm{supp}\,\varphi_{i}\subset\mathrm{supp}\,f_{i}\right\}$, then

[TABLE]

for all $f\in V$.

Proof.

Let us start with $q>1$. The implication (i)$\implies$(ii) is true for each $f$ separately. Indeed, if (3.13) holds, then by self-adjointness of $P_{V_{n}}$ and Lemma 3.3 applied to $\|\cdot\|_{X}$ pointwise to $f$,

[TABLE]

We will prove the implication (ii)$\implies$(i) now. Let us assume $q>1$ first. We are in a fully finite-dimensional setting. It is enough to prove

[TABLE]

for any $g=\left(g_{1},\ldots,g_{N}\right)\in V$ such that

[TABLE]

By homogeneity, we can set

[TABLE]

Since $\left(\mathbb{E}\|\cdot\|_{X}^{q}\right)^{\frac{1}{q}}$ is a norm on $V$, under the constraint (3.24) it is a convex function going to $\infty$ in infinity, so it attains a minimum. Let $f$ be a minimizer of $\mathbb{E}\|\cdot\|_{X}^{q}$. Suppose for a moment $q>1$. For any $h\in V$ such that $\langle h,g\rangle=0$ we have

[TABLE]

In other words,

[TABLE]

when both are treated as functionals on $V$. Therefore

[TABLE]

for some $\lambda\in\mathbb{R}$, which by (3.13) and (3.23) gives

[TABLE]

Ultimately, utilising Lemma 3.3 again,

[TABLE]

Therefore $\lambda\geq 0$ and thus $\lambda=|\lambda|\geq C\|f\|_{Y}^{q-1}$ as desired.

Now we will prove that (3.15) is sufficient for (3.16). We will proceed in an analogous manner, but we have to take care of nondifferentiability of $\|\cdot\|_{X}$ in [math]. As previously, we pick $g\in V$ such that

[TABLE]

and normalize $f$ to satisfy

[TABLE]

If $f$ is a minimizer of $\mathbb{E}\|f\|_{X}$ given (3.37), then for any $h\in V\cap F$, the function $t\mapsto\|f+th\|_{X}$ is in each point of $\Omega$ either differentiable in [math] (when $f\neq 0$) or identically [math] (when $f=0$ and consequently $h=0$). Thus, for any $h\in V\cap F$ such that $\langle h,g\rangle=0$,

[TABLE]

Therefore $\ker g\subset\ker P_{V\cap F}\left(\nabla\|\cdot\|_{X}(f)\right)$ as functionals on $V\cap F$, so

[TABLE]

in particular

[TABLE]

so $\left|\lambda\right|\geq C$. Thus

[TABLE]

∎

Now, we are ready for the main result of this section. The parameter $\varepsilon$ is for technical reasons and we will make most use of the case $\varepsilon=0$ and $\{0,1\}$-valued $w$, in which case the inequality is true with the constant $\frac{1}{2}\kappa^{2}$. Also, as will be noted in the proof of Theorem 3.1, the decomposition in the interpolation norm on the right hand side may always be chosen to be of disjoint supports at the cost of constant 2. Sometimes we will use a weaker version of Theorem 3.5 with the norm on the right hand side replaced by a smaller norm $L^{1}\left(\Omega,\ell^{p}(I\times J)\right)$ of the sequence $\left(\mathbbm{1}_{\left\{\mathbb{E}_{i}\left(w_{i,j}\vee\varepsilon\right)\geq\kappa\right\}}f_{i,j}\right)_{i,j}$.

Theorem 3.5**.**

Let $\Omega$ be a finite probability space, $I,J$ be finite sets, $\left(\mathcal{F}_{i}\right)_{i\in I}$ be an independent family of sigma-algebras, $f_{i,j}$ be $\mathcal{F}_{i}$-measurable, $w_{i,j}$ be $[0,1]$-valued functions on $\Omega$ and $p\geq 1$, $0\leq\varepsilon<\kappa\leq 1$. Then

[TABLE]

where $C_{p,\kappa}=\kappa^{p}2^{-\frac{1}{p^{\prime}}}$. Moreover, if $w_{i,j}$ are $\{0,1\}$-valued, the inequality holds with constant $C_{p,\kappa}=\left(\kappa-\varepsilon\right)^{2-\frac{1}{p}}2^{-\frac{1}{p^{\prime}}}$.

Proof.

Every time we encounter a fraction that can have [math] denominator, it will also have [math] numerator and it is to be treated as [math]. The notation $t^{r}$ stands for $|t|^{r}\mathrm{sgn}t$ so that $\frac{\mathrm{d}|t|^{p}}{\mathrm{d}t}=pt^{p-1}$. For $p=1$ the inequality is elementary, so we assume $p>1$. We may also assume $\varepsilon>0$ and get $\varepsilon=0$ by taking limits.

Let $A_{i,j}=\left\{\mathbb{E}_{i}\left(w_{i,j}\vee\varepsilon\right)\geq\kappa\right\}$ and $V_{i,j}=\left\{\varphi\in L^{1}\left(\mathcal{F}_{i}\right):\mathrm{supp}\,\varphi\subset A_{i,j}\right\}$. We can without loss of generality assume $f_{i,j}\in V_{i,j}$, because otherwise we replace $f_{i,j}$ by $\mathbbm{1}_{A_{i,j}}f_{i,j}$. We are now in the setting of Lemma 3.4, with $\|f\|_{X}=\left\|\left(\left(w_{i,j}\vee\varepsilon\right)f_{i,j}\right)_{i,j}\right\|_{\ell^{p}(I\times J)}$ and $Y=\left(L^{1}+L^{p}\right)\left(\bigsqcup_{i}\left(\Omega,\mathcal{F}_{i},\mu\right),\ell^{p}(J)\right)$. For a given $f\in V$, we have

[TABLE]

therefore the projection $P_{V\cap F}$ is just

[TABLE]

because the projections onto adapted sequences and onto sequences supported on $\mathrm{supp}\,f_{i,j}$ are respectively $\mathbb{E}_{i}$ and multiplication by $\mathbbm{1}_{\mathrm{supp}\,f_{i,j}}$ applied coordinatewise (in particular they commute). Moreover, $P_{V\cap F}$ is a contraction on $Y^{*}=\left(L^{\infty}\cap L^{p^{\prime}}\right)\left(\bigsqcup_{i}\left(\Omega,\mathcal{F}_{i},\mu\right),\ell^{p^{\prime}}(J)\right)$. One easily calculates

[TABLE]

By Lemma 3.4, it is enough to prove

[TABLE]

because $\frac{\left(w_{i,j}\vee\varepsilon\right)^{p}f_{i,j}^{p-1}}{\left\|\left(w\vee\varepsilon\right)f\right\|_{\ell^{p}(I\times J)}^{p-1}}$ is already supported on $\mathrm{supp}\,f_{i,j}$. Let us fix $i,j$ for a moment. On each atom of $\mathcal{F}_{i}$ contained in $\mathrm{supp}\,f_{i,j}\subset A_{i,j}$ we have by Hölder the inequality

[TABLE]

By rearranging the terms and multiplying by $\left|f_{i,j}\right|^{p-1}$,

[TABLE]

This inequality has been proved on $\mathrm{supp}\,f_{i,j}$, but outside of it both sides are [math], so it is true everywhere. Moreover, on $A_{i,j}$ we have

[TABLE]

If additionally $w_{i,j}$ is $\{0,1\}$-valued,

[TABLE]

Plugging bounds for the factor involving $w$ to (3.53) and then to (3.51), we see that it remains to prove

[TABLE]

We have

[TABLE]

so in (3.58) we can replace $J$ with a singleton and write $f_{i}$ in the place of $\left\|\left(f_{i,j}\right)_{j\in J}\right\|_{\ell^{p}(J)}$, which transforms the inequality into

[TABLE]

for nonnegative and $\mathcal{F}_{i}$-measurable $f_{i}$. Raising both sides to the power $\frac{1}{p-1}$, we end up with

[TABLE]

Suppose

[TABLE]

Then for all $i$,

[TABLE]

so

[TABLE]

Ultimately,

[TABLE]

∎

As a byproduct, we obtain another proof of Theorem 3.1, which we present for the sake of completeness.

Proof of Theorem 3.1. For a fixed $n$, both sides of (3.1) are norms of the vector-valued function $\left(f_{i}\right)_{i=1,\ldots,n}$, dominated by $\sum_{i}\left\|f_{i}\right\|_{L^{1}\left(\Omega\right)}$. We can assume that $f_{i}$’s attain finitely many values due to density of such $\left(f_{i}\right)_{i\in I}$ in $\bigoplus_{i}L^{1}\left(\Omega\right)$. Let $\mathcal{F}$ be the sigma-algebra generated by $f_{1},\ldots,f_{n}$, or equivalently by the partition of $\Omega$ into intersections of their level sets. We can restrict the infimum on the right side of (3.1) to $g_{i},h_{i}$ being $\mathcal{F}$-measurable, because for any decomposition $f_{i}=g_{i}+h_{i}$, the decomposition $f_{i}=\mathbb{E}\left(g_{i}\mid\mathcal{F}\right)+\mathbb{E}\left(h_{i}\mid\mathcal{F}\right)$ produces a smaller result. Thus we can think of $\Omega$ being a finite space, the atoms being elements of the partition. Since both sides of (3.1) are continuous in $\mu$, we can assume that the atoms have measures being multiples of $\frac{1}{N}$ for some $N\in\mathbb{N}$. This enables us to split each of them into atoms of size $\frac{1}{N}$, getting a new probability space $\tilde{\Omega}$ equipped with the normalized counting measure and a new sigma-algebra $\tilde{\mathcal{F}}$, with respect to which $f_{i}$’s are still measurable. By the same argument as before, we can extend the infimum to decompositions measurable with respect to $\tilde{\mathcal{F}}$. Therefore, we can assume without loss of generality that $\Omega$ is finite with normalized counting measure.

The $\lesssim$ inequality of (3.1) is elementary and holds with constant $1$, because for any decomposition $f_{i}=g_{i}+h_{i}$ we have

[TABLE]

The other inequality, is precisely Lemma 3.5 with $J$ being a singleton, $I=\{1,\ldots,n\}$, $w\equiv 1$, $\kappa=1$ and $\varepsilon=0$. It remains to prove that we can choose the desired decomposition so that the summands have disjoint supports. Without loss of generality, $f=\bigsqcup_{i}f_{i}\geq 0$. Let $f=g+h$ be some decomposition satisfying $\int_{\bigsqcup_{i}\Omega}g+\left(\int_{\bigsqcup_{i}\Omega}h^{p}\right)^{\frac{1}{p}}\leq 1$. Then for any $x\in\bigsqcup\Omega$ we have either $g(x)\geq\frac{1}{2}f(x)$ or $h(x)\geq\frac{1}{2}f(x)$. In the former case we put $\tilde{g}(x)=f(x)$, $\tilde{h}(x)=0$ and in the latter $\tilde{g}(x)=0$, $\tilde{h}(x)=f(x)$, choosing arbitrarily if $g(x)=h(x)$. This way we have $\tilde{g}+\tilde{h}=f$ and $\|\tilde{g}\|_{L^{1}}\leq 2\|g\|_{L^{1}}$ and $\|\tilde{h}\|_{L^{p}}\leq 2\|h\|_{L^{p}}$.

In the original version of Theorem 3.1, the decomposition was defined in terms of decreasing rearrangement of $f$. There is also another way of constructing a more explicit decomposition. Define a function $\Phi:\mathbb{R}_{+}\to\mathbb{R}_{+}$ by

[TABLE]

and take

[TABLE]

By convexity of $\Phi$ and the inequality $\Phi(t)\leq\min\left(t^{p},pt\right)$,

[TABLE]

as desired. ∎

4. Decomposition theorem for $U^{p}_{m}\left(\Omega^{\infty}\right)$

We would like to extend Theorem 3.5 to moments of $m$-th order $U$-statistics, and as a result extend Corollary 3.2 to treat $f\in\sum_{m=0}^{M}U_{m}^{p}=U_{\leq M}^{p}$. Let us note the following theorem due to Bourgain [7], which generalizes Marcinkiewicz-Zygmund inequality and in particular implies that $U^{p}_{m}$ is complemented in $U^{p}_{\leq M}$ for $m\leq M$.

Theorem 4.1** (Bourgain).**

Let $f\in U_{\leq M}^{p}$. Then

[TABLE]

Corollary 4.2**.**

Let $f\in U_{\leq M}^{p}$ and $P_{{i_{1}},\ldots,i_{m}}f\left(x\right)=f_{i_{1},\ldots,i_{m}}\left(x_{i_{1}},\ldots,x_{i_{m}}\right)$. Then

[TABLE]

Proof.

This is just a combination of Theorem 4.1 and Corollary 2.5. ∎

Again, for $p\geq 2$ this has been handled by multivariate extension of Rosenthal inequality, see [12]. For $1\leq p\leq 2$, by calling $\left|f_{i}\right|^{p}$ the new $f_{i}$ and $\frac{2}{p}$ the new $p$, we arrive at expressions of the form

[TABLE]

which the main object of our interest will be. For this, we can relax the condition on $p$ in (4.1) to $1\leq p<\infty$ and the sum may be run over $i\in[1,n]^{m}$ instead of $1\leq i_{1}<\ldots<i_{m}\leq n$.

4.1. The case $m=2$

We are going to single out the case $m=2$, because we believe that it will significantly improve the legibility of the proof.

Theorem 4.3**.**

Let $f_{i,j}\in L^{1}\left(\Omega^{2}\right)$ for $i,j=1,\ldots,n$ and $1\leq p<\infty$. Then

[TABLE]

with a constant not dependent on $p$. In more explicit terms, the inequality ‘$\gtrsim$’ means that if

[TABLE]

then there is a decomposition

[TABLE]

such that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Moreover, it can be chosen in such a way that for any $i,j$, supports of $a_{i,j},b_{i,j},c_{i,j},d_{i,j}$ are disjoint.

From this, we immediately derive

Corollary 4.4**.**

For $1\leq p\leq 2$ and $f\in U^{p}_{2}\left(\Omega^{n}\right)$ such that $f(x)=\sum_{i<j}f_{i,j}\left(x_{i},x_{j}\right)$,

[TABLE]

Moreover, the decomposition can be chosen such that $a_{i,j},b_{i,j},c_{i,j},d_{i,j}$ are mean zero in each variable.

Proof.

Just as indicated above, we use Corollary 4.2 and then, in a routine convexification argment, Theorem 4.3 for $\frac{2}{p}$ and $\left|f_{i,j}\right|^{p}$ to get the desired equivalence. The resulting decomposition is then improved by putting all summands to [math] for $i\geq j$ and replacing $a_{i,j}$ etc. by $P_{2}a_{i,j}$ etc. (here, $P_{2}$ acts on functions on $\Omega^{2}$, namely $P_{2}=\left(\mathrm{id}-\mathbb{E}\right)\otimes\left(\mathrm{id}-\mathbb{E}\right)$), which is legal because

[TABLE]

∎

Proof of Theorem 4.3..

Since all the norms involved, and consequently their interpolation sums, depend only on the modulus of a function, we may assume $f_{i,j}\geq 0$. By $\|\cdot\|_{\ell^{p}}\leq\|\cdot\|_{\ell^{1}}$ and $\|\cdot\|_{L^{1}}\leq\|\cdot\|_{L^{p}}$ we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Thus, the $\lesssim$ inequality of (4.2) follows with constant $1$ and the only interesting part is the $\gtrsim$ inequality, i.e. the existence of a decomposition satisfying (4.5)-(4.8).

We perform the discretization procedure as described in the proof of Theorem 3.1, with the only changes being that as the dense set we choose functions that attain finitely many values, each (possibly treated with repetitions) of them on a set of product form, and our atoms also have to be products of atoms in $\Omega$. This allows us to assume that $\Omega$ is finite and equipped with the counting measure. Also, because both sides are lattice norms, we will be content with $a,b,c,d$ satisfying the desired ineqaulity and $a_{i,j}+b_{i,j}+c_{i,j}+d_{i,j}\geq f_{i,j}$ instead of $=f_{i,j}$.

Let $F_{i}$ be a function on $\Omega\times\Omega^{n}$ defined by the formula

[TABLE]

We can write (4.3) in the form

[TABLE]

By fixing $y$ and applying Theorem 3.1 to the sequence of functions $F_{i}\left(\cdot,y\right)$, we get a decomposition

[TABLE]

such that

[TABLE]

[TABLE]

Utilising the condition $\mathrm{supp}\,G_{i}\left(\cdot,y\right)\cap\mathrm{supp}\,H_{i}\left(\cdot,y\right)=\emptyset$ for all $y$ gives

[TABLE]

for some sequence of functions $w_{i}:\Omega\times\Omega^{n}\to\{0,1\}$. Plugging into (4.15) and integrating with respect to $y$ we get

[TABLE]

Let us fix $i$, $\xi$ and denote

[TABLE]

Applying Theorem 3.5 in a setting where $|J|=1$,

[TABLE]

Theorem 3.1 applied to functions $W_{i,j}f_{i,j}\left(x_{i},\cdot\right)$ for $j=1,\ldots,n$ provides $\{0,1\}$-valued functions $u_{i,j}$ such that

[TABLE]

[TABLE]

We may now put

[TABLE]

The desired inequality for $a_{i,j}$ is obtained by integrating (4.20) and (4.19) with respect to $\xi$, summing over $i$ and plugging into (4.17), and analogously for $c_{i,j}$.

By integrating (4.16) with respect to $y$ we get

[TABLE]

We may now incorporate $i$ and $x_{i}$ into one variable running through the space $\bigsqcup_{i}\Omega$. Since $\Omega$ was equipped with the counting measure, $\bigsqcup_{i}\Omega$ also is, up to a constant. This puts us in the setting of Theorem 3.5 and allows to split the sequence of functions $y\mapsto\left(1-w_{i}\right)\left(\xi,y\right)f_{i,j}\left(\xi,y_{j}\right)$ into an $L^{1}\left(\bigsqcup_{j}\Omega,\ell^{p}\left(\bigsqcup_{i}\Omega\right)\right)$ part and an $L^{p}\left(\bigsqcup_{j}\Omega,\ell^{p}\left(\bigsqcup_{i}\Omega\right)\right)$ one. Bearing in mind that

[TABLE]

we may apply Theorem 3.5 to obtain functions $s_{i,j}:\Omega^{2}\to\{0,1\}$ such that

[TABLE]

This allows us to take

[TABLE]

The desired inequalities for $b_{i,j}$ and $d_{i,j}$ are directly verified. By definition of $a_{i,j},b_{i,j},c_{i,j},d_{i,j}$, we have $a_{i,j}+b_{i,j}+c_{i,j}+d_{i,j}=f_{i,j}$. They are also disjointly supported due to $W_{i,j},u_{i,j},s_{i,j}$ being $\{0,1\}$-valued, which ends the proof.∎

4.2. The general case

We will now prove Theorem 4.3 in full generality. For brevity, we will denote $\bigsqcup_{i=1}^{n}\Omega$ by $\overline{\Omega}$, variables running through $\overline{\Omega}$ by $\overline{x},\overline{y},\ldots$ and write $\mathrm{d}\overline{x}=\mathrm{d}\left(\bigsqcup_{i}\mu\right)\left(\overline{x}\right)$. For example, if $\varphi_{i}\in L^{1}\left(\Omega\right)$, then

[TABLE]

Theorem 4.5**.**

For $i=\left(i_{1},\ldots,i_{m}\right)\in\{1,\ldots,n\}^{m}$, let $f_{i}\in L^{1}\left(\Omega^{m}\right)$. Suppose that

[TABLE]

Then, treating $\bigsqcup_{i}f_{i}$ as a function on $\overline{\Omega}^{m}$, we have

[TABLE]

and $a^{(J)}=\bigsqcup_{i}a^{(J)}_{i}$ can be chosen such that for any $i$, the supports of $a^{(J)}_{i}$ for different $J$ are disjoint.

Proof. As previously, $\Omega$ is a finite set with counting measure and $f_{i}$ are nonnegative. In order to show the first inequality of (4.27), we just need to check that each of the norms $\left\|\cdot\right\|_{L^{1}\left(\overline{\Omega}^{J^{\prime}},L^{p}\left(\overline{\Omega}^{J}\right)\right)}$ on functions on $\overline{\Omega}^{m}$ dominates the norm on the left hand side of (4.26). Indeed, if $\overline{a}=\bigsqcup_{i}a_{i}$,

[TABLE]

Here, we used abbreviations $i_{J}=\left(i_{j}\right)_{j\in J}$, $x^{(J)}=\left(x^{(j)}\right)_{j\in J}$, $x^{(J)}_{i_{J}}=\left(x^{(j)}_{i_{j}}\right)_{j\in J}$). The inequality (4.29) is $\|\cdot\|_{L^{1}}\leq\|\cdot\|_{L^{p}}$ with respect to $x^{(J^{\prime})}$, (4.30) is $\left(\sum_{i_{J^{\prime}}}c_{J^{\prime}}\right)^{\frac{1}{p}}\leq\sum_{i_{J^{\prime}}}\left(c_{i_{J^{\prime}}}\right)^{\frac{1}{p}}$, and (4.31) comes from the fact that for fixed $i_{J}$, the integrand depends on $x^{(J)}$ only through $x^{(J)}_{i_{J}}$.

We will now prove the second inequality of (4.27) by induction with respect to $m$. Let us assume that the theorem is true for some $m\geq 1$. The discretization procedure is performed as previously. Let us take $f_{i,k}\in L^{1}\left(\Omega^{m+1}\right)$ for $i_{1},\ldots,i_{m},k\in[1,n]$, $i=\left(i_{j}\right)_{1\leq j\leq m}$. We define $F_{i}\in L^{1}\left(\Omega^{m}\times\Omega^{n}\right)$ by

[TABLE]

for $x_{1},\ldots,x_{m}\in\Omega$ and $y\in\Omega^{n}$. For a fixed $y$, applying the induction hypothesis to the functions $F_{i}\left(\cdot,y\right)\in L^{1}\left(\Omega^{m}\right)$, yields a family of functions $w^{(J)}_{i}:\Omega^{m}\times\Omega^{n}\to\{0,1\}$ such that

[TABLE]

[TABLE]

(we just take $w^{(J)}_{i}=a^{(J)}_{i}/F_{i}$, because without loss of generality $a^{(J)}_{i}\left(x,y\right)=0$ for all $J$ if $F_{i}(x,y)=0$). We used another abbreviation: $x^{(\leq m)}_{i_{\leq m}}=\left(x^{(j)}_{i_{j}}\right)_{j\leq m}$. Let us define

[TABLE]

where $\mathbb{E}_{k}$ is taken with respect to the second variable running through $\Omega^{n}$.

Denote $\overline{w}^{(J)}\left(\cdot,y\right)=\bigsqcup_{i}w^{(J)}_{i}\left(\cdot,y\right)$, $\overline{W}^{(J)}_{k}\left(\cdot,y\right)=\bigsqcup_{i}W^{(J)}_{i,k}\left(\cdot,y\right)$, $\overline{f}_{k}\left(\cdot,y\right)=\bigsqcup_{i}f_{i,k}\left(\cdot,y\right)$. Applying (4.35) at each $y$, and then Theorem 3.5 at each $J$ and $\overline{x}^{(J^{\prime})}$ (treating the integral over $\overline{\Omega}^{J}$ as summation over a finite set) we get

[TABLE]

where

[TABLE]

For each $J$ and $\overline{x}^{(J^{\prime})}$, we can apply Theorem 3.1 to functions $\overline{\varphi}_{k}^{(J)}\left(\overline{x}^{(J^{\prime})},y_{k}\right)$. This produces $u^{(J)}_{k}:\overline{\Omega}^{J^{\prime}}\times\Omega\to\{0,1\}$ such that

[TABLE]

Plugging this into the previous inequality, we get

[TABLE]

where the functions $A^{(J)}_{i,k}:\Omega^{m}\times\Omega\to\mathbb{R}$ for $J\subset[1,m+1]$ are defined by

[TABLE]

Let us recall that by (4.34),

[TABLE]

for all $i,k$ and consequently

[TABLE]

for any $i,k$ and $x\in\Omega^{m}$, which by (4.36) implies

[TABLE]

Therefore

[TABLE]

Ultimately, using the fact that $\sum_{J\subset[1,m+1]}L^{1}\left(\overline{\Omega}^{[1,m+1]\setminus J},L^{p}\left(\overline{\Omega}^{J}\right)\right)$ is a lattice, we get

[TABLE]

as desired, with $C_{m+1}=C2^{2m}C_{m}$ for some numerical constant $C$. Once we have a decomposition verifying (4.27), for each $\overline{x}\in\overline{\Omega}^{m}$ we select $J_{\overline{x}}$ such that

[TABLE]

Now, the functions

[TABLE]

are dominated by $a^{(J)}$ up to the constant $2^{m}$ and their sum over $J$ is $\overline{f}$. ∎

By an identical reasoning as previously, we get

Corollary 4.6**.**

Let $1\leq p\leq 2$ and $f\in U^{p}_{m}\left(\Omega^{n}\right)$ have a representation

[TABLE]

Then

[TABLE]

where the infimum is taken over decompositions of $f$ into summands $a^{(J)}\in U^{p}_{m}$ and

[TABLE]

5. Interpolation of $U^{1}_{m}\left(\Omega^{\infty},L^{p}(\mathbb{R})\right)$

We can extend the definition of spaces $U_{m}^{p}$ to the vector-valued setting. Let $B$ be a Banach space. Then we define $U_{m}^{p}\left(\Omega^{n},B\right)$ as the closure of $U_{m}^{p}\left(\Omega^{n}\right)\otimes B$ in the Bochner space $L^{p}\left(\Omega^{n},B\right)$. In particular, $U_{1}^{p}$ consists of functions of the form $f(x)=\sum_{i}f\left(x_{i}\right)$, where $f_{i}\in L^{p}\left(\Omega,B\right)$ and $\int_{\Omega}f_{i}\left(x_{i}\right)\mathrm{d}x_{i}=0$. By combining Corollary 4.2 with Lemma 2.3, we get

Corollary 5.1**.**

Let $f\in U_{\leq M}^{1}\left(\Omega^{n},B\right)$, where $B$ is a Hilbert space. Then

[TABLE]

Definition 5.2**.**

Let $X\subset L^{1}(S)$ and $X\left(B\right)$ be generated by $X\otimes B$ in $L^{1}\left(S,B\right)$. The subspace $X$ is said to have Bourgain-Pisier-Xu (BPX) property if $\left(X\left(\ell^{1}\right),X\left(\ell^{p}\right)\right)$ is $K$-closed in $\left(L^{1}\left(S,\ell^{1}\right),L^{1}\left(S,\ell^{p}\right)\right)$ for some $1<p<\infty$.

The following is a result of Xu [30, Proposition 11], based on pieces of reasoning by Bourgain [5] and Pisier [27].

Theorem 5.3** (Bourgain, Pisier, Xu).**

If $X\subset L^{1}$ has BPX property, then $L^{1}/X$ is of cotype 2 and every operator $L^{1}/X\to\ell^{2}$ is 1-summing.

5.1. The case $m=2$

As previously, we single out $m=2$.

Theorem 5.4**.**

The couple

[TABLE]

is $K$-closed in

[TABLE]

with a constant independent of $n$.

Proof. For $f\in L^{0}\left(\Omega^{n}\times\mathbb{R}\right)$ and $\alpha>0$, let $\left(f\circ\alpha\right)(x,s)=f(x,\alpha s)$. To shorten the notation, we will denote the underlying couples by $\left(U^{1}_{1}\left(L^{1}\right),U^{1}_{1}\left(L^{2}\right)\right)$ and $\left(L^{1}\left(L^{1}\right),L^{1}\left(L^{2}\right)\right)$. Since

[TABLE]

we have

[TABLE]

Analogously

[TABLE]

Therefore we only have to prove

[TABLE]

for $f\in U^{1}_{1}\left(L^{1}\right)+U^{1}_{1}\left(L^{2}\right)$.

For any such $f$, the decomposition $f(x)=\sum_{i}f_{i}\left(x_{i}\right)$ is unique, because $f_{i}=\mathbb{E}_{i}f$. Let $f_{i,j}:\Omega\times[0,1]\to\mathbb{R}$, $f_{j}:\Omega^{n}\times[0,1]\to\mathbb{R}$ be defined by

[TABLE]

Then $f_{j}(x)=\sum_{i}f_{i,j}\left(x_{i}\right)$ and we can treat $f_{i}$ as $\bigsqcup_{j}f_{i,j}$ and $f$ as $\bigsqcup_{j}f_{j}$, by identification of $(0,\infty)$ with $\bigsqcup_{j}[0,1]$. Using the trivial part of Theorem 3.1 at each $x$ separately, we get

[TABLE]

where the last equivalence is Lemma 2.2 applied at each $s$ to the sequence of $\ell^{2}$-valued independent mean [math] functions $\left(f_{i,j}\left(\cdot,s_{j}\right)\right)_{j}$. Let

[TABLE]

be the decomposition given by Theorem 4.3. We can ensure that $a_{i,j},b_{i,j},c_{i,j},d_{i,j}$ are of mean [math] in the first variable, because $f_{i,j}$ are and subtracting the underlying conditional expectation preserves (4.5)-(4.8). Let

[TABLE]

We have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Ultimately, we put

[TABLE]

By definition of $a_{i},b_{i},c_{i},d_{i}$, we have $f(x)=\sum_{i}g_{i}\left(x_{i}\right)+\sum_{i}h_{i}\left(x_{i}\right)$. Combining the above inequalities,

[TABLE]

as desired.∎

5.2. The general case

We are now prepared for the proof of the following, which by [30], is going to imply that $L^{1}\left(\Omega^{\infty}\right)/U^{1}_{\leq M}\left(\Omega^{\infty}\right)$ is of cotype 2. In this subsection, $\lesssim,\gtrsim,\simeq$ are understood to depend on $M$ or $m$.

Theorem 5.5**.**

The couple

[TABLE]

is $K$-closed in

[TABLE]

with a constant independent of $n$.

Proof. Let us take $f\in U^{1}_{\leq M}\left(\Omega^{n},\left(L^{1}\left(\mathbb{R}\right)\right)\right)+U^{1}_{\leq M}\left(\Omega^{n},L^{2}\left(\mathbb{R}\right)\right)$ with a decomposition

[TABLE]

where

[TABLE]

By density argument, we may assume that $f_{i_{1},\ldots,i_{m}}(x)$ vanishes outside of $[-N,N]$, which allows $f_{i_{1},\ldots,i_{m}}$ to formally be treated as an element of $U^{1}_{m}\left(\Omega^{n},L^{1}\left(\mathbb{R}\right)\right)$. Just as in the proof of Theorem 5.4, we reduce the problem to

[TABLE]

Again, we denote the restriction of $f$ to $[k,k+1]$ in the variable running through $\mathbb{R}$ by $f_{k}$ and an analogous restriction of $f_{i}$ by $f_{i,k}$. As previously, we calculate the right hand side in preparation to use Theorem 4.5.

[TABLE]

Here, the ‘$\simeq$’ inequality is an application of Corollary 5.1 at every $\left(s_{k}\right)_{k\in Z}$. Since the $K$-functional

[TABLE]

is a norm, we can without loss of generality fix $m$, assume that

[TABLE]

and aim at proving

[TABLE]

If we treat $s$ as the $m+1$-th set of variables, Theorem 4.5 applied to the sequence $\left(f_{i,k}\right)_{(i,k)\in[1,n]^{m}\times\mathbb{Z}}$ of functions in $L^{1}\left(\Omega^{m}\times[0,1]\right)$ (we take $f_{i,k}=0$ unless $i_{1}<\ldots<i_{m}$) gives a decomposition

[TABLE]

such that

[TABLE]

where $a_{i,k}^{(J)}$ correspond to subsets of $[1,m+1]$ not containing $m+1$ and $b_{i,k}^{(J)}$ correspond to subsets of $[1,m+1]$ containing $m+1$. Just as remarked after the formulation of Theorem 4.3, we can modify $a_{i,k}^{(J)}$, $b_{i,k}^{(J)}$ by setting

[TABLE]

and replacing $a_{i,k}^{(J)}$ by $\left(P_{m}\otimes\mathrm{id}\right)a_{i,k}^{(J)}$ and $b_{i,k}^{(J)}$ by $\left(P_{m}\otimes\mathrm{id}\right)b_{i,k}^{(J)}$, where $P_{m}$ acts on the first $m$ variables and $\mathrm{id}$ acts on the last. This operation is legal, because $P_{m}=\left(\mathrm{id}-\mathbb{E}\right)^{\otimes m}$ is a finite combination of conditional expectations. Moreover, by (5.2) and (5.6) it produces a decomposition of $f_{i,k}$ into summands that are in $U^{1}_{m}$ with respect to the first $m$ variables. Also, due to boundedness of $P_{m}$ in all mixed $L^{1}\left(L^{2}\right)$ norms, the left hand side of (5.7) gets smaller up to a constant. Therefore, we can without loss of generality assume (5.8) and

[TABLE]

making them suitable for forming

[TABLE]

[TABLE]

both of which are in $U^{1}_{m}$ with respect to the first $n$ variables thanks to (5.9) and satisfy

[TABLE]

because of (5.6). Now, making use of Corollary 4.2 in (5.11) and (4.33) in (5.2),

[TABLE]

Similarly, using Corollary 5.1 in (5.13) and (4.33) in (5.2),

[TABLE]

Summing up the last two inequalities and connecting them with (5.7), we see that (5.10) defines a decomposition of $f$ verifying (5.5), which ends the proof. ∎

6. Interpolation of $U^{p}_{m}\left(\Omega^{\infty},L^{p}(\mathbb{R})\right)$

For $m\geq 2$, the projection $P_{m}$ is not a Calderón–Zygmund operator because its norm on $L^{p}$ behaves as $\left(\frac{p}{\log p}\right)^{2}$ for $p\to\infty$ and we know of no way to represent $P_{1}$ as such an operator. Nonetheless, we are able to show that Bourgain’s result [6] about $K$-closedness of an image of a C-Z projection in $\left(L^{1},L^{2}\right)$ holds for $P_{m}$ as well. Here, we present only the case of $m=1$, while for the general $m$ the proof is, as previously, analogous but more technical.

Theorem 6.1**.**

The couple

[TABLE]

is $K$-closed in

[TABLE]

One can easily recover $K$-closedness of respective scalar-valued spaces by restricting to the subspace consisting of functions with values in $\mathrm{span}\left\{\mathbbm{1}_{[0,1]}\right\}$. It is of independent interest that the proof below will show that a sum of only 3 of 4 interpolation summands appearing in Theorem 4.3 for $p=2$ also has a natural interpretation.

Proof.

Let $f\in U^{1}_{1}\left(\Omega^{\infty},L^{1}\left(\mathbb{R}\right)\right)+U^{2}_{1}\left(\Omega^{\infty},L^{2}\left(\mathbb{R}\right)\right)$. Then $f(x)=\sum_{i}f_{i}\left(x_{i}\right)$, where $f_{i}:\Omega\times\mathbb{R}\to\mathbb{R}$. Denoting by $f_{i,j}$ the restriction of $f_{i}$ to $\Omega\times[j,j+1]$, we can put $f_{i}=\bigsqcup_{j}f_{i,j}$. Our goal is to prove

[TABLE]

Just like in the proof of the theorem about inteprolation of $U^{1}_{m}\left(\ell^{p}\right)$, by means of scaling in $\mathbb{R}$ we can without loss of generality assume that $t=1$. Then the right hand side is the $L^{1}+L^{2}$ norm on the space $\bigsqcup_{j}\left(\Omega^{\infty}\times[0,1]\right)$, so by the trivial part of Johnson-Schechtman inequality

[TABLE]

Let us fix $s$ for a moment. The right hand side equals $\int\mathrm{d}x\left\|\sum_{i\in\mathbb{N}}\left(f_{i,j}\left(x^{(j)}_{i},s_{j}\right)\right)_{j\in\mathbb{Z}}\right\|_{\ell^{2}\left(\mathbb{Z}\right)}$. Therefore, by Marcinkiewicz-Zygmund inequality, the sum over $i$ is unconditional, which allows us to write

[TABLE]

The variables $x^{(j)}_{i}$ are independent, so by Johnson-Schechtman inequality, there are $a_{i,j}:\Omega\times[0,1]^{\infty}\to\{0,1\}$ such that

[TABLE]

By a standard application of Theorem 3.5 for the second summand and trivially for the first,

[TABLE]

[TABLE]

for $\tilde{a}_{i,j}=\mathbbm{1}_{\left\{\mathbb{E}_{j}a_{i,j}\geq\frac{1}{2}\right\}}$, where $\mathbb{E}_{j}$ is taken with respect to $s$. Since

[TABLE]

the first summand no longer features integration over an infinite product. Applying $L^{2}\left(\bigsqcup_{i}\Omega\right)$-valued Johnson-Schechtman inequality to the second summand gives $b_{j}:[0,1]\to\{0,1\}$ such that

[TABLE]

Ultimately, taking $\alpha_{i,j}=\tilde{a}_{i,j}f_{i,j}$, $\beta_{i,j}=b_{i,j}\left(1-\tilde{a}_{i,j}\right)f_{i,j}$, $\gamma_{i,j}=\left(1-b_{i,j}\right)\left(1-\tilde{a}_{i,j}\right)f_{i,j}$ we get

[TABLE]

and

[TABLE]

In order to get a decomposition for the $K$-functional of Hoeffding subspaces, we put $g=\bigsqcup_{j}g_{j}$, $h=\bigsqcup_{j}h_{j}$, where

[TABLE]

Obviously,

[TABLE]

because $\gamma_{i,j}$ are orthogonal. Moreover, by the trivial part of Johnson-Schechtman inequality,

[TABLE]

which plugged into (6.13) proves that $f=g+h$ satisfies the desired inequality.∎