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On a property of the Rademacher system and S. V. Astashkinabcd, E. M. Semenove a Samara National Research University, Samara, Russia b Lomonosov Moscow State University, Moscow, Russia c Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia d Bahcesehir University, Istanbul, Turkey e Voronezh State University, Voronezh, Russia
Russian version:
Matematicheskii Sbornik, 2024, Volume 215, Number 3, DOI: https://doi.org/10.4213/sm9922 
Bibliographic databases:
   IntroductionBy Khintchine’s classical inequality (for instance, see [1], Theorem V.8.4), for each $p\geqslant 1$ and any sequence of real numbers $(a_k)_{k=1}^\infty\in \ell^2$ we have
$$
\begin{equation}
\biggl\|\sum_{k=1}^\infty a_k r_k\biggr\|_{L^p[0, 1]} \leqslant \sqrt{p}\, \|(a_k)\|_{\ell^2},
\end{equation}
\tag{0.1}
$$
where the $r_k$ are the Rademacher functions: $r_k(t)=\operatorname{sign} (\sin 2^k \pi t)$, $k \in \mathbb N$, $t \in [0,1]$, and $\|(a_k)\|_{\ell^2}:=\bigl(\sum_{k=1}^\infty a_k^2\bigr)^{1/2}$. Using (0.1) and expanding the function $M(u)=e^{u^2}-1$ in the Taylor series it is easy to show that the distribution of a function $f:=\sum_{k=1}^\infty a_k r_k$ satisfies
$$
\begin{equation*}
\lim_{\tau\to\infty}m\{t\in [0,1]\colon |f(t)|>\tau\} e^{\tau^2}=0
\end{equation*}
\notag
$$
($m$ is the Lebesgue measure on $[0,1]$). Using the terminology of function spaces we can equivalently express this result as follows. Let $\operatorname{Exp}L^2$ be the Orlicz space constructed for the function $M(u)=e^{u^2}-1$, and let $G$ be the closure of $L^\infty$ in $\operatorname{Exp}L^2$ (for all definitions, see § 1). If $R$ is the closure of the linear span of the Rademacher system in $L^2[0,1]$, then there is a continuous embedding $R\subset G$, that is, for some $C>0$ and any sequence $(a_k)_{k=1}^\infty\in \ell^2$
$$
\begin{equation}
\biggl\|\sum_{k=1}^\infty a_k r_k\biggr\|_{G} \leqslant C\|(a_k)\|_{\ell^2}.
\end{equation}
\tag{0.2}
$$
Furthermore, as a theorem established in [2] shows (also see [3], Theorem 2.b.4, (i), or [4], Theorem 2.3), the latter result is sharp in the following sense: if the inequality
$$
\begin{equation}
\biggl\|\sum_{k=1}^\infty a_k r_k\biggr\|_{X} \leqslant C\|(a_k)\|_{\ell^2},
\end{equation}
\tag{0.3}
$$
holds in a rearrangement invariant space $X$ on $[0,1]$ for some $C>0$ and all $(a_k)_{k=1}^\infty\in \ell^2$, then a continuous embedding $X\supset G$ exists. The first aim of this paper is to improve the above result by showing that functions with arbitrarily large distributions whose ratio to the distribution of a standard normal variable tends to zero exist in the closed linear span $R$ itself (not only in the rearrangement invariant space $X$ satisfying condition (0.3), as follows from the theorem in [2] mentioned above). More precisely, we show that for each measurable function $x(t)$ satisfying
$$
\begin{equation*}
\lim_{\tau\to\infty}m\{t\in [0,1]\colon |x(t)|>\tau\}e^{\tau^2}=0
\end{equation*}
\notag
$$
there exists a function $f=\sum_{k=1}^\infty a_kr_k$, where $(a_k)_{k=1}^\infty\,{\in}\, \ell^2$, such that for some ${C\,{>}\,0}$ we have
$$
\begin{equation*}
m\{t\in [0,1]\colon |x(t)|>\tau\} \leqslant Cm\{t\in [0,1]\colon |f(t)|>\tau\}, \qquad \tau>0.
\end{equation*}
\notag
$$
Turning to the terminology of function spaces and concrete constants, we can state this result as follows. Theorem 1. For each function $x\in G$ there exists a Rademacher sum $f= \sum_{k=1}^\infty a_kr_k$, $(a_k)_{k=1}^\infty\in \ell^2$, such that In particular, it follows from (0.5) that $\|x\|_X\leqslant 2^{7}\|f\|_X$ for each rearrangement invariant space $X$ on $[0,1]$. Note that an initial version of Theorem 1 was established by the authors in [5] (see Theorem 4 there). The subspace $R$ spanned by the Rademacher system in $L^2$ is a model example of a $\Lambda(2)$-space. A subspace $H$ of $L^p[0,1]$, $p\geqslant 1$, is called a $\Lambda(p)$-space if convergence in the $L^p$-norm in $H$ is equivalent to convergence in measure. It is easy to verify that in this case the $L^p$- and $L^1$-norms are equivalent in $H$ (also see [6], Proposition 4.5). Recall that the investigations of $\Lambda(p)$-spaces were initiated in Rudin’s classical paper [7] on Fourier analysis on the circle $[0,2\pi)$, and then they were continued by many authors (in particular, see Bourgain’s deep results in [8]). The second problem treated in this paper concerns the possible extension of the result of Theorem 1 to $\Lambda(2)$-spaces. Its statement has been prompted by a question of Kashin addressed to one of the authors: does each infinite-dimensional subspace $H$ of $L^2[0,1]$ contain functions with ‘almost’ normal (Gaussian) distribution? It is easy to see (cf. Proposition 2) that without some extra assumptions on the subspace the answer is negative. On the other hand, using well-known results on the comparison of sequences of functions with the Rademacher system (see [4], Theorem 8.2, and [9], Theorem V.4.4), we managed to obtain affirmative results for two classes of $\Lambda(2)$-spaces. Namely, when $H$ contains either an orthonormal sequence of functions which are uniformly bounded on some set of positive measure, or a sequence of independent functions distinct from identical constants, then we obtain results similar to Theorem 1 (see Corollaries 1 and 2). In view of the above results, the following problem looks natural. Problem. Let $H$ be a $\Lambda(2)$-space. Do there exist positive constants $C_1$ and $C_2$ such that for each function $x\in G$ there exists a function $g\in H$ such that ${\|g\|_{L^2}\leqslant C_1\|x\|_G}$ and AcknowledgementThe authors are grateful to the referee for her (or his) useful critical observations, which enabled them to improve significantly the presentation of the main results. § 1. PreliminariesWe present some necessary facts on the theory of rearrangement invariant spaces. For more information, see [3] and [10]–[13]. A Banach space $X$ of measurable functions on $[0,1]$ is called rearrangement invariant if the following hold:
We assume without loss of generality that $\|\chi_{[0,1]}\|_X=1$, where $\chi_E (t)$ is the characteristic function of the measurable subset $E$ of $ [0,1]$. Then $L^\infty \subset X \subset L^1$ and $\|x\|_{L^1} \leqslant\|x\|_{X} \leqslant \| x \|_{L^\infty}$ for each rearrangement invariant space $X$ and any $x\in L^\infty$. Let $X_0$ denote the closure of $L^\infty$ in $X$. If $X \neq L^\infty$, then $X_0$ is separable. For the rearrangement invariant space $X$ to be separable it is necessary and sufficient that for each function $x\in X$ we have where $x^*$ is the nonincreasing left-continuous rearrangement of $|x|$, which is equimeasurable with $x$ and defined by
$$
\begin{equation*}
x^*(t)=\inf\{\tau>0\colon m\{s\in [0,1]\colon |x(s)|>\tau\}<t\}, \qquad 0<t\leqslant 1.
\end{equation*}
\notag
$$
The Köthe dual (or associated) space $X'$ of the rearrangement invariant space $X$ consists of all measurable functions $y$ with finite norm
$$
\begin{equation*}
\|y\|_{X'}=\sup_{\| x \|_{X} \leqslant 1} \int_{0}^1 x(t) y (t ) \,dt.
\end{equation*}
\notag
$$
The space $X'$ is also rearrangement invariant and is a subspace of the Banach dual space $X^*$. Moreover, $X'=X^*$ if and only if $X$ is separable. The natural embedding of $X$ in $X''$ is an isometry. The space $X'$ is maximal: if $x_n\in X'$, $n=1,2,\dots$, $\sup_{n=1,2,\dots}\|x_n\|_{X'}<\infty$ and $x_n\to{x}$ almost everywhere on $[0,1]$, then
$$
\begin{equation*}
x\in X'\quad\text{and} \quad \|x\|_{X'}\leqslant \liminf_{n\to\infty}{\|x_n\|_{X'}}.
\end{equation*}
\notag
$$
Here are some examples of rearrangement invariant spaces. One natural generalization of the spaces $L^p$, ${1 \leqslant p < \infty}$, is Orlicz spaces. Let $M(u)$ be a continuous convex increasing function on $[0,\infty)$ such that $M(0)=0$ and $M(1)=1$. Then the Orlicz space $L_M$ can be defined as the set of measurable functions on $[0,1]$ with finite Luxemburg norm
$$
\begin{equation*}
\|x\|_{L_M}:=\inf \biggl\{\lambda>0\colon \int_{0}^{1} M\biggl(\frac{|x (t)|}{\lambda}\biggr) \,dt \leqslant 1\biggr\}.
\end{equation*}
\notag
$$
In particular, if $M(u)=u^p$, then $L_M=L^p$ with coincidence of the norms. A space $L_M$ is separable if and only if $M$ satisfies the $\Delta_2^\infty$-condition, that is, $M(2u) \leqslant CM(u)$ for some $C>0$ and all $u\geqslant 1$. In particular, a convex increasing function $M_p(u)$ that is equivalent to $e^{u^p}-1$, $p>0$, for large values of $u$ does not satisfy this condition. Hence the corresponding exponential Orlicz space $L_{M_p}$, usually denoted by $\operatorname{Exp}L^p$, is not separable. For many problems of the theory of rearrangement invariant spaces (including the questions considered in our paper) the separable space $(\operatorname{Exp}L^2)_0$ is of importance; it is usually denoted by $G$. Let $\varphi$ be a continuous increasing concave function on $[0,1]$ such that $\varphi(0)=0$, $\varphi(1)=1$ and $\lim_{t \to 0} \varphi(t)/t=\infty$. The Lorentz space $\Lambda(\varphi)$ consists of all measurable functions on $[0,1]$ that satisfy
$$
\begin{equation*}
\|x\|_{\Lambda(\varphi)}:=\int_{0}^{1} x^*(t) \,d \varphi(t)<\infty.
\end{equation*}
\notag
$$
Each Lorentz space is separable. The Banach dual space of $\Lambda(\varphi)$ is the Marcinkiewicz space $M(\varphi)$ with the norm
$$
\begin{equation*}
\|x\|_{M (\varphi)}:=\sup_{0<s \leqslant 1} \frac{1}{\varphi(s)} \int_{0}^{s} x^*(t) \, dt.
\end{equation*}
\notag
$$
All Marcinkiewicz spaces are nonseparable, and $M(\varphi)'=\Lambda(\varphi)$ with coincidence of the norms. The function $\phi_X(s):=\|\chi_E\|_X$, where $E$ is a measurable subset of $[0,1]$ and $m(E)=s$, is called the fundamental function of the rearrangement invariant space $X$. It is quasiconcave, that is, $\phi_X(0)=0$, $\phi_X(s)$ is nondecreasing, and $\phi_X(s)/s$ is nonincreasing. In particular, $\phi_{\Lambda(\varphi)}(s)=\varphi(s)$ and $\phi_{M(\varphi)}(s)=s/\varphi(s)$. § 2. Auxiliary resultsWe present a few simple auxiliary results which we need in what follows. Proof. Since the space $\operatorname{Exp}L^2$ is rearrangement invariant, it follows from the definition of the norm in it that for each $t$, $0<t\leqslant 1$,
$$
\begin{equation*}
t\exp\biggl(\frac{x^*(t)}{\|x\|_{\operatorname{Exp}L^2}}\biggr)^2 \leqslant\int_0^t\exp\biggl(\frac{x^*(s)}{\|x\|_{\operatorname{Exp}L^2}}\biggr)^2\,ds\leqslant 2.
\end{equation*}
\notag
$$
After some elementary transformations this yields (2.1).
Now, as mentioned in § 1, a rearrangement invariant space $X$ is separable if and only if equality (1.1) holds for each function $x\in X$. Hence if $x\in G$, then $\lim _{t \to 0}\|x^{*} \chi_{(0, t)}\|_{\operatorname{Exp}L^2}=0$. Therefore, as by (2.1) we have
$$
\begin{equation*}
\frac{x^*(t)}{\log^{1/2}(2/t)}\leqslant \|x^{*} \chi_{(0,t)}\|_{\operatorname{Exp}L^2}, \qquad 0<t\leqslant 1,
\end{equation*}
\notag
$$
we obtain (2.2).
Conversely, assume that (2.2) holds for $x\in \operatorname{Exp}L^2$. Then it is easy to show that
$$
\begin{equation}
\lim_{v\to 0} \frac{1}{v\log^{1/2}(2/v)}\int_0^v x^*(s)\,ds=0
\end{equation}
\tag{2.3}
$$
(for the proof of a more general result, see [10], Theorem II.5.3). In addition, a direct verification shows that $\log^{1/2}(2/t)$ is a function in $\operatorname{Exp}L^2$. Hence the inequality
$$
\begin{equation*}
x^{*}(t) \chi_{(0,u)}(t)\leqslant \log^{1/2}\frac2t\cdot \sup_{0<v\leqslant u} \frac{1}{v\log^{1/2}(2/v)}\int_0^v x^*(s)\,ds, \qquad 0<t,u\leqslant 1,
\end{equation*}
\notag
$$
yields
$$
\begin{equation*}
\|x^{*} \chi_{(0,u)}\|_{\operatorname{Exp}L^2}\leqslant \sup_{0<v\leqslant u} \frac{1}{v\log^{1/2}(2/v)}\int_0^v x^*(s)\,ds\cdot \biggl\|\log^{1/2}\frac{2}{t}\biggr\|_{\operatorname{Exp}L^2}, \quad 0<u\leqslant 1.
\end{equation*}
\notag
$$
As a result, it follows from this and (2.3) that $\lim _{u \to 0}\|x^{*} \chi_{(0,u)}\|_{\operatorname{Exp}L^2}=0$, that is, $x\in G$.
The lemma is proved. The proof of the following result is standard, and we leave it out. Lemma 2. Let $\{f_n\}_{n=1}^\infty$ be a sequence of nonincreasing functions on $[a,b]$ such that $\lim_{n\to\infty}f_n(t)=f(t)$ for all $t\in [a,b]$, where $f$ is continuous on $[a,b]$. Then the sequence $\{f_n\}$ converges to $f$ uniformly on $[a,b]$. Finally, below we require an upper bound for Rademacher sums in the dyadic $\mathrm{BMO}$-space, usually denoted by $\mathrm{BMO}_d$. Recall its definition. Let $\mathbb D$ be the family of all dyadic subintervals of $[0,1]$, that is, of intervals of the form $((k-1)2^{-n},k2^{-n})$, $1\leqslant k\leqslant2^n$, $n=0,1,\dots$ . The space $\mathrm{BMO}_d=\mathrm{BMO}_d[0,1]$ consists of all functions $x \in L^2[0, 1]$ such that
$$
\begin{equation*}
\| x \|_{d}:=\sup_I \frac{1}{| I|} \int_I | x(s) - x_I | \, ds < \infty,
\end{equation*}
\notag
$$
where the supremum is taken over all intervals $I\in\mathbb D$ such that $|I|=m(I)$, and $\displaystyle x_I:=\frac{1}{|I|} \int_I x(s) \, ds$. Proof. For $x \in L^2[0, 1]$ we have $\|x\|_d\leqslant \|x\|_d'$, where
$$
\begin{equation*}
\|x\|_d':=\sup_{I\in\mathbb D}\biggl(\frac{1}{|I|}\int_I | x(s) - x_I |^2 \, ds \biggr)^{1/2}.
\end{equation*}
\notag
$$
A straightforward verification shows that for the function $f$ under consideration we have $\|f\|_d'=\|a\|_{\ell^2}$ (also see [5], formula (1.2) and Proposition 3).
The lemma is proved. § 3. Proof of Theorem 1Applying the central limit theorem to the sequence of functions
$$
\begin{equation*}
u_n(t):=\frac{1}{\sqrt{n}}\sum_{k=1}^nr_k(t), \qquad n=1,2,\dots,
\end{equation*}
\notag
$$
on the interval $[0,1]$ and bearing in mind that the $u_n$, $n=1,2,\dots$, have a symmetric distribution we obtain
$$
\begin{equation*}
\lim_{n\to\infty}m\{t\in [0,1]\colon |u_n(t)|>\tau\}=\Psi(\tau), \qquad \tau >0,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\Psi(\tau):=\frac{2}{\sqrt{2\pi}} \int_{\tau}^\infty e^{-t^2/2}\,dt.
\end{equation*}
\notag
$$
Hence passing to rearrangements (for instance, see [4], Lemma 2.1) we obtain
$$
\begin{equation}
\lim_{n\to{\infty}}u_n^*(s)=u(s), \qquad 0<s\leqslant 1,
\end{equation}
\tag{3.1}
$$
where $u(s):=\Psi^{-1}(s)$ is the inverse function of $\Psi$. Furthermore, since the inequality
$$
\begin{equation*}
\int_{\tau}^\infty e^{-t^2/2}\,dt\geqslant \int_{\tau}^{\infty}te^{-t^2}\,dt =\frac12\, e^{-\tau^2}, \qquad \tau\geqslant0,
\end{equation*}
\notag
$$
implies that $\Psi(\tau)\geqslant (1/\sqrt{2\pi})\cdot e^{-{\tau}^2}$, $\tau\geqslant 0$, passing to inverse functions we conclude that
$$
\begin{equation*}
u(s)\geqslant \log^{1/2}\frac{1}{s\sqrt{2\pi}}, \qquad 0<s\leqslant \frac{1}{\sqrt{2\pi}}.
\end{equation*}
\notag
$$
Thus, setting $v(s):=\log^{1/2}({2}/{s})$, for each $0<t\leqslant 1$ we obtain
$$
\begin{equation}
\int_0^t v(s)\,ds=2\sqrt{2\pi}\int_0^{t/(2\sqrt{2\pi})} \log^{1/2}\frac{1}{s\sqrt{2\pi}}\,ds \leqslant 6\int_0^t u(s)\,ds.
\end{equation}
\tag{3.2}
$$
On the other hand, by Lemma 1 (see (2.1) and (2.2)), if $x\in G$, then
$$
\begin{equation}
x^*(t)\leqslant \|x\|_G v(t), \quad 0<t\leqslant 1, \qquad \lim_{t\to 0}\frac{x^*(t)}{v(t)}=0.
\end{equation}
\tag{3.3}
$$
Now we construct a sequence of points $t_1>t_2>\dotsb$ in $[0,1]$ such that $t_k\downarrow 0$,
$$
\begin{equation}
x^*(t)\leqslant \|x\|_G v(t)\cdot\sum_{k=1}^\infty 2^{1-k}\chi_{(t_{k+1},t_k]}(t),\qquad 0<t\leqslant 1,
\end{equation}
\tag{3.4}
$$
and
$$
\begin{equation}
\int_{t_{k+1}}^\tau u(t)\,dt\geqslant \frac12\int_{0}^\tau u(t)\,dt \quad \text{for all } \tau\in [t_k,t_{k-1}], \quad k=2,3,\dots\,.
\end{equation}
\tag{3.5}
$$
We set $t_1=1$ and using the limit relation in (3.3) choose $t_2<1$ so that
$$
\begin{equation}
x^*(t)\leqslant \frac12 \|x\|_G v(t) \quad \text{for all } 0<t\leqslant t_2.
\end{equation}
\tag{3.6}
$$
For each $\tau\in [t_2,t_1]$ there exists $b(\tau)\in (0,t_1)$ such that
$$
\begin{equation*}
\int_{b(\tau)}^\tau u(t)\,dt=\frac12\int_0^\tau u(t)\,dt,
\end{equation*}
\notag
$$
and since the integral is absolutely continuous, the function $\tau\mapsto b(\tau)$ is continuous on $[t_2,t_1]$. Hence there exists $t_3':=\min_{t_2\leqslant\tau\leqslant t_1} b(\tau)>0$. Let $t_3\in (0,t_3']$ be a point such that
$$
\begin{equation*}
x^*(t)\leqslant \frac{1}{2^2}\|x\|_G v(t) \quad \text{for } 0<t\leqslant t_3.
\end{equation*}
\notag
$$
Then, by the first inequality in (3.3) and by (3.6) inequality (3.4) holds for $t\in[t_3,t_1]$, and the choice of $t_3$ ensures inequality (3.5) for $k=2$. We choose the other points $t_k$ in a similar way. By Lemma 2 convergence in (3.1) is uniform on each interval $(\lambda,1]$, where ${\lambda\in (0,1)}$. Hence setting $m_1=0$ we can select $m_2>m_1$ so that
$$
\begin{equation*}
\biggl(\frac{1}{(m_2-m_1)^{1/2}}\sum_{k=m_1+1}^{m_2} r_k\biggr)^*(t)\geqslant \frac12\, u(t)
\end{equation*}
\notag
$$
for all $t\in[t_2,t_1]$. If we have already found $m_1<m_2<\dots<m_i$, then by the elementary properties of the Rademacher functions (for instance, see [4], Proposition 2.2) we have
$$
\begin{equation*}
\lim_{n\to\infty}\biggl(\frac{1}{(n-m_i)^{1/2}}\sum_{k=m_i+1}^{n} r_k\biggr)^*(s)=u(s), \qquad 0< s\leqslant 1.
\end{equation*}
\notag
$$
Hence, using Lemma 2 again, we can select $m_{i+1}>m_i$ so that
$$
\begin{equation}
\biggl(\frac{1}{(m_{i+1}-m_i)^{1/2}}\sum_{k=m_i+1}^{m_{i+1}} r_k\biggr)^*(t)\geqslant \frac12\, u(t) \quad\text{for all } t\in [t_{i+1},t_1].
\end{equation}
\tag{3.7}
$$
We set
$$
\begin{equation*}
\begin{gathered} \, a_k:=\frac{2^{7-i} \|x\|_G}{(m_{i+1}-m_i)^{1/2}}, \qquad m_i<k\leqslant m_{i+1}, \quad i\in\mathbb N, \\ z_i:=\sum_{k=m_i+1}^{m_{i+1}} a_kr_k, \quad i\in\mathbb N,\quad\text{and} \quad g:=\sum_{k=1}^\infty a_kr_k. \end{gathered}
\end{equation*}
\notag
$$
Hence, if $a=(a_k)_{k=1}^\infty$, then
$$
\begin{equation*}
\|a\|_{\ell^2}\leqslant 128\|x\|_G\biggl(\sum_{i=1}^\infty 4^{-i}\biggr)^{1/2}=128\cdot 3^{-1/2}\|x\|_G< 80 \|x\|_G,
\end{equation*}
\notag
$$
so that $g\in L^2$ and Since the Rademacher system is unconditional with constant $1$ in any rearrangement invariant space (see [4], Proposition 2.2) and, in particular, in $L^1$ with the norm $\displaystyle\int_0^\tau x^*(s)\,ds$, $0<\tau\leqslant 1$, for all $i=2,3,\dots$ and $\tau\in [t_i,t_{i-1}]$ we have
$$
\begin{equation*}
\int_0^\tau g^*(t)\,dt\geqslant \int_0^\tau z_i^*(t)\,dt\geqslant \int_{t_{i+1}}^\tau z_i^*(t)\,dt.
\end{equation*}
\notag
$$
Furthermore, in view of (3.7), (3.5) and (3.2), for all $\tau\in [t_i,t_{i-1}]$ it follows that
$$
\begin{equation*}
\begin{aligned} \, \int_{t_{i+1}}^\tau z_i^*(t)\,dt &\geqslant 2^{6-i}\|x\|_G \int_{t_{i+1}}^\tau u(t)\,dt \\ &\geqslant 2^{5-i}\|x\|_G \int_{0}^\tau u(t)\,dt \geqslant 2^{2-i}\|x\|_G \int_{0}^\tau v(t)\,dt. \end{aligned}
\end{equation*}
\notag
$$
At the same time, as inequality (3.4) shows, on the interval $(0,t_{i-1}]$ we have $2^{2-i}\|x\|_G v(t)\geqslant x^*(t)$. Thus, since $\bigcup_{i=2}^\infty [t_i,t_{i-1}]=(0,1]$, it follows from the above inequalities that
$$
\begin{equation}
\int_0^\tau x^*(t)\,dt\leqslant \int_0^\tau g^*(t)\,dt, \qquad 0<\tau\leqslant 1.
\end{equation}
\tag{3.9}
$$
Now recall that for each function $w\in \mathrm{BMO}_d$ we have
$$
\begin{equation*}
\frac{1}{t}\int_0^t w^*(s)\,ds-w^*(t)\leqslant 32\|w\|_d, \qquad 0<t\leqslant \frac 14
\end{equation*}
\notag
$$
(see [14] and [15], Theorem 3.3, where this inequality was deduced for the ordinary space $\mathrm{BMO}$, but a simple analysis of the proof shows that its analogue also holds for $\mathrm{BMO}_d$). Hence by (3.9) and Lemma 3 for $0<t\leqslant 1/4$,
$$
\begin{equation}
\begin{aligned} \, x^*(t) &\leqslant \frac{1}{t}\int_0^t x^*(s)\,ds\leqslant \frac{1}{t}\int_0^t g^*(s)\,ds \nonumber \\ &\leqslant g^*(t)+32\|g\|_{d}\leqslant g^*(t)+32\|a\|_{\ell^2}. \end{aligned}
\end{equation}
\tag{3.10}
$$
Note that the last inequality also holds for $1/4<t\leqslant 1$, because in this case, using (3.9) again we obtain
$$
\begin{equation*}
x^*(t)\leqslant \frac{1}{t}\int_0^t x^*(s)\,ds\leqslant 4\int_0^t g^*(s)\,ds\leqslant 4\|g\|_{L^2}=4\|a\|_{\ell^2}.
\end{equation*}
\notag
$$
Since $\|g\|_{L^p}\leqslant \sqrt{p}\, \|g\|_{L^2}$, $p\geqslant 1$, by Khintchine’s inequality (see (0.1)), standard arguments (for instance, see Theorem 2.7 in [16] and its proof) yield
$$
\begin{equation*}
m\biggl\{t\in [0,1]\colon |g(t)|\geqslant \frac{\|a\|_{\ell^2}}{2}\biggr\}\geqslant \frac{1}{128}
\end{equation*}
\notag
$$
or, by the definition of the nonincreasing rearrangement of a function,
$$
\begin{equation*}
g^*\biggl(\frac{t}{128}\biggr)\geqslant \frac{\|a\|_{\ell^2}}{2}, \qquad 0<t\leqslant 1.
\end{equation*}
\notag
$$
Hence it follows from (3.10) that or, going over to the distribution functions,
$$
\begin{equation*}
\begin{aligned} \, m\{t\in [0,1]\colon |x(t)|>\tau\} &\leqslant m\biggl\{t\in [0,1]\colon 65\biggl|g\biggl(\frac{t}{128}\biggr)\biggr|>\tau\biggr\} \\ &=128m\{t\in [0,1]\colon 65|g(t)|>\tau\}, \qquad \tau>0. \end{aligned}
\end{equation*}
\notag
$$
As a result, relations (0.4) and (0.5) follow from the last inequality and (3.8) once we set $f:=65g$. Theorem 1 is proved. Remark. By Theorem 1 there exists an operator $A$ from $G$ to $\ell^2$ such that ${\|Ax\|_{\ell^2}\asymp \|x\|_G}$, $x\in G$. We can see from the proof that $A$ is not a linear operator; moreover, there exists no linear operator with this property, for otherwise $G$ is isomorphic to a Hilbert space. In particular, the orthogonal projection
$$
\begin{equation*}
Qx(t):=\sum_{k=1}^\infty \int_0^1x(s)r_k(s)\,ds\cdot r_k(t)
\end{equation*}
\notag
$$
has norm $1$ in $L^2$; a fortiori, $\|Qx\|_{L^2}\leqslant \|x\|_G$ for each $x\in G$. Thus, the function $f:=Qx$ satisfies an analogue of (0.4). On the other hand it is easy to verify that, by contrast, an analogue of (0.5) does not hold for an arbitrary $x\in G$ and such $f$. In the next section we transfer the result of Theorem 1 to some more general subspaces. § 4. Subspaces containing an orthonormal sequence which is uniformly bounded on a set of positive measureTheorem 2. Let $\{y_n\}\subset L^2$ be a sequence satisfying at least one of the following sets of conditions: or Then there exist a subsequence $\{y_{n_k}\}\subset\{y_n\}$ and positive constants $C_1$ and $C_2$ such that for each function $x\in G$ there exists $g:=\sum_{k=1}^\infty a_ky_{n_k}$, where $(a_k)\in \ell^2$, that satisfies the conditions $\|g\|_{L^2}\leqslant C_1\|x\|_G$ and
$$
\begin{equation*}
m\{t\in [0,1]\colon |x(t)|>\tau\}\leqslant C_2 m\{t\in [0,1]\colon |g(t)|>\tau\} \quad \textit{for all } \tau>0.
\end{equation*}
\notag
$$
In particular, this holds for an arbitrary uniformly bounded orthonormal sequence of functions on $[0,1]$. Proof. First assume that conditions (a1) and (b1) are satisfied. Since $\{y_n\}$ satisfies the assumptions of Theorem 8.2 in [4] (also see [17]), there exists a subsequence $\{y_{n_k}\}\subset \{y_n\}$ that is equivalent to the Rademacher system in distribution. This means that for some $C\geqslant 1$, for all $\tau>0$ and $a_k\in\mathbb{R}$ we have
$$
\begin{equation}
\begin{aligned} \, \notag &C^{-1}m\biggl\{t\colon \biggl|\sum_{k=1}^\infty a_kr_{k}\biggr|>C\tau\biggr\} \\ &\qquad\leqslant m\biggl\{t\colon \biggl|\sum_{k=1}^\infty a_ky_{n_k}\biggr|>\tau\biggr\} \leqslant Cm\biggl\{t\colon \biggl|\sum_{k=1}^\infty a_kr_{k}\biggr|>\frac{\tau}{C}\biggr\}. \end{aligned}
\end{equation}
\tag{4.1}
$$
By Theorem 1, for each function $x\in G$ there exists a Rademacher sum $f=\sum_{k=1}^\infty a_kr_k$, $(a_k)_{k=1}^\infty\in \ell^2$, such that (0.4) and (0.5) hold. Hence, by the above inequalities for $g':=\sum_{k=1}^\infty a_ky_{n_k}$ we have and for all $\tau>0$. Thus, setting $g:=Cg'$, $C_1=5200C^3$ and $C_2=2^7C$ we obtain the required relations.
Now assume that conditions (a2) and (b2) hold. Since $y_n\chi_E\to 0$ weakly in $L^2$ by (a2), the sequence of functions $u_n:=y_n\chi_E$, $n=1,2,\dots$, satisfies conditions (a1) and (b1). Hence, as before, there exists a subsequence $\{u_{n_k}\}\subset \{u_n\}$ that is equivalent to the Rademacher system in distribution, so that (4.1) holds for $y_{n_k}$ replaced by $u_{n_k}$. Let $x\in G$, and let $f=\sum_{k=1}^\infty a_kr_k$, $(a_k)_{k=1}^\infty\in \ell^2$, satisfy (0.4) and (0.5). Set $g=C\sum_{k=1}^\infty a_ky_{n_k}$ and $g'':=\sum_{k=1}^\infty a_ku_{n_k}$. Then, as $\{y_n\}$ is orthonormal, Furthermore, from the definition of $g$ and $g''$ we obtain
$$
\begin{equation*}
\begin{aligned} \, &m\{t\in [0,1]\colon |x(t)|>\tau\} \\ &\qquad\leqslant2^7 m\{t\in [0,1]\colon |f(t)|>\tau\}\leqslant 2^7C m\biggl\{t\in [0,1]\colon |g''(t)|>\frac{\tau}{C}\biggr\} \\ &\qquad= 2^7C m\{t\in E\colon |g(t)|>\tau\} \leqslant 2^7C m\{t\in [0,1]\colon |g(t)|>\tau\} \end{aligned}
\end{equation*}
\notag
$$
for all $\tau>0$.
Theorem 2 is proved. Corollary 1. Assume that the subspace $H$ of $L^2$ contains a sequence $\{y_n\}$ satisfying conditions (a1) and (b1) or (a2) and (b2) in Theorem 2. Then there exist positive constants $C_1$ and $C_2$ such that for each function $x\in G$ there exists $g\in H$ such that $\|g\|_{L^2}\leqslant C_1\|x\|_G$ and
$$
\begin{equation*}
m\{t\in [0,1]\colon |x(t)|>\tau\}\leqslant C_2 m\{t\in [0,1]\colon |g(t)|>\tau\} \quad \textit{for all } \tau>0.
\end{equation*}
\notag
$$
In particular, this holds when $H$ is a $\Lambda(2)$-space containing an orthonormal sequence $\{z_n\}_{n=1}^\infty$ such that Proof. Only the second assertion requires a proof. Furthermore, it is sufficient to show that there exists a sequence $\{y_n\}$ satisfying conditions (a2) and (b2) from Theorem 2.
First of all, since $H$ is a $\Lambda(2)$-space, there exists $D\geqslant 1$ such that
$$
\begin{equation}
\|u\|_{L^2}\leqslant D\|u\|_{L^1}\quad \text{for all } u\in H.
\end{equation}
\tag{4.2}
$$
We show that for some $\delta=\delta(D)$ and all $u\in H$ we have
$$
\begin{equation}
m\{t\in [0,1]\colon |u(t)|>\delta\|u\|_{L^2}\}\geqslant\delta.
\end{equation}
\tag{4.3}
$$
Assume that $0<\eta<1$, $u\in H$, $u\ne 0$ and $m(E_u)<\eta$, where $E_u:=\{{t\in [0,1]}\colon |u(t)|>\eta\|u\|_{L^1}\}$. By Hölder’s inequality and (4.2) we have
$$
\begin{equation*}
\begin{aligned} \, \|u\|_{L^{3/2}} &=\biggl(\int_{E_u} |u(t)|^{3/2}\,dt+\int_{[0,1]\setminus E_u} |u(t)|^{3/2}\,dt\biggr)^{2/3} \\ &\leqslant \biggl\{m(E_u)^{1/4}\biggl(\int_{E_u} |u(t)|^{2}\,dt\biggr)^{3/4}+\eta^{3/2}\|u\|_{L^1}^{3/2}\biggr\}^{2/3} \\ &\leqslant(\eta^{1/4}D^{3/2}\|u\|_{L^1}^{3/2}+\eta^{3/2}\|u\|_{L^1}^{3/2})^{2/3}= (D^{3/2}\eta^{1/4}+\eta^{3/2})^{2/3}\|u\|_{L^1}. \end{aligned}
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
\|u\|_{L^{1}}\leqslant (D^{3/2}\eta^{1/4}+\eta^{3/2})^{2/3}\|u\|_{L^1}< 2^{2/3}D\eta^{1/6}\|u\|_{L^1},
\end{equation*}
\notag
$$
so that $\eta>\frac{1}{16}D^{-6}$. Thus, if $0<\eta\leqslant\frac{1}{16}D^{-6}$, then
$$
\begin{equation*}
m\{t\in [0,1]\colon |u(t)|>\eta\|u\|_{L^1}\}\geqslant\eta.
\end{equation*}
\notag
$$
Therefore, in view of (4.2), for $u\in H$ we obtain
$$
\begin{equation*}
m\biggl\{t\in [0,1]\colon |u(t)|>\frac{\eta}{D}\|u\|_{L^2}\biggr\}\geqslant m\{t\in [0,1]\colon |u(t)|>\eta\|u\|_{L^1}\}\geqslant\eta,
\end{equation*}
\notag
$$
which means that inequality (4.3) holds, provided that $0<\delta\leqslant\frac{1}{16}D^{-7}$. We fix such $\delta$.
Now let $\{z_n\}_{n=1}^\infty$ be a sequence in $H$ satisfying the assumptions of the corollary. Since $\|z_n\|_{L^2}=1$, $n=1,2,\dots$, it follows from (4.3) that
$$
\begin{equation}
m\{t\in [0,1]\colon |z_n(t)|>\delta\}\geqslant\delta, \qquad n=1,2,\dots\,.
\end{equation}
\tag{4.4}
$$
Moreover, by assumption there exist $A$ and a subsequence $\{z_{n_k}\}\subset \{z_n\}$ such that
$$
\begin{equation*}
m\biggl([0,1]\setminus\bigcap_{k=1}^\infty\{t\in [0,1]\colon |z_{n_k}(t)|\leqslant A\}\biggr)\leqslant\sum_{k=1}^\infty(1-m\{t\in [0,1]\colon |z_{n_k}(t)|\leqslant A\})\leqslant \frac{\delta}{2}.
\end{equation*}
\notag
$$
Hence from (4.4) we obtain
$$
\begin{equation*}
m\biggl(\bigcap_{k=1}^\infty\{t\in [0,1]\colon \delta<|z_{n_k}(t)|\leqslant A\}\biggr)\geqslant\frac{\delta}{2}.
\end{equation*}
\notag
$$
Thus, conditions (a2) and (b2) in Theorem 2 hold for $y_k:=z_{n_k}$, $k=1,2,\dots$, and $E:=\bigcap_{k=1}^\infty\{t\in [0,1]\colon \delta\leqslant |z_{n_k}(t)|\leqslant A\}$, and thus the corollary is proved. Now we look at a different situation. § 5. Subspaces containing a sequence of independent functionsWe say that a sequence of measurable functions $\{y_n\}_{n=1}^\infty$ on $[0,1]$ majorizes the Rademacher system in distribution if for some $C\geqslant 1$ and all $\tau>0$ and $a_n\in\mathbb{R}$ we have
$$
\begin{equation*}
m\biggl\{t\colon \biggl|\sum_{n=1}^\infty a_nr_{n}\biggr|>\tau\biggr\}\leqslant Cm\biggl\{t\colon \biggl|\sum_{n=1}^\infty a_ny_{n}\biggr|>\frac{\tau}{C}\biggr\}.
\end{equation*}
\notag
$$
We start with the case when the functions $y_n$, $n=1,2,\dots$, are mean zero on $[0,1]$. Proposition 1. Let $\{y_n\}_{n=1}^\infty$ be a sequence of independent functions on $[0,1]$ such that $\displaystyle\int_0^1 y_n(t)\,dt=0$, $\|y_n\|_{L^2}=1$ and $\|y_n\|_{L^1}\geqslant\alpha$ for some $\alpha>0$ and all $n=1,2,\dots$ . Then $\{y_n\}_{n=1}^\infty$ majorizes the Rademacher system in distribution. Proof. Consider the natural ‘symmetrization’ of the sequence $\{y_n\}_{n=1}^\infty$, namely, the functions $z_n(s,t):=y_n(s)-y_n(t)$, $(s,t)\in [0,1]^2:=[0,1]\times [0,1]$, $n=1,2,\dots$ . We show that the sequences $\{r_n\}_{n=1}^\infty$ and $\{z_n\}_{n=1}^\infty$ satisfy the assumptions of the well-known Kwapien–Rychlik test (for instance, see [9], Theorem V.4.4). Since both sequences consist of independent symmetrically distributed functions, we must only verify that for some $C_1>0$, for all $n\in\mathbb{N}$ we have
First, by assumption, for all $n=1,2,\dots$
$$
\begin{equation}
\begin{aligned} \, \|z_n\|_{L^1}&=\int_0^1\int_0^1 |y_n(s)-y_n(t)|\,ds\,dt \nonumber \\ &\geqslant \int_0^1\biggl|\int_0^1 (y_n(s)-y_n(t))\,ds\biggr|\,dt =\|y_n\|_{L^1}\geqslant\alpha. \end{aligned}
\end{equation}
\tag{5.2}
$$
Now proceeding as Kadec and Pelczyński in their well-known paper [18] (see the proof of Theorem 1e), we show that if $0<\varepsilon\leqslant\alpha^2/8$, then
$$
\begin{equation}
m\{(s,t)\in [0,1]^2\colon |z_n(s,t)|>\varepsilon\|z_n\|_{L^2}\}\geqslant \varepsilon.
\end{equation}
\tag{5.3}
$$
In fact, assume that $0<\varepsilon<1$ and $m(E_n)<\varepsilon$, where
$$
\begin{equation*}
E_n:=\{(s,t)\in [0,1]^2\colon |z_n(s,t)|>\varepsilon\|z_n\|_{L^2}\}.
\end{equation*}
\notag
$$
Then using the Cauchy–Schwarz–Bunyakovsky inequality we obtain
$$
\begin{equation*}
\|z_n\|_{L^1}=\int_{E_n}|z_n(s,t)|\,ds\,dt +\int_{[0,1]^2\setminus E_n}|z_n(s,t)|\,ds\,dt\leqslant (m(E_n)^{1/2}+\varepsilon)\|z_n\|_{L^2}.
\end{equation*}
\notag
$$
Hence by (5.2) and the equality $\|z_n\|_{L^2}=\sqrt{2}\, \|y_n\|_{L^2}=\sqrt{2}$ the assumption that $m(E_n)<\varepsilon$ leads to the inequality $\alpha<2\sqrt{2}\, \varepsilon^{1/2}$, that is, $\varepsilon>\alpha^2/8$. Thus, for $\varepsilon\leqslant\alpha^2/8$ we obtain (5.3).
In particular, it follows from (5.3) that for all $0<\tau\leqslant 1$
$$
\begin{equation*}
m\biggl\{(s,t)\in [0,1]^2\colon |z_n(s,t)|>\frac{\alpha^2\tau}{4\sqrt{2}}\biggr\}\geqslant \frac{\alpha^2}{8}, \qquad n=1,2,\dots,
\end{equation*}
\notag
$$
hence, taking the definition of Rademacher functions into account, for all ${n=1,2,\dots}$ and $\tau>0$ we obtain
$$
\begin{equation*}
m\{t\in [0,1]\colon |r_n(t)|>\tau\}\leqslant \frac{8}{\alpha^2}m\biggl\{(s,t)\in [0,1]^2\colon |z_n(s,t)|>\frac{\alpha^2\tau}{4\sqrt{2}}\biggr\}.
\end{equation*}
\notag
$$
Thus, inequality (5.1) holds for $C_1={8}/{\alpha^2}$, so that by Theorem V.4.4 in [9] there exists a constant $C'=C'(\alpha)$ such that for arbitrary $\tau>0$ and $a_n\in\mathbb{R}$ we have
$$
\begin{equation*}
\begin{aligned} \, m\biggl\{t\in [0,1]\colon \biggl|\sum_{n=1}^\infty a_nr_n(t)\biggr|>\tau\biggr\} &\leqslant C'm\biggl\{(s,t)\in [0,1]^2\colon \biggl|\sum_{n=1}^\infty a_nz_n(s,t)\biggr|>\frac{\tau}{C'}\biggr\} \\ &\leqslant2C'm\biggl\{t\in [0,1]\colon \biggl|\sum_{n=1}^\infty a_ny_n(t)\biggr|>\frac{\tau}{2C'}\biggr\}, \end{aligned}
\end{equation*}
\notag
$$
where the second inequality is an immediate consequence of the definition of the functions $z_n$.
The proof is complete. To drop the mean zero condition we need the following lemma. It reflect a simple and well-known fact: if the projection of a vector $u$ to a vector $v$ of norm one in a Hilbert space is equal to $\|u\|$, then $u$ and $v$ are collinear. Lemma 4. Let $u\in L^2[0,1]$ and $\displaystyle\|u\|_{L^2}\,{=}\int_0^1 u(t)\,dt$. Then $u(t)= c$ for some $c\geqslant 0$ and almost all $t\in [0,1]$. Theorem 3. If a $\Lambda(2)$-space $H$ contains a sequence of not identically constant independent functions, then $H$ contains a sequence majorizing the Rademacher system in distribution. Proof. It is sufficient to show that $H$ contains a sequence $\{y_n\}_{n=1}^\infty$ satisfying the assumptions of Proposition 1.
Let $\{u_n\}_{n=1}^\infty$ be a sequence of not identically constant independent functions on $[0,1]$ such that $u_n\in H$, $n=1,2,\dots$ . We assume without loss of generality that $\displaystyle I_n:=\int_0^1 u_n(t)\,dt\ne 0$ for all $n=1,2,\dots$ . Consider the functions $v_n:={a_nu_{2n}-u_{2n-1}}$, where $a_n:=I_{2n-1}/I_{2n}$, $n=1,2,\dots$ . Clearly, the $v_n$ are independent, $v_n\in H$ and $\displaystyle\int_0^1 v_n(t)\,dt=0$, $n=1,2,\dots$ . Furthermore, since $u_n$, $n=1,2,\dots$, are independent functions, from the assumptions of the theorem and Lemma 4 we obtain
$$
\begin{equation*}
\begin{aligned} \, \|v_n\|_{L^2}^2 &=a_n^2\|u_{2n}\|_{L^2}^2+\|u_{2n-1}\|_{L^2}^2-2a_nI_{2n-1}I_{2n} \\ &=\frac{I_{2n-1}^2}{I_{2n}^2}\|u_{2n}\|_{L^2}^2+\|u_{2n-1}\|_{L^2}^2-2I_{2n-1}^2 \geqslant \|u_{2n-1}\|_{L^2}^2-I_{2n-1}^2>0. \end{aligned}
\end{equation*}
\notag
$$
Now if $y_n:={v_n}/{\|v_n\|_{L^2}}$, $n=1,2,\dots$, then $y_n\in H$, and it is easy to see that these functions satisfy all the assumptions of Proposition 1.
The proof is complete. Using Theorems 1 and 3 and arguing as in the proof of Theorem 2 we obtain the following result. Corollary 2. Assume that a $\Lambda(2)$-space $H$ contains a sequence of not identically constant independent functions. Then there exist positive constants $C_1$ and $C_2$ such that for each function $x\in G$ there exists $g\in H$ such that $\|g\|_{L^2}\leqslant C_1\|x\|_G$ and § 6. A counterexampleLet $X$ be an arbitrary rearrangement invariant space on $[0,1]$ such that ${X\ne L^\infty}$. We present an example showing that if a subspace $H$ of $X$ does not contain any sequence of functions with (uniformly) equivalent $L_1$- and $L^2$-norms, we cannot hope that $H$ contains functions with distribution majorizing that of an arbitrary function in $X$. So in the statements proved above (also see the problem presented in the introduction) $H$ must be a $\Lambda(2)$-space in a certain sense. Proposition 2. Let $F\colon (0,\infty)\to (0,1)$ be a nonincreasing function. Then there exist a sequence of pairwise disjoint intervals $\Delta_n\subset [0,1]$, $n=1,2,\dots$, and a sequence of positive numbers $\tau_n\uparrow\infty$ as $n\to\infty$ such that Hence if $H$ is a closed linear span of the functions $\chi_{\Delta_n}$, $n=1,2,\dots$, in $L^2[0,1]$, then Proof. We can take as $\Delta_1$ an arbitrary interval in $[0,1]$ such that $m(\Delta_1)<1$. Also let $\tau_1:=m(\Delta_1)^{-1/2}$. Assuming that we have already chosen disjoint intervals ${\Delta_k\subset [0,1]}$, $k=1,\dots,n$, satisfying $\sum_{k=1}^nm(\Delta_k)<1$, we set $\tau_n:=\max_{k=1,\dots,n}m(\Delta_k)^{-1/2}$ and $\alpha_n:=F(\tau_n)/n$.
Now let $\Delta_{n+1}$ be an interval such that $\Delta_{n+1}\subset [0,1]\setminus \bigcup_{k=1}^n \Delta_k$ and
$$
\begin{equation}
m(\Delta_{n+1})<\min\biggl(\min_{k=1,\dots,n}2^{k-n-1}\alpha_k,\,1-\sum_{i=1}^nm(\Delta_i)\biggr).
\end{equation}
\tag{6.1}
$$
Setting below $\tau_{n+1}:=\max_{k=1,\dots,n+1}m(\Delta_k)^{-1/2}$ and continuing in a similar way, we obtain sequences of pairwise disjoint intervals $\Delta_n\subset [0,1]$, $n=1,2,\dots$, and positive numbers $\tau_n\uparrow\infty$.
Let $(a_k)\in \ell^2$, $\|(a_k)\|_{\ell^2}\leqslant 1$. Then $|a_k|\leqslant 1$, $k=1,2,\dots$, and so for each $n\in\mathbb{N}$, by the choice of the $\tau_n$ and the pairwise disjointness of the $\Delta_k$ we have
$$
\begin{equation*}
\biggl|\sum_{k=1}^{n} \frac{a_k}{m(\Delta_k)^{1/2}}\chi_{\Delta_k}(t)\biggr|\leqslant \max_{k=1,\dots,n}{m(\Delta_k)^{-1/2}}=\tau_{n}.
\end{equation*}
\notag
$$
Hence by (6.1) and the definition of the $\alpha_n$ we have
$$
\begin{equation*}
\begin{aligned} \, &m\biggl\{t\in [0,1]\colon \biggl|\sum_{k=1}^\infty a_k\frac{\chi_{\Delta_k}(t)}{m(\Delta_k)^{1/2}}\biggr|>\tau_n\biggr\} \\ &\qquad=m\biggl\{t\in [0,1]\colon \biggl|\sum_{k=n+1}^\infty a_k\frac{\chi_{\Delta_k}(t)}{m(\Delta_k)^{1/2}}\biggr|>\tau_n\biggr\} \\ &\qquad\leqslant \sum_{k=n+1}^\infty m(\Delta_k)\leqslant \sum_{k=1}^\infty \frac{\alpha_n}{2^{k}}=\frac{F(\tau_n)}{n}. \end{aligned}
\end{equation*}
\notag
$$
This proves the first assertion of the proposition. Since the sequence $\{\chi_{\Delta_n}/m(\Delta_n)^{1/2}\}_{n=1}^\infty$ is an orthonormal basis of $H$ and $\tau_n\uparrow\infty$ as $n\to\infty$, the second assertion follows directly form the first.
The proof is complete. 
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