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This article is cited in 1 scientific paper (total in 1 paper) On subspaces of Orlicz spaces spanned by independent copies of a mean zero function S. V. Astashkinabcd a Samara National Research University b Lomonosov Moscow State University c Moscow Center for Fundamental and Applied Mathematics d Bahçesehir University, Turkey
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2024, Volume 88, Issue 4, DOI: https://doi.org/10.4213/im9531 
Bibliographic databases:
 § 1. IntroductionAccording to the classical Khintchine inequality (see, for example, Theorem V.8.4 in [1]), for each $0<p<\infty$, there exist constants $A_p>0$ and $B_p>0$ such that, for any sequence of real numbers $(c_k)_{k=1}^\infty$,
$$
\begin{equation}
A_p \|(c_k)\|_{\ell^2}\leqslant \biggl\|\sum_{k=1}^\infty c_k r_k\biggr\|_{L^p[0, 1]} \leqslant B_p \|(c_k)\|_{\ell^2},
\end{equation}
\tag{1}
$$
where $r_k$ are the Rademacher functions, $r_k(t) = \operatorname{sign} (\sin 2^k \pi t)$, $k \in \mathbb{N}$, $t \in [0,1]$, and $\|(c_k)\|_{\ell^2}:=\bigl(\sum_{k=1}^\infty c_k^2 \bigr)^{1/2}$. So, for every $0<p<\infty$, the sequence $\{r_k\}_{k=1}^\infty$ is equivalent in $L^p$ to the canonical basis for the space $\ell^2$. This example demonstrates a certain general phenomenon, which is reflected in the following concept. A closed linear subspace $H$ of the space $L^p=L^p[0,1]$, $1\leqslant p<\infty$, is called a $\Lambda(p)$-space if on $H$ the $L^p$-convergence is equivalent to the convergence in measure, or equivalently, for each (or some) $q\in (0,p)$ there is a constant $C_q>0$ such that
$$
\begin{equation}
\|f\|_{L^p}\leqslant C_q\|f\|_{L^q}\quad\text{for all}\quad f\in H
\end{equation}
\tag{2}
$$
(see Proposition 6.4.5 in [2]). Consequently, inequality (1) shows that the span $[r_k]$ in $L^p$ is a $\Lambda(p)$-space for any $1\leqslant p<\infty$. The starting point for the notion of a $\Lambda(p)$-space was the classical Rudin’s paper [3] on Fourier analysis on the circle $[0,2\pi)$, in which the following related concept was studied. Let $0<p<\infty$. A set $E\subset \mathbb{Z}$ is called a $\Lambda(p)$-set if, for some $0<q<p$, there is a constant $C_q>0$ such that inequality (2) holds for every trigonometric polynomial $f$ with spectrum (that is, the support of its Fourier transform) contained in $E$. As is easy to see, this is equivalent to the fact that the subspace $L_E$ spanned by the set of exponentials $\{e^{2\pi int},\, n\in E\}$ is a $\Lambda(p)$-space. In particular, in [3], for all integers $n>1$, Rudin constructed $\Lambda(2n)$-sets that are not $\Lambda(q)$-sets for any $q>2n$. In 1989, Bourgain [4] strengthened this result by extending Rudin’s theorem to all $p>2$. In view of the well-known Vallée Poussin criterion (see Lemma 7 below), this implies, for each $p>2$, the existence of a $\Lambda(p)$-set $E$ such that functions of the unit ball of the subspace $L_E$ fail to have equicontinuous norms in $L^p$ (for the definitions, see § 2). On the “other side” of $L^2$, as often happens, the picture is completely different. Even earlier, in 1974, Bachelis and Ebenshtein showed in [5] that, for $p\in (1,2)$, every $\Lambda(p)$-set is a $\Lambda(q)$-set for some $q>p$ (for a detailed exposition of the theory of $\Lambda(p)$-sets, see the survey [6]). Moreover, in the same direction, Rosenthal (see Theorem 13 in [7]) proved that, for every $1<p<2$, a (closed linear) subspace $H$ of the space $L^p$ is a $\Lambda(p)$-space if and only if functions of the unit ball of $H$ have equicontinuous norms in $L^p$. The recent author’s paper [8] deals with an extension of Rosenthal’s theorem to the class of Orlicz function spaces $L_M$. Generalizing the concept of a $\Lambda(p)$-space (see [2], Definition 6.4.4), a (closed) subspace $H$ of an Orlicz space $L_M$ (or of a symmetric space $X$) on $[0 ,1]$ will be called strongly embedded in $L_M$ (respectively, in $X$) if the convergence in the $L_M$-norm (respectively, in the $X$-norm) on $H$ is equivalent to the convergence in measure. The condition $1<p<2$ from Rosenthal’s theorem in this more general setting turns into the inequality $1<\alpha_M^\infty\leqslant \beta_M^\infty<2$ for the Matuszewska–Orlicz indices of the function $M$. As shown in [8], unlike $L^p$, the last condition does not guarantee that an analogue of Rosenthal’s theorem is valid in $L_M$. In particular, the norms of functions of the unit ball of any subspace strongly embedded in the space $L_M$ and isomorphic to some Orlicz sequence space are equicontinuous in $L_M$ if and only if the function $t^{-1/\beta_M^\infty}$ does not belong to $L_M$ (see Theorem 3 in [8]). Thus, if this condition is not fulfilled, an analogue of Rosenthal’s theorem does not hold even for this special class of subspaces of Orlicz spaces. The family of subspaces of the space $L_M$ which are isomorphic to Orlicz sequence spaces includes, in particular, the subspaces spanned in $L_M$ by independent copies of mean zero functions from this space (see § 3.4 below). The present paper is mainly devoted to a detailed study of subspaces of this type. Note that the research related to the class of subspaces of $L^p$-spaces with a symmetric basis spanned by sequences $\{f_k\}_{k=1}^\infty$ of independent functions was started quite a long time ago. The interest in this topic has increased after 1958, when Kadec [9] “put an end” to the solution of the well-known Banach problem proving that, for every pair of numbers $p$ and $q$ such that $1\leqslant p<q<2$, a sequence $\{\xi^{(q)}_k\}_{k=1}^\infty$ of independent copies of a $q$-stable random variable $\xi^{(q)}$ spans a subspace in $L^p$ isomorphic to $\ell^q$. Following this, in 1969, Bretagnolle and Dacunha-Castelle showed (see [10]–[12]) that, for any function $f\in L^p$ such that $\int_0^1 f(t)\,dt=0$, a sequence $\{f_k\}_{k=1}^\infty$ of independent copies of $f$ is equivalent in $L^p$, $1\leqslant p<2$, to the canonical basis in some Orlicz sequence space $\ell_\psi$, where the function $\psi$ is $p$-convex and $2$-concave (see [12], Theorem 1, p. X.8). Later, somewhat closed results were obtained by Braverman (see Corollary 2.1 in [13] and [14]). In the opposite direction, as shown in [11], if $\psi$ is a $p$-convex and $2$-concave Orlicz function such that $\lim_{t\to 0}\psi(t)t^{-p}=0$, then a sequence of independent copies of some mean zero function $f\in L^p$ is equivalent in $L^p$ to the canonical basis in $\ell_\psi$. This research was continued by Astashkin and Sukochev [15], who found, among other things, direct connections between an Orlicz function $\psi$ and the distribution of a function $f\in L^p$, whose independent copies span in $L^p$ a subspace isomorphic to the space $\ell_\psi$. This had led to a natural question of whether a given $\psi$ determines uniquely (up to equivalence for large values of the argument) the distribution of a mean zero function $f\in L^p$ whose independent copies generate a subspace of $L^p$ isomorphic to $\ell^p$. A partial solution of this problem was obtained in subsequent papers [16] and [17]. In particular, according to Theorem 1.1 in [16], if an Orlicz function $\psi$ is sufficiently “far” from the “extreme” functions $t^p$ and $t^2$, $1\leqslant p<2$, such uniqueness exists, and the distribution of such a function $f$ is equivalent (for large values of the argument) to that of the function ${1}/{\psi^{-1}}$. In [17], some of these results were extended to general symmetric function spaces on $[0,1]$ satisfying certain conditions. The present paper depends substantially on the above facts. Other important ingredients in the proofs are a version of the famous Vallée Poussin criterion, and the author’s results [18], which imply that an Orlicz space $L_M$ such that $1<\alpha_M^\infty\leqslant \beta_M^\infty<2$ contains the function $1/\psi^{-1}$ provided that there is a strongly embedded subspace in $L_M$ isomorphic to the Orlicz sequence space $\ell_\psi$. The paper is organized as follows. In § 2 and § 3, we give necessary preliminary information and some auxiliary results related to symmetric spaces, Orlicz functions, and Orlicz spaces. The main results are presented in § 4. In § 4.1, we find, in terms of dilations of a function $f\in L_M$, $\int_0^1 f(t)\,dt=0$, conditions under which the subspace $[f_k]$ spanned by independent copies of $f$ is strongly embedded in $L_M$ (see Proposition 1). Here, we also obtain conditions ensuring that the unit ball of the subspace $[f_k]$ of the above type consists of functions with equicontinuous norms in $L_M$ (see Proposition 2). In § 4.2, these results are applied to the question of whether strong embedding of the subspace $[f_k]$ of this type in $L_M$ implies the equicontinuity in $L_M$ of the norms of functions of the unit ball of this subspace (see Theorem 2). The most complete results are obtained in § 4.3, where $t^{-1/\beta_M^\infty}\notin L_M$ (in particular, this condition holds for $L^p$). Namely, if $1<\alpha_M^\infty\leqslant \beta_M^\infty<2$ and the subspace $[f_k]$ is isomorphic to the Orlicz sequence space $\ell_\psi$, then the above properties of this subspace can be characterized via the Matuszewska–Orlicz indices of the functions $M$ and $\psi$ as follows: the unit ball of the subspace $[f_k]$ consists of functions with equicontinuous norms in $L_M$ $\Longleftrightarrow$ the subspace $[f_k]$ is strongly embedded in $L_M$ $\Longleftrightarrow$ $\alpha_\psi^0>\beta_M^\infty$ (see Theorem 3). In the concluding section § 4.4 of the paper, it is shown that the unit ball of any subspace of the $L^2$-space spanned by mean zero identically distributed independent functions consists of functions with equicontinuous norms in $L^2$ (see Theorem 4). Some results of this paper were announced in the note [19]. § 2. PreliminariesGiven two nonnegative functions (quasinorms) $F_1$ and $F_2$ defined on a set $T$, we write $F_1\preceq F_2$ if there exists a constant $C>0$ such that $F_1(t)\leqslant CF_2( t)$ for all $t\in T$. If $F_1\preceq F_2$ and $F_2\preceq F_1$, we say that $F_1$ and $F_2$ are equivalent on $T$ (written $F_1\asymp F_2$). For $T=(0,\infty)$, we will also speak about the equivalence for large (respectively, small) values of the argument. This means that $F_1\asymp F_2$ for all $t\geqslant t_0$ (respectively, $0<t\leqslant t_0$), where $t_0$ is sufficiently large (respectively, sufficiently small). The notation $X\approx Y$ means that the Banach spaces $X$ and $Y$ are linearly and continuously isomorphic. A subspace of a Banach space is always assumed to be linear and closed. In what follows, $C$, $C_1,\dots$ are positive constants, which are not necessarily the same at different places. 2.1. Symmetric spacesFor a detailed exposition of the theory of symmetric spaces, see the monographs [20]–[22]. A Banach space $X$ of real-valued functions measurable on the space $(I,m)$, where $I=[0,1]$, or $(0,\infty)$, and $m$ is the Lebesgue measure, is called symmetric (or rearrangement invariant) if $x\in X$ and ${\|x\|}_X \leqslant {\|y\|}_X$ whenever $y \in X$ and $x^*(t)\leqslant y^*(t)$ almost everywhere (a.e.) on $I$. Here and in what follows, $x^*(t)$ denotes the right-continuous nonincreasing rearrangement of a function $|x(s)|$ defined by
$$
\begin{equation*}
x^*(t):=\inf \{ \tau\geqslant 0\colon n_x(\tau)\leqslant t \},\qquad 0<t<m(I),
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
n_x(\tau):=m\{s\in I\colon |x(s)|>\tau\},\qquad\tau>0.
\end{equation*}
\notag
$$
In particular, every symmetric space $X$ is a Banach lattice of measurable functions. This means that if $x$ is measurable on $I$, $y \in X$, and $|x(t)|\leqslant |y(t)|$ a.e. on $I$, then $x\in X$ and ${\|x\|}_X \leqslant {\|y\|}_X$. In addition, by definition, if $x$ and $y$ are equimeasurable functions, that is, $n_x(\tau)=n_y(\tau)$ for all $\tau>0$, and $y\in X$, then $x\in X$ and ${\|x\|}_X ={\|y\|}_X$. Note that every measurable function $x(t)$ is equimeasurable with its rearrangement $x^*(t)$. For each symmetric space $X$ on $[0,1]$ (respectively, on $(0,\infty)$) we have the continuous embeddings $L^\infty[0,1] \subseteq X \subseteq L^1[0,1]$ (respectively, $(L^1\cap L^\infty)(0,\infty)\subseteq X \subseteq (L^1+L^\infty)(0,\infty))$. In what follows, it will be always assumed the normalization condition $\|\chi_{[0,1]}\|_X=1$. In this case, the constant in each of the preceding embeddings is equal to $1$. The fundamental function $\phi_X$ of a symmetric space $X$ is defined by $\phi_X(t):=\|\chi_A\|_X$, where $\chi_A$ is the characteristic function of a measurable set $A\subset I$ such that $m(A)=t$. The function $\phi_X$ is quasi-concave (that is, $\phi_X(0)=0$, $\phi_X$ does not decrease, and $\phi_X(t)/t$ does not increase on $I$). Let $X$ be a symmetric space on $[0,1]$. For any $\tau>0$, the dilation operator ${\sigma}_\tau x(t):=x(t/\tau)\chi_{(0,\min\{1,\tau\})}(t)$, $0\leqslant t\leqslant 1$, is bounded in $X$ and $\|{\sigma}_\tau\|_{X\to X}\leqslant \max(1,\tau)$ (see, for example, Theorem II.4.4 in [20]). To avoid any confusion, we will not introduce a special notation for the dilation operator $x(t)\mapsto x(t/\tau)$, $\tau>0$, defined on the set of functions $x(t)$ measurable on $(0,\infty)$. The norm of this operator in any symmetric space $X$ on the semi-axis satisfies exactly the same estimate as that for the above operator ${\sigma}_\tau$. Given a symmetric space $X$ on $[0,1]$, the associated space $X'$ consists of all measurable functions $y$ such that
$$
\begin{equation*}
\|y\|_{X'}:=\sup\biggl\{\int_{0}^1{x(t)y(t)\,dt}\colon \|x\|_X\,\leqslant{1}\biggr\}<\infty.
\end{equation*}
\notag
$$
The associated space $X'$ is also a symmetric space, which embeds isometrically into the dual space $X^*$. In addition, $X'=X^*$ if and only if $X$ is separable. A symmetric space $X$ is called maximal if, from the conditions $x_n\in X$, $n=1,2,\dots$, $\sup_{n=1,2,\dots}\|x_n\|_X<\infty$ and $x_n\to{x}$ a.e., it follows that $x\in X$ and $||x||_X\leqslant \liminf_{n\to\infty}{||x_n||_X}$. The space $X$ is maximal if and only if the canonical embedding of $X$ into its second associated space $X''$ is an isometric surjection. Symmetric sequence spaces can be defined similarly (see, for instance, § II.8 in [20]). In particular, if $X$ is a symmetric sequence space, then the fundamental function of $X$ is defined by $\phi_X(n):=\bigl\|\sum_{k=1}^n e_k\bigr\|_X$, $n=1,2,\dots$ . In what follows, $e_k$ are canonical unit vectors in sequence spaces, that is, $e_k=(e_k^i)_{i=1}^\infty$, $e_k^i=0$, $i\ne k$, and $e_k^k=1$, $k,i=1,2,\dots$ . The family of symmetric spaces includes many classical spaces that play an important role in analysis, in particular, the $L^p$-spaces, Orlicz, Lorentz, Marcinkiewicz spaces, and many other spaces. The next part of this section gives some preliminaries for the theory of Orlicz spaces, which are the main subject of the study in this paper. 2.2. Orlicz functions and Orlicz spacesOrlicz spaces are the most natural and important generalization of $L^p$-spaces. A detailed exposition of their properties can be found in the monographs [23]–[25]. Let $M$ be an Orlicz function, that is, an increasing convex continuous function on the semi-axis $[0, \infty)$ such that $M(0) = 0$. Without loss of generality we will assume in what follows that $M(1) = 1$. The Orlicz space $L_M:=L_M(I)$ consists of all functions $x(t)$ which are measurable on $I$ and have finite Luxemburg norm
$$
\begin{equation*}
\| x \|_{L_M}: = \inf \biggl\{\lambda > 0 \colon \int_I M\biggl(\frac{|x(t)|}{\lambda}\biggr) \, dt \leqslant 1 \biggr\}.
\end{equation*}
\notag
$$
In particular, if $M(u)=u^p$, $1\leqslant p<\infty$, we obtain the space $L^p$ with the usual norm. Note that the definition of the space $L_M[0,1]$ depends (up to equivalence of norms) only on the behaviour of the function $M(u)$ for large values of $u$. The fundamental function of this space can be calculated by the formula $\phi_{L_M}(u)=1/M^{-1}(1/u)$, $0<u\leqslant 1$, where $M^{-1}$ is the inverse function of $M$. If $M$ is an Orlicz function, then the complementary (or Yang conjugate) function $\widetilde{M}$ for $M$ is defined as follows: As is easy to see, $\widetilde{M}$ is also an Orlicz function, and the complementary function for $\widetilde{M}$ is $M$.Every Orlicz space $L_M(I)$ is maximal; $L_M[0,1]$ (respectively, $L_M(0,\infty)$) is separable if and only if $M$ satisfies the so-called $\Delta_2^\infty$-condition ($M\in \Delta_2^\infty$) (respectively, the $\Delta_2$-condition ($M\in \Delta_2$)), that is, $\sup_{u\geqslant 1} (M(2u)/M(u))<\infty$ (respectively, $\sup_{u>0} (M(2u)/M(u))<\infty)$. In this case, $L_M(I)^*=L_M(I)'=L_{\widetilde{M}}(I)$. An important characteristic of an Orlicz space $L_M[0,1]$ are the Matuszewska–Orlicz indices at infinity $\alpha_M^{\infty}$ and $\beta_M^{\infty}$ defined by
$$
\begin{equation*}
\alpha_M^{\infty}: = \sup \biggl\{ p\colon \sup_{t,\, s \geqslant 1} \frac{M(t)s^p}{M(ts)} < \infty \biggr\}, \qquad \beta_M^{\infty}: = \inf \biggl\{ p \colon \inf_{t,\, s \geqslant 1} \frac{M(t)s^p}{M(ts)} > 0 \biggr\}
\end{equation*}
\notag
$$
(see [26] or [27], Proposition 5.3). It can be easily checked that $1 \leqslant \alpha_M^{\infty} \leqslant \beta_M^{\infty} \leqslant \infty$. Moreover, $M\in \Delta_2^\infty$ (respectively, $\widetilde{M}\in \Delta_2^\infty$) if and only if $\beta_M^{\infty} <\infty$ (respectively, $\alpha_M^{\infty}>1$). The Matuszewska–Orlicz indices are a special case of the so-called Boyd indices, which can be defined for any symmetric space on $[0,1]$ or $(0,\infty)$ (see, for example, Definition 2.b.1 in [21] or § II.4.3 in [20]). Similarly, one can define an Orlicz sequence space. Namely, if $\psi$ is an Orlicz function, then the space $\ell_{\psi}$ consists of all sequences $a=(a_{k})_{k=1}^{\infty}$ such that
$$
\begin{equation*}
\| a\|_{\ell_{\psi}} := \inf\biggl\{\lambda>0: \sum_{k=1}^{\infty} \psi \biggl( \frac{|a_{k}|}{\lambda} \biggr)\leqslant 1\biggr\}<\infty.
\end{equation*}
\notag
$$
If $\psi(u)=u^p$, $p\geqslant 1$, then $\ell_\psi=\ell^p$ isometrically. The fundamental function of an Orlicz space $\ell_{\psi}$ may be found by the formula
$$
\begin{equation}
\phi_{\ell_\psi}(n)=\frac{1}{\psi^{-1}(1/n)},\qquad n=1,2,\dots\,.
\end{equation}
\tag{3}
$$
A space $\ell_{\psi}$ is separable if and only if $\psi$ satisfies the $\Delta_2^0$-condition ($\psi\in \Delta_2^0$), that is,
$$
\begin{equation*}
\sup_{0<u\leqslant 1} \frac{\psi(2u)}{\psi(u)}<\infty.
\end{equation*}
\notag
$$
In this case, $\ell_{\psi}^*=\ell_{\psi}'=\ell_{\widetilde{\psi}}$, where $\widetilde{\psi}$ is the complementary function for $ \psi$. As is easy to check (see also Proposition 4.a.2 in [28]), the unit vectors $e_n$, $n=1,2,\dots$, form a symmetric basis in any Orlicz sequence space $\ell_{\psi}$ if $\psi\in \Delta_{2}^{0}$. Recall that a basis $\{x_n\}_{n=1}^\infty$ of a Banach space $X$ is called symmetric if there exists a constant $C>0$ such that, for an arbitrary permutation $\pi$ of the set of positive integers and any $a_n\in\mathbb{R}$,
$$
\begin{equation*}
C^{-1}\biggl\|\sum_{n=1}^{\infty}a_nx_n\biggr\|_X\leqslant \biggl\|\sum_{n=1}^{\infty} a_nx_{\pi(n)} \biggr\|_X \leqslant C\biggl\|\sum_{n=1}^{\infty}a_nx_n\biggr\|_X.
\end{equation*}
\notag
$$
The definition of an Orlicz sequence space $\ell_{\psi}$ depends (up to equivalence of norms) only on the behaviour of the function $\psi$ for small values of the argument. More precisely, if $\varphi,\psi \in \Delta_2^0$, then the following conditions are equivalent: 1) $\ell_{\psi}=\ell_{\varphi}$ (with equivalence of the norms); 2) the canonical vector bases in the spaces $\ell_{\psi}$ and $\ell_{\varphi}$ are equivalent; 3) the functions $\psi$ and $\varphi$ are equivalent for small values of the argument (see Proposition 4.a.5 in [28] or Theorem 3.4 in‘[25]). If $\psi$ is a degenerate Orlicz function, that is, $\psi(u)=0$ for some $u> 0$, we have $\ell_{\psi}=\ell_\infty$ (with equivalence of the norms). Let $\psi$ be an Orlicz function, $\psi\in \Delta_2^0$, $A>0$. Consider the following subsets of $C[0, 1]$:
$$
\begin{equation*}
E_{\psi, A}^0 = \overline{\biggl\{ \frac{\psi(st)}{\psi(s)} \colon 0<s<A \biggr\}},\qquad C_{\psi, A}^0 = \overline{\operatorname{conv} E_{\psi, A}^0},
\end{equation*}
\notag
$$
where the closure is taken in the $C[0,1]$-norm, and $\operatorname{conv} F$ denotes the convex hull of a set $F\subset C[0,1]$. All these sets are nonempty compact subsets of the space $C[0,1]$ (see Lemma 4.a.6 in [28]). According to a well-known result of Lindenstrauss and Tsafriri (see, for example, Theorem 4.a.8 in [28]), the Orlicz space $\ell_\varphi$ is isomorphic to some subspace of the space ${\ell_\psi}$ if and only if $\varphi\in C_{\psi, 1}^0$. For any Orlicz function $\psi$, the Matuszewska–Orlicz indices at zero $\alpha_{\psi}^0$ and $\beta_{\psi}^0$ are defined by
$$
\begin{equation*}
\alpha_{\psi}^0: = \sup \biggl\{ p \colon \sup_{0<t, s \leqslant 1} \frac{\psi(st)}{s^p\psi(t)} < \infty \biggr\}, \qquad \beta_{\psi}^0: = \inf \biggl\{ p \colon \inf_{0<t, s \leqslant 1} \frac{\psi(st)}{s^p\psi(t)} > 0 \biggr\}.
\end{equation*}
\notag
$$
As for the Matuszewska–Orlicz indices at infinity, we have the following inequalities: $1 \leqslant \alpha_{\psi}^{\infty} \leqslant \beta_{\psi}^{\infty} \leqslant \infty$ (see, for example, Chapter 4 in [28]). Moreover, the space $\ell^p$ or $c_0$ if $p=\infty$, is isomorphic to some subspace of an Orlicz space $\ell_\psi$ if and only if $p\in [\alpha_{\psi}^0,\beta_{\psi}^0]$ (see Theorem 4.a.9 in [28]).
§ 3. Auxiliary results3.1. Strongly embedded subspaces and sets of functions with equicontinuous normsLet $X$ be a symmetric space on $[0,1]$. Recall (see § 1) that a subspace $H\subset X$ is strongly embedded if the convergence in the $X$-norm on $H$ is equivalent to the convergence in measure. The following result is known in one form or another (for the case of $L^p$-spaces see Proposition 6.4.5 in [2]). For the reader’s convenience, we present its proof. Lemma 1. Suppose $X$ is a symmetric space on $[0,1]$ such that $X\ne L^1$ and $H$ is a subspace of $X$. If the norms of $X$ and $L^1$ are equivalent on $H$, then $H$ is strongly embedded in $X$. Proof. Assuming the contrary, we find a sequence $\{x_n\}\subset X$ such that $\{x_n\}$ converges to zero in measure, but $\|x_n\|_X\not\to 0$. Passing to a subsequence, we can take for granted that $\{x_n\}$ converges to zero a.e. on $[0,1]$ and $\|x_n\|_X=1$, $n=1,2,\dots$ . For any $A>0$, we have
$$
\begin{equation}
\begin{aligned} \, \|x_n\|_{L^1} &= \int_{\{|x_n|\geqslant A\}} |x_n(t)|\,dt+ \int_{\{|x_n|<A\}} |x_n(t)|\,dt \nonumber \\ &\leqslant \|x_n\|_X\|\chi_{\{|x_n|\geqslant A\}}\|_{X'}+\int_{\{|x_n|<A\}} |x_n(t)|\,dt \nonumber \\ &= \phi_{X'}(m\{|x_n|\geqslant A\})+\int_{\{|x_n|<A\}} |x_n(t)|\,dt, \end{aligned}
\end{equation}
\tag{4}
$$
where $X'$ is the associated space for $X$ and $\phi_{X'}$ is the fundamental function of $X'$ (see § 2.1). Since $X\ne L^1$, we have $X'\ne L_\infty$, and hence, as one can easily check, $\lim_{u\to 0+}\phi_{X'}(u)=0$.
Let $\delta>0$ be arbitrary. First, for all $n=1,2,\dots$ we have
$$
\begin{equation*}
m\{|x_n|\geqslant A\}\leqslant \frac{\|x_n\|_{L^1}}{A}\leqslant \frac{\|x_n\|_X}{A}= \frac{1}{A},
\end{equation*}
\notag
$$
and, consequently, there is $A_0>0$ such that
$$
\begin{equation*}
\sup_{n=1,2,\dots}\phi_{X'}(m\{|x_n|\geqslant A_0\})\leqslant\frac{\delta}{2}.
\end{equation*}
\notag
$$
Second, by the Lebesgue dominated convergence theorem, there is a positive integer $n_0$ such that, for the above $A_0$ and for all $n\geqslant n_0$,
$$
\begin{equation*}
\int_{\{|x_n|<A_0\}} |x_n(t)|\,dt\leqslant\frac{\delta}{2}.
\end{equation*}
\notag
$$
As a result, applying the last two inequalities and estimate (4) for $A=A_0$, we find that $\|x_n\|_{L^1}\leqslant\delta$ for $n\geqslant n_0$. Since $\delta>0$ is arbitrary, the norms of the spaces $X$ and $L^1$ are not equivalent on $H$. This contradiction to the assumption proves the lemma. Let $X$ be a symmetric space on $[0,1]$. The functions of a set $K\subset X$ are said to have equicontinuous norms in $X$ if
$$
\begin{equation*}
\lim_{\delta\to 0}\sup_{m(E)<\delta}\, \sup_{x\in K}\|x\chi_{E}\|_X=0.
\end{equation*}
\notag
$$
Let $H$ be a subspace of a symmetric space $X$. In what follows, by $B_H$ we denote the closed unit ball of $H$, that is, $B_H:=\{x\in H\colon \|x\|_X\leqslant 1\}$. Lemma 2. Let $X$ be a symmetric space on $[0,1]$ and $H$ be a subspace of $X$, $X\ne L^1$. If the $X$-norms of functions of the set $B_H$ are equicontinuous, then $H$ is strongly embedded in $X$. Proof. First, by the assumption and the definition of the rearrangement $x^*$, for every $\varepsilon>0$, there exists $\delta>0$ such that, for any function $x\in H$, $\|x\|_X\leqslant 1$,
Next, for an arbitrary measurable function $x(t)$ on $[0,1]$ and each $\delta>0$, consider the set
$$
\begin{equation*}
Q_x(\delta):=\{t\in [0,1]\colon |x(t)|\geqslant\delta\|x\|_X\}.
\end{equation*}
\notag
$$
Let us show that if $\delta>0$ is sufficiently small, then
$$
\begin{equation}
H\subset \{x\in L^1\colon m(Q_x(\delta))\geqslant \delta\}.
\end{equation}
\tag{6}
$$
Indeed, assuming that this is not the case, for each $\delta>0$, we find a function $x_\delta\in H$ such that $m(Q_{x_\delta}(\delta))<\delta$. By the definition of the rearrangement $x_\delta^*$ and since $\|\chi_{[0,1]}\|_X=1$, we have
$$
\begin{equation*}
\begin{aligned} \, \|x_\delta^*\chi_{[0,\delta]}\|_X &\geqslant \|x_\delta^*\chi_{[0,m(Q_{x_\delta}(\delta))]}\|_X\geqslant \|x_\delta\chi_{Q_{x_\delta}(\delta)}\|_X \geqslant \|x_\delta\|_X- \|x_\delta\chi_{[0,1]\setminus Q_{x_\delta}(\delta)}\|_X \\ &\geqslant \|x_\delta\|_X- \delta\|x_\delta\|_X \|\chi_{[0,1]}\|_X = (1-\delta)\|x_\delta\|_X. \end{aligned}
\end{equation*}
\notag
$$
Since $\delta>0$ and $\varepsilon>0$ are arbitrary, the last inequality contradicts (5) if we take for $x$ in this inequality the function $x_\delta/\|x_\delta\|_{L_M}$ for sufficiently small $\delta$. Thus, (6) is proved.
Now, let $\delta>0$ satisfy (6). Then for all $x\in H$ we have
$$
\begin{equation*}
\|x\|_{L^1}\geqslant \int_{Q_x(\delta)} |x(t)|\, dt\geqslant \delta\|x\|_Xm(Q_x(\delta))\geqslant\delta^2 \|x\|_X.
\end{equation*}
\notag
$$
Since the opposite inequality $\|x\|_{L^1}\leqslant \|x\|_X$, $x\in X$, is fulfilled for any symmetric space $X$ (see § 2.1), we conclude that the norms of $X$ and $L^1$ are equivalent on $H$. The required result now follows from Lemma 1. Remark 1. Slightly modifying the proof, one can show that Lemma 2 is valid for $X=L^1$ as well. At the same time, the converse statement to this lemma does not hold in general (see Remark 6 below or, for more detail, [29], Example 2). 3.2. $P$-convex and $q$-concave Orlicz functions and Matuszewska–Orlicz indicesLet $1\leqslant p<\infty$. An Orlicz function $M$ is said to be $p$-convex (respectively, $p$-concave) if the mapping $t \mapsto M(t^{1/p})$ is convex (respectively, concave). It is easy to check that an Orlicz space $L_M[0,1]$ is $p$-convex (respectively, $p$-concave) if and only if the function $M$ is equivalent to some $p$-convex (respectively, $p$-concave) Orlicz function for large values of the argument. Similarly, an Orlicz sequence space $\ell_\psi$ is $p$-convex (respectively, $p$-concave) if and only if the function $\psi$ is equivalent to some $p$-convex (respectively, $p$-concave) Orlicz function for small values of the argument. Recall that a Banach lattice $X$ is called $p$-convex, respectively, $p$-concave, where $1 \leqslant p \leqslant\infty$, if there exists $C>0$ such that, for any $n\in\mathbb{N}$ and arbitrary elements $x_1, x_2, \dots, x_n$ from $X$,
$$
\begin{equation*}
\biggl\|\biggl(\sum_{k=1}^n |x_k|^p\biggr)^{1/p}\biggr\|_X \leqslant C \biggl(\sum_{k=1}^n \|x_k\|_X^p\biggr)^{1/p},
\end{equation*}
\notag
$$
respectively,
$$
\begin{equation*}
\biggl(\sum_{k=1}^n\|x_k\|_X^p\biggr)^{1/p} \leqslant C \biggl\| \biggl(\sum_{k=1}^n |x_k|^p\biggr)^{1/p}\biggr\|_X
\end{equation*}
\notag
$$
(with the natural modification of expressions for $p=\infty$). Obviously, every Banach lattice is $1$-convex and $\infty$-concave with constant $1$. The space $L^p$ is $p$-convex and $p$-concave with constant $1$. From the definition of Matuszewska–Orlicz indices and Lemma 20 from [30] (see also Lemma 5 in [15]) we obtain the following characterization of the above properties. Lemma 3. Let $1\leqslant p<\infty$ and let $\psi$ be an Orlicz function on $[0,\infty)$. Then (i) $\psi$ is equivalent to a $p$-convex (respectively, $p$-concave) function for small values of the argument $\Longleftrightarrow$ $\psi(st)\leqslant C s^p\psi(t)$ (respectively, $s^p\psi(t)\leqslant C \psi(st)$) for some $C>0$ and all $0<t,s\leqslant 1$; (ii) $\psi$ is equivalent to a $(p+\varepsilon)$-convex (respectively, $(p-\varepsilon)$-concave) function for small values of the argument and some $\varepsilon>0$ $\Longleftrightarrow$ $\alpha_\psi^0>p$ (respectively, $\beta_\psi^0<p$). The proof of the following technical result is analogous to that of Lemma 6 given in [8] and hence omitted. Lemma 4. Let $\psi$ and $\varphi$ be Orlicz functions, $\varphi\in C_{\psi,1}^0$. Then $\alpha_\psi^0\leqslant \alpha_\varphi^0\leqslant \beta_\varphi^0\leqslant \beta_\psi^0$. The following lemma is a direct consequence of the results proved in [18]. Lemma 5. Let $M$ be an Orlicz function, $1<\alpha_M^\infty\leqslant \beta_M^\infty<2$. Suppose that $H$ is a strongly embedded subspace of the Orlicz space $L_M$ such that $H\approx \ell_\psi$, where $\beta_\psi^0<2$. Let $\varphi\in C_{\psi,1}^0$. Then $1/\varphi^{-1}\in L_M$. In particular, $t^{-1/\alpha_\psi^0}\in L_M$. Therefore, if $t^{-1/\beta_M^\infty}\notin L_M$, then $\alpha_\psi^0>\beta_M^\infty$. Proof. First, we note that $\ell^{\alpha_\psi^0}$ is isomorphic to some subspace of the Orlicz space $\ell_\psi$ (see Theorem 4.a.9 in [28] or § 2.2). Consequently, by the assumption, $L_M$ contains a subspace isomorphic to $\ell^{\alpha_\psi^0}$. On the other hand, since $1<\alpha_M^\infty\leqslant \beta_M^\infty<2$, we have $L_M\in\Delta_2^\infty$ and $L_M^*=L_{\widetilde{M}}\in\Delta_2^\infty$ (see § 2.2). Hence, the spaces $L_M$ and $L_M^*$ are maximal and separable. Next, by the well-known Ogasawara theorem (see, for example, Theorem X.4.10 in [31]), $L_M$ is reflexive. Therefore, $L_M$ does not contain subspaces isomorphic to $\ell^1$, whence $\alpha_\psi^0>1$. Thus, from the condition and Lemma 4 it follows that $1<\alpha_\varphi^0\leqslant \beta_\varphi^0<2$.
Further, applying Lemma 3, we find that if $\varepsilon>0$ is sufficiently small, then the function $\varphi$ is $(1+\varepsilon)$-convex and $(2-\varepsilon)$-concave for small values of the argument. Moreover, since $\varphi\in C_{\psi,1}^0$, by Theorem 4.a.8 in [28] (see also § 2.2), the space $\ell_{\varphi}$ is isomorphic to some subspace of the space $\ell_\psi$. Thus, $L_M$ contains a strongly embedded subspace isomorphic to $\ell_{\varphi}$, and we can apply Corollary 3.3 from [18] to conclude that $1/\varphi^{-1}\in L_M$. To prove the second statement of the lemma, note that the function $\varphi(t)=t^{\alpha_\psi^0}$ belongs to the set $C_{\psi,1}^0$ (see § 2.2). Therefore, by the above, $t^{-1/\alpha_\psi^0}\in L_M$. Hence, if additionally $t^{-1/\beta_M^\infty}\notin L_M$, then $\alpha_\psi^0>\beta_M^\infty$, proving Lemma 5. 3.3. A version of the Vallée Poussin criterionThe following simple fact will be used below. Lemma 6. Let $N$ be an increasing continuous function on the half-axis $[0, \infty)$ such that $N(u)/u$ increases for $u>0$ and $N(0) = 0$. Let $N\in \Delta_2$ (respectively, $N\in \Delta_2^\infty$). Then $N$ is equivalent to the Orlicz function $M(t)=\int_0^t (N(u)/u)\,du$ for $t>0$, and $M(0)=0$ on $[0,\infty)$ (respectively, for large values of the argument). Proof. Assume that $N\in \Delta_2$ (the case $N\in \Delta_2^\infty$ is treated similarly).
Note that $M$ is an increasing continuous function on the half-axis $[0, \infty)$. The function $M'(t)=N(t)/t$ is increasing, and hence $M$ is an Orlicz function, and $M(t)\leqslant N(t)$, $t>0$. The opposite estimate follows from the condition $N\in \Delta_2$: where $K$ is the $\Delta_2$-constant of $N$. Thus, $M$ and $N$ are equivalent on $[0,\infty)$, and the proof is completed.The proof of the following result, which is a variant of the famous Vallée Poussin criterion (see, for example, [32]–[34]), can be found in [8]. Lemma 7. Let $M$ be an Orlicz function such that $M\in \Delta_2^\infty$ and $\widetilde{M}\in \Delta_2^\infty$. For any $f\in L_M$, there exists a function $N$ equivalent to some Orlicz function for large values of the argument and such that $N(1)=1$, $N\in \Delta_2^\infty$, $\widetilde{N}\in \Delta_2^\infty$, and Moreover, if in addition $M$ is $p$-convex for large values of the argument, then $N$ is also equivalent to some $p$-convex Orlicz function for large values of the argument. 3.4. A description of subspaces of Orlicz spaces generated by mean zero identically distributed independent functionsRecall (see, for instance, Chap. 2 in [35]) that a set of functions $\{f_k\}_{k=1}^n$ measurable on $[0,1]$ is called independent if, for any intervals $I_k\subset \mathbb{R}$,
$$
\begin{equation*}
m\{t\in [0,1]\colon f_k(t)\in I_k,\,k=1,2,\dots,n\}= \prod_{k=1}^n m\{t\in [0,1]\colon f_k(t)\in I_k\}.
\end{equation*}
\notag
$$
It is said that $\{f_k\}_{k=1}^\infty$ is a sequence of independent functions if the set $\{f_k\}_{k=1}^n$ is independent for each $n\in\mathbb{N}$. Let $M$ be an Orlicz function, $M\in \Delta_2^\infty$, $L_M=L_M[0,1]$ be the Orlicz space, $\{f_k\}_{k=1}^\infty$ be a sequence of mean zero independent functions equimeasurable with a function $f \in L_M$, $\int_0^1 f_k(t)\,dt\,{=}\,0$, $k=1,2,\dots$ . Then (see [36], p. 794, or [37]), with equivalence constants independent of $a_k\in\mathbb{R}$, $k=1,2,\dots$, we have
$$
\begin{equation*}
\biggl\|\sum_{k=1}^\infty a_kf_k\biggr\|_{L_M}\asymp \biggl\|\biggl(\sum_{k=1}^\infty a_k^2f_k^2\biggr)^{1/2}\biggr\|_{L_M}.
\end{equation*}
\notag
$$
In turn, if $\theta(u)=u^2$ for $0\leqslant u\leqslant 1$, $\theta(u)=M(u)$ for $u\geqslant 1$ and $\ell_\psi$ is the Orlicz sequence space generated by the function
$$
\begin{equation}
\psi(u):=\int_0^1\theta(u|f(t)|) \, dt,\qquad u\geqslant 0,
\end{equation}
\tag{7}
$$
then by Theorem 8 in [38]
$$
\begin{equation*}
\biggl\|\biggl(\sum_{k=1}^\infty a_k^2f_k^2\biggr)^{1/2}\biggr\|_{L_M} \asymp \|(a_k)\|_{\ell_\psi}.
\end{equation*}
\notag
$$
Hence,
$$
\begin{equation}
\biggl\|\sum_{k=1}^\infty a_kf_k\biggr\|_{L_M}\asymp\|(a_k)\|_{\ell_\psi},
\end{equation}
\tag{8}
$$
which means that the sequence $\{f_k\}_{_{k=1}}^\infty$ is equivalent in $L_M$ to the canonical basis $\{e_k\}_{_{k=1}}^\infty$ in the Orlicz sequence space ${\ell_\psi}$, where $\psi$ is defined by (7). Observe that, in general, $\theta$ is not an Orlicz function. However, $\theta(t)/t$ is an increasing continuous function, and $\theta\in\Delta_2$ since $M\in \Delta_2^\infty$. Therefore, by Lemma 6, $\theta$ is equivalent on $(0,\infty)$ to the Orlicz function $\widetilde{\theta}(t):=\int_0^t (\theta(u)/u) \,du$. This and (7) imply that $\psi$ is also equivalent to some Orlicz function. Next, for any measurable function $x(t)$ on $[0,1]$ and any sequence $a = (a_k)_{k=1}^\infty$ of reals, we set
$$
\begin{equation*}
(a\mathbin{\overline\otimes} x)(s):= \sum_{k=1}^\infty a_kx(s-k+1)\chi_{(k,k+1)}(s), \qquad s>0.
\end{equation*}
\notag
$$
As is easy to see, the distribution function of the function $a \mathbin{\overline\otimes} x$ is equal to the sum of the distribution functions of the terms $a_kx$, $k=1,2,\dots$, that is,
$$
\begin{equation*}
n_{a\mathbin{\overline\otimes} x}(\tau)= \sum_{k=1}^\infty n_{a_k x}(\tau),\qquad \tau>0.
\end{equation*}
\notag
$$
As above, suppose that $M$ is an Orlicz function, $\{f_k\}_{k=1}^\infty$ be a sequence of mean zero independent functions equimeasurable with some function $f\in L_M$. According to the well-known Johnson–Schechtman theorem (see Theorem 1 in [36]), with constants that do not depend on $a_k\in\mathbb{R}$, $k=1,2,\dots$, we have
$$
\begin{equation*}
\biggl\|\sum_{k=1}^\infty a_kf_k\biggr\|_{L_M}\asymp \|(a \mathbin{\overline\otimes} f)^*\chi_{[0,1]}\|_{L_M}+ \|(a \mathbin{\overline\otimes} f)^*\chi_{[1,\infty)}\|_{L^2}.
\end{equation*}
\notag
$$
Combining this together with (8), we obtain
$$
\begin{equation}
\|(a_k)\|_{\ell_\psi}\asymp \|(a \mathbin{\overline\otimes} f)^*\chi_{[0,1]}\|_{L_M}+ \|(a \mathbin{\overline\otimes} f)^*\chi_{[1,\infty)}\|_{L^2}.
\end{equation}
\tag{9}
$$
In particular, the function
$$
\begin{equation*}
\biggl(\biggl(\sum_{k=1}^n e_k\biggr)\mathbin{\overline\otimes} f\biggr)(s) =\sum_{k=1}^n f(s-k+1)\chi_{(k,k+1)}(s)
\end{equation*}
\notag
$$
is equimeasurable with the function $f(t/n)$, $t>0$. Thus, if $f=f^*$, then, taking into account that the fundamental function $\phi_{\ell_\psi}$ satisfies (3) (see § 2.2), by (9) and the definition of the dilation operator $\sigma_\tau$ (see § 2.1), we get
$$
\begin{equation}
\begin{aligned} \, \frac{1}{\psi^{-1}(1/n)}&\asymp \|\sigma_nf\|_{L_M}+ \biggl\|f\biggl(\frac{\cdot}{n}\biggr)\chi_{[1,\infty)}\biggr\|_{L^2} \nonumber \\ &=\|\sigma_nf\|_{L_M}+\biggl(n\int_{1/n}^1f(s)^2\,ds\biggr)^{1/2},\qquad n\in\mathbb{N}. \end{aligned}
\end{equation}
\tag{10}
$$
Let us illustrate the above discussion with two examples, which show that the studied properties of the subspace $[f_k]:=[f_k]_{L_M}$ spanned by a sequence of independent copies of a mean zero function $f\in L_M$ and isomorphic to some Orlicz sequence space ${\ell_\psi}$ (see (7)), depend not only on the “degree of proximity” of the function $\psi$ to the function $M$, but also on whether the function $t^{-1/\beta_M^\infty}$ belongs to the space $L_M$ (see [8]). Example 1. Let $1<p<2$, $M(u)=u^p$, $f(t):=t^{-1/p}\ln^{-3/(2p)}(e/t)$, $0 < t \leqslant 1$. Then $f=f^*$, and, if $[f_k]_{L^p}=\ell_\psi$ and $[f_k]_{L^1}=\ell_{\varphi}$, by (10) (see also Proposition 2.4 in [16]), we have In the next example, as in the preceding one, the function $\psi$ is “close” to $M$, differing only by a power of the logarithm. However, now $t^{-1/\beta_M^\infty}\in L_M$ (in contrast, in Example 1, $\beta_M^\infty=p$, and hence $t^{-1/\beta_M^\infty}\,{\notin}\, L_M=L^p$), and, as a result, the subspace $[f_k]_{L_M}$, isomorphic to the space $\ell_\psi$, is strongly embedded in $L_M$. Example 2. Let $1<p<2$, $0<\alpha<1/p$, $M(u)$ be an Orlicz function equivalent to the function $u^p\ln^{-2} u$ for large values of $u$, $f(t):=t^{-1/p}\ln^{\alpha}(e/t)$, $0<t\leqslant 1$. We have and hence $f\in L_M$ by to the choice of parameters $p$ and $\alpha$. Consider an Orlicz function $\psi$ such that $\psi(s)\asymp s^p\ln^{p\alpha}(e/s)$ for small values of the argument. On the one hand, it is immediately verified that $1/\psi^{-1}(t)\asymp f(t)$, $0<t\leqslant 1$. On the other hand, for some $C>0$
$$
\begin{equation*}
\psi(st)\leqslant C\psi(s)\psi(t),\qquad 0\leqslant s,t\leqslant 1.
\end{equation*}
\notag
$$
Therefore, by Theorem 4.1 in [17], for every symmetric space $X$ such that $f\in X$, we have $[f_k]_X\approx\ell_\psi$; here, as above, $\{f_k\}$ is a sequence of mean zero independent functions equimeasurable with $f$. In particular, $[f_k]_{L_M}\approx[f_k]_{L^1}\approx \ell_\psi$, and hence the subspace $[f_k]_{L_M}$ is strongly embedded in $L_M$. Moreover, as we will see in Theorem 2, due to the submultiplicativity of $\psi$, the unit ball of this subspace consists of functions with equicontinuous norms in $L_M$. In what follows, we will repeatedly use the following statement, which follows from the results of [16] on the uniqueness of the distribution of a function whose independent copies span a given subspace in the $L^p$-space. Lemma 8. Let $M$ be an Orlicz function, $M \in \Delta_2^\infty$, $f \in L_M$. Suppose that the subspace $[f_k]_{L_M}$, where $\{f_k\}$ is a sequence of independent functions equimeasurable with $f$ and such that $\int_0^1 f_k(t)\,dt=0$, is strongly embedded in $L_M$. Let $[f_k]_{L_M}=\ell_\psi$, where $1<\alpha_\psi^0\leqslant\beta_\psi^0<2$. Then $n_f(\tau)\asymp n_{1/\psi^{-1}}(\tau)$ for large $\tau>0$. Proof. By the assumption, with constants independent of $n\in\mathbb{N}$ and $a_k\in\mathbb{R}$,
$$
\begin{equation*}
\biggl\|\sum_{k=1}^n a_kf_k\biggr\|_{L_M} \asymp \biggl\|\sum_{k=1}^n a_kf_k\biggr\|_{L^1}.
\end{equation*}
\notag
$$
Furthermore, since $[f_k]_{L_M}\approx \ell_\psi$, we have by (8)
$$
\begin{equation*}
\frac{1}{\psi^{-1}(1/n)}=\biggl\|\sum_{k=1}^n e_k\biggr\|_{\ell_\psi}\asymp \biggl\|\sum_{k=1}^n f_k\biggr\|_{L_M},\qquad n\in\mathbb{N}.
\end{equation*}
\notag
$$
Thus, with constants independent of $n\in\mathbb{N}$,
$$
\begin{equation*}
\frac{1}{\psi^{-1}(1/n)}\asymp\biggl\|\sum_{k=1}^n f_k\biggr\|_{L^1}.
\end{equation*}
\notag
$$
Next, we have $1<\alpha_\psi^0\leqslant\beta_\psi^0<2$, and therefore by Lemma 3, the function $\psi$ is $(1+\varepsilon)$-convex and $(2-\varepsilon)$-concave for small values of the argument if $\varepsilon>0$ is sufficiently small. Now the result of the lemma is a direct consequence of the last equivalence and Theorem 1.1 from [16] applied to the case $p=1$. This proves the lemma. § 4. The main results4.1. A characterization of properties of subspaces generated by independent copies of a mean zero function $f$ in terms of dilations of $f$Let us start with a sufficient (and necessary in many cases) condition under which a sequence of independent copies of a mean zero function $f\in L_M$ spans a strongly embedded subspace in a given Orlicz space $L_M$. Proposition 1. Let $M$ be an Orlicz function, $f\in L_M$. (i) If $\lim_{t\to\infty}M(t)/t=\infty$ and
$$
\begin{equation}
\|\sigma_n f\|_{L_M}\preceq \|\sigma_n f\|_{L^1},\qquad n\in\mathbb{N},
\end{equation}
\tag{13}
$$
then the subspace $[f_k]$ spanned by a sequence of mean zero independent functions $\{f_k\}$ equimeasurable with $f$ is strongly embedded in $L_M$. (ii) Conversely, if such a sequence $\{f_k\}$ as in (i) spans in $L_M$ a strongly embedded subspace isomorphic to an Orlicz space ${\ell_\psi}$, with $1<\alpha_\psi^0\leqslant\beta_\psi^0<2$, then inequality (13) holds. Proof. Without loss of generality we assume that $f=f^*$.
(i) According to the discussion in § 3.4, the sequence $\{f_k\}$ is equivalent in the space $L_M$ (respectively, $L^1$) to the canonical basis in some Orlicz sequence space ${\ell_\psi}$ (respectively, $\ell_{\theta}$). Since $\lim_{t\to\infty}M(t)/t=\infty$, we have $L_M\ne L^1$. Consequently, by Lemma 1, it suffices to show that ${\ell_\psi}= \ell_{\theta}$, or what is the same, that the fundamental functions of these spaces are equivalent for small $t>0$ (see § 2.2). By (10),
$$
\begin{equation}
\frac{1}{\psi^{-1}(1/n)}\asymp \|\sigma_nf\|_{L_M} + \biggl(n\int_{1/n}^1f(s)^2\,ds\biggr)^{1/2},\qquad n\in\mathbb{N},
\end{equation}
\tag{14}
$$
and, similarly,
$$
\begin{equation*}
\frac{1}{\theta^{-1}(1/n)}\asymp \|\sigma_nf\|_{L^1} + \biggl(n\int_{1/n}^1f(s)^2\,ds\biggr)^{1/2},\qquad n\in\mathbb{N},
\end{equation*}
\notag
$$
and now the required equivalence follows from condition (13), formula (3) for the fundamental function of an Orlicz space, and from the convexity of $\psi$ and ${\theta}$.
(ii) It suffices to show that inequality (13) holds for all $n$ sufficiently large. Since $\psi^{-1}$ is an increasing concave function on $(0,1]$, it follows that $\psi^{-1}(t)\leqslant\psi^{-1}(Ct)\leqslant C\psi^{-1}(t)$ for any $C\geqslant 1$ and all $0<t\leqslant 1$; in addition, the function $1/\psi^{-1}$ coincides with its nonincreasing rearrangement. Next, by Lemma 8, the distribution functions $n_f(\tau)$ and $n_{1/\psi^{-1}}(\tau)$ are equivalent for large $\tau>0$. Combining this together with the definition of the nonincreasing rearrangement of a measurable function (see § 2.1), we have, for some $t_0\in (0,1]$,
$$
\begin{equation*}
f(t)\asymp \frac1{\psi^{-1}(t)}, \qquad 0<t\leqslant t_0.
\end{equation*}
\notag
$$
Thus, since (14) is satisfied by the assumption, we have, for a sufficiently large $n_0\in \mathbb{N}$,
$$
\begin{equation*}
\|\sigma_nf\|_{L_M}\preceq f\biggl(\frac1{n}\biggr),\qquad n\geqslant n_0.
\end{equation*}
\notag
$$
Now, inequality (13) for $n\geqslant n_0$ is a direct consequence of the last estimate and the inequality
$$
\begin{equation*}
f\biggl(\frac1{n}\biggr)\leqslant n\int_0^{1/n} f(u)\,du= \int_0^{1} f\biggl(\frac{u}{n}\biggr)\,du = \|\sigma_nf\|_{L^1},\qquad n\in\mathbb{N}.
\end{equation*}
\notag
$$
This completes the proof of Proposition 1. In the same terms we can also state a condition for equicontinuity of $L_M$-norms of functions of the unit ball of such a subspace of $L_M$. Proposition 2. Let $M$ be an Orlicz function, $\lim_{t\to\infty}M(t)/t=\infty$, $f\in L_M$, and let $\{f_k\}$ be a sequence of mean zero independent functions equimeasurable with $f$. Consider the following conditions: (a) the unit ball of the subspace $[f_k]$ consists of functions with equicontinuous norms in $L_M$; (b) there is a convex nondecreasing function $N$ on $[0,\infty)$ such that $N(0)=0$, $N\in\Delta_2^\infty$, $\lim_{u\to\infty}{N(u)}/{M(u)}=\infty$, and
$$
\begin{equation}
\|\sigma_n f\|_{L_N}\preceq\|\sigma_n f\|_{L_M},\qquad n\in\mathbb{N}.
\end{equation}
\tag{15}
$$
Then (b) $\Rightarrow$ (a). If, in addition, $[f_k]_{L_M}\approx {\ell_\psi}$, where $1<\alpha_\psi^0\leqslant\beta_\psi^0<2$, then the inverse implication (a) $\Rightarrow$ (b) also holds. Proof. (b) $\Rightarrow$ (a). First, from (15) and the assumption ${f\in L_M}$ it follows that $f{\in L_N}$. Next,
$$
\begin{equation*}
\lim_{u\to\infty}\frac{M(u)}{u}=\lim_{u\to\infty}\frac{N(u)}{u}=\infty,
\end{equation*}
\notag
$$
and so, arguing exactly as in the proof of Proposition 1(i), we can show that the sequence $\{f_k\}$ in both spaces $L_M$ and $L_N$ is equivalent to the canonical basis in the same Orlicz sequence space. Hence, the norms of these spaces are equivalent on the subspace $H:=[f_k]_{L_M}$, that is, for some $C>0$
$$
\begin{equation}
B_H\subset \{x\in L_N\colon \|x\|_{L_N}\leqslant C\}.
\end{equation}
\tag{16}
$$
Moreover, due to the conditions and Lemma 3 from [29], we infer that the embedding $L_N\subset L_M$ is strict. This means that
$$
\begin{equation*}
\lim_{\delta\to 0}\sup_{\|x\|_{L_N}\leqslant 1,\, m(\operatorname{supp}x)\leqslant\delta}\|x\|_{L_M}=0
\end{equation*}
\notag
$$
(for more details related to properties of strict embeddings of symmetric spaces, see [39]). As a result, and now $(a)$ follows.
(a) $\Rightarrow$ (b). Let $H:=[f_k]$. According to the condition and the Vallée Poussin criterion (see, for example, Theorem 3.2 from [34]), there exists a nondecreasing convex function $Q$ on $[0,\infty)$ such that $Q(0)=0$, $Q\in\Delta_2^\infty$, $\lim_{u\to\infty}{Q(u)}/{u}=\infty$, and $\sup_{x\in B_H}\|Q(|x|)\|_{L_M}<\infty$. The last relation means that, for some $C\geqslant 1$ and all $x\in B_H$,
$$
\begin{equation*}
\int_0^1 M\biggl(\frac{Q(|x(t)|)}{C}\biggr)\,dt\leqslant 1.
\end{equation*}
\notag
$$
The function $Q$ is convex, and hence $Q(|x(t)|)/C\geqslant Q(|x(t)|/C)$. Consequently,
$$
\begin{equation*}
\int_0^1 M\biggl(Q\biggl(\frac{|x(t)|}{C}\biggr)\biggr)\,dt\leqslant 1
\end{equation*}
\notag
$$
for all $x\in B_H$. Setting $N(u):=M(Q(u))$ and taking into account the properties of the functions $M$ and $Q$, it is easy to verify that the function $N$ satisfies all the conditions in (b). In addition, by the last inequality embedding (16) still holds. Thus, the $L_M$- and $L_N$-norms are equivalent on the subspace $H$. Since $H$ is strongly embedded in the space $L_M$ by the condition and Lemma 2, it follows that $H$ is also strongly embedded in $L_N$ (see also Lemma 1). Now by Proposition 1(ii)
$$
\begin{equation*}
\|\sigma_n f\|_{L_N}\preceq \|\sigma_n f\|_{L^1}\leqslant\|\sigma_n f\|_{L_M},\qquad n\in\mathbb{N},
\end{equation*}
\notag
$$
which proves inequality (15), and, therefore, Proposition 2. 4.2. Subspaces of the space $L_M$ spanned by independent copies of mean zero functions and whose unit ball consists of functions with equicontinuous $L_M$-normsLet $h\colon [0,1]\to [0,\infty)$, $h(t)>0$ if $0<t\leqslant 1$. Recall that the dilation function $\mathcal M_h$ of $h$ is defined as follows:
$$
\begin{equation*}
\mathcal M_h(t):=\sup_{0<s\leqslant \min(1,1/t)}\frac{h(st)}{h(s)},\qquad t>0.
\end{equation*}
\notag
$$
Proposition 3. Let $\psi\colon [0,1]\to [0,1]$ is an increasing and continuous function, $\psi(0)=0$, $\psi(1)=1$, let $f(t):=1/\psi^{-1}(t)$, $0<t\leqslant 1$, and let $g$ be a nonincreasing nonnegative function on $(0,1]$ such that Then, for any sequence $c=(c_k)\in {\ell_\psi}$, Proof. Without loss of generality we will further assume that $\|c\|_{\ell_\psi}=1$.
We first observe that, thanks to the properties of $\psi$, the function from the right-hand side of equality (17) is nonnegative, continuous, and nonincreasing. In addition, it does not exceed $1$, and tends to zero as $\tau$ goes off to infinity. Therefore, there exists a nonincreasing function $g\colon (0,1]\to [0,\infty)$ satisfying (17). Since $\psi$ does not decrease and $\psi(0)=0$, we have, for each $\tau\geqslant 1$,
$$
\begin{equation*}
n_f(\tau) = m\biggl\{ u\in (0,1] \colon \frac{1}{\psi^{-1}(u)} > \tau\biggr\} = m\biggl\{ u\in (0,1] \colon \psi \biggl(\frac{1}{\tau}\biggr) > u\biggr\}= \psi\biggl(\frac{1}{\tau}\biggr).
\end{equation*}
\notag
$$
Therefore, by the definition of the function $c \mathbin{\overline\otimes} f$ (see § 3.4),
$$
\begin{equation}
n_{c \mathbin{\overline\otimes} f} (\tau) = \sum_{k=1}^\infty n_{c_k f} (\tau) = \sum_{k=1}^\infty \psi\biggl(\frac{|c_k|}{\tau}\biggr).
\end{equation}
\tag{18}
$$
In addition, since $\|c\|_{\ell_\psi} = 1$, we have, for any $k=1,2,\dots$,
$$
\begin{equation*}
\psi(|c_k|) \leqslant \sum_{i=1}^\infty \psi(|c_i|)=1= \psi(1).
\end{equation*}
\notag
$$
Using again the monotonicity of $\psi$, we find that $|c_k| \leqslant 1$ for all $k=1,2,\dots$ . Hence, by the definition of the function $\mathcal{M}_\psi$, we have, for each $\tau \geqslant 1$ and all $k=1,2,\dots$,
$$
\begin{equation*}
\psi\biggl(\frac{|c_k|}{\tau}\biggr)\leqslant \psi(|c_k|)\mathcal{M}_\psi \biggl(\frac{1}{\tau}\biggr).
\end{equation*}
\notag
$$
Now from (17) and (18) we have
$$
\begin{equation}
n_{c \mathbin{\overline\otimes} f} (\tau) \leqslant \mathcal{M}_\psi\biggl(\frac{1}{\tau}\biggr) \sum_{k=1}^\infty \psi(|c_k|) \leqslant \mathcal{M}_\psi\biggl(\frac{1}{\tau}\biggr)=n_g(\tau), \qquad \tau\geqslant 1,
\end{equation}
\tag{19}
$$
since $\|c\|_{\ell_\psi}=1$ and because $\psi$ is increasing.
Now, let us check that, for each $s \in (0,1)$,
$$
\begin{equation}
\{ \tau>0\colon n_g(\tau)\leqslant s\}\subset \{ \tau>0\colon n_{c \mathbin{\overline\otimes} f} (\tau)\leqslant s\}.
\end{equation}
\tag{20}
$$
Indeed, $n_g(1)=\mathcal{M}_\psi(1)=1$, whence $g(t)> 1$ a.e. on $(0,1]$. Hence
$$
\begin{equation*}
\{ \tau>0\colon n_{g}(\tau)\leqslant s\}\subset (1,\infty),
\end{equation*}
\notag
$$
and, therefore, by (19), the inequality $n_{g}(\tau)\leqslant s$ implies that $n_{c \mathbin{\overline\otimes} f} (\tau)\leqslant s$. This proves embedding (20).
Since $g$ is not increasing, it follows from the definition of a nonincreasing rearrangement and (20) that
$$
\begin{equation*}
(c\mathbin{\overline\otimes} f)^*\cdot \chi_{(0,1)} \leqslant g,
\end{equation*}
\notag
$$
which completes the proof of Proposition 3. Remark 2. Suppose that the function $\mathcal{M}_\psi(t)$ strictly increases on $(0,1]$. Then, as is easy to check, the function $g$, as defined by (17), coincides with the inverse function $\mathcal{M}_\psi^{-1}(t)$. The following result is a consequence of Proposition 3 and the definition of a symmetric space. Corollary 1. Let $\psi\colon [0,1]\to [0,1]$ be an increasing continuous function, $\psi(0)=0$, $\psi(1)=1$, $f(t):=1/\psi^{-1}(t)$, $0<t\leqslant 1$, and $g$ be a nonincreasing nonnegative function on $(0,1]$ such that its distribution function $n_g(\tau)$ is defined by (17). If $X$ is a symmetric space on $[0,1]$ such that $g\in X$, then for any sequence $c=(c_k)\in {\ell_\psi}$, We next need the following technical lemma. Lemma 9. If a function $\psi\colon [0,1]\to [0,1]$ increases, $\psi(0)=0$, $\psi(1)=1$, and $h(t)=\mathcal M_{1/\psi^{-1}}(t)$, $0<t\leqslant 1$, then Proof. Since $\psi$ is increasing, $\mathcal{M}_\psi(1)=1$, and $h$ is nonincreasing, it suffices to show that, for any $\tau\geqslant 1$ and arbitrarily small $\varepsilon>0$,
We set $t:=\mathcal{M}_\psi({1}/{\tau})-\varepsilon$. By the definition of $h$, we have
$$
\begin{equation*}
h(t)=\sup_{0<s\leqslant 1}\frac{\psi^{-1}(s)}{\psi^{-1}(st)}= \sup_{0<u\leqslant t\leqslant 1} \frac{\psi^{-1}(u/t)}{\psi^{-1}(u)}.
\end{equation*}
\notag
$$
Thus, (21) holds if and only if there is $u>0$ such that $0<u\leqslant t\leqslant 1$ and
$$
\begin{equation*}
\psi^{-1}\biggl(\frac{u}{t}\biggr)>\tau \psi^{-1}(u),
\end{equation*}
\notag
$$
or, equivalently, Note that $\tau \psi^{-1}(u)\leqslant 1$. Therefore, changing to $\psi^{-1}(u)=v$, we find that the last inequality holds if and only if
$$
\begin{equation*}
\mathcal{M}_\psi\biggl(\frac{1}{\tau}\biggr):= \sup_{0<v\leqslant 1} \frac{\psi(v/\tau)}{\psi(v)}>t.
\end{equation*}
\notag
$$
But by the choice of $t$ this inequality holds. This proves inequality (21), and, therefore, Lemma 9. Theorem 1. Let $M$ be an Orlicz function such that $1<\alpha_M^\infty\leqslant \beta_M^\infty<2$. Assume also that $f\in L_M$ and $\mathcal{M}_{f^*}\in L_M$. Let $\{f_k\}$ be a sequence of mean zero independent functions equimeasurable with $f$ and $[f_k]_{L_M}\approx\ell_\psi$, where $1<\alpha_\psi^0\leqslant \beta_\psi^0<2$. Then the unit ball of the subspace $[f_k]_{L_M}$ consists of functions with equicontinuous norms in $L_M$. Proof. Without loss of generality we can assume that $f=f^*$. Let us first prove that the subspace $[f_k]_{L_M}$ is strongly embedded in $L_M$.
From the definition of the dilation function $\mathcal{M}_f$ we have
$$
\begin{equation*}
\sigma_{1/s}f(t)=f(st)\leqslant \mathcal{M}_f (t)f(s),\qquad 0<s,t\leqslant 1.
\end{equation*}
\notag
$$
Since ${\mathcal{M}}_f\in L_M$ by condition and $f$ is a nonnegative nonincreasing function, this inequality implies that, for all $0<s\leqslant 1$,
$$
\begin{equation*}
\|\sigma_{1/s}f\|_{L_M}\leqslant \|\mathcal{M}_f\|_{L_M}f(s)\leqslant \|\mathcal{M}_f\|_{L_M}\cdot \frac1s\int_0^s f(u)\,du= \|\mathcal{M}_f\|_{L_M}\|\sigma_{1/s}f\|_{L^1}.
\end{equation*}
\notag
$$
Now the required result is secured by Proposition 1(i).
Let us now proceed with the proof of the theorem. The subspace $[f_k]_{L_M}$ is strongly embedded in $L_M$, and hence by Lemma 8 we have $n_f(\tau)\asymp n_{1/\psi^{-1}}(\tau)$ for large $\tau>0$. The functions $f$ and $1/\psi^{-1}$ do not increase and $\psi^{-1}(1)=1$, and hence, proceeding as in the proof of Proposition 1(ii), we have, for some $0<t_0\leqslant 1$,
$$
\begin{equation*}
f(t)\asymp \frac1{\psi^{-1}(t)},\quad 0<t\leqslant t_0,\quad\text{and}\quad f(t)\preceq \frac1{\psi^{-1}(t)},\quad 0<t\leqslant 1.
\end{equation*}
\notag
$$
Consequently,
$$
\begin{equation*}
\mathcal M_{1/\psi^{-1}}(t)=\sup_{0<s\leqslant 1}\frac{\psi^{-1}(s)}{\psi^{-1}(st)} \preceq \sup_{0<s\leqslant 1}\frac{f(st)\psi^{-1}(s)f(s)}{f(s)}\preceq \mathcal M_f(t),\qquad 0<t\leqslant t_0.
\end{equation*}
\notag
$$
The function $\mathcal M_{1/\psi^{-1}}$ does not increase, and, by the condition, $\mathcal M_f\in L_M$, and hence, from the latter inequality, Lemma 9, and the definition of the function $g$ (see Proposition 3) it follows that $g$ lies in the space $L_M$.
Next, by using Lemma 7, we find a function $N$ equivalent to some Orlicz function such that $N(1)=1$, $N\in \Delta_2^\infty$, $\widetilde{N}\in \Delta_2^\infty$, $\lim_{u\to\infty}{N(u)}/{M(u)}=\infty$, and $g\in L_N$. Assuming that $N$ is an Orlicz function itself, it follows from Corollary 1 that, for any sequence $c=(c_k)\in {\ell_\psi}$,
$$
\begin{equation*}
\|(c\mathbin{\overline\otimes} f)^*\cdot\chi_{(0,1)}\|_{L_N}\leqslant \|g\|_{L_N}\|c\|_{\ell_\psi}.
\end{equation*}
\notag
$$
Since (see § 3.4)
$$
\begin{equation*}
\begin{aligned} \, \biggl\|\sum_{k=1}^\infty c_kf_k\biggr\|_{L_N} &\asymp \|(c\mathbin{\overline\otimes} f)^*\cdot\chi_{(0,1)}\|_{L_N}+ \|(c\mathbin{\overline\otimes} f)^*\cdot\chi_{(1,\infty)}\|_{L^2}, \\ \biggl\|\sum_{k=1}^\infty c_kf_k\biggr\|_{L_M} &\asymp \|(c\otimes f)^*\cdot\chi_{(0,1)}\|_{L_M} + \|(c\otimes f)^*\cdot\chi_{(1,\infty)}\|_{L^2} \asymp\|c\|_{\ell_\psi} \end{aligned}
\end{equation*}
\notag
$$
and since $L_N\subset L_M$, we have
$$
\begin{equation*}
\biggl\|\sum_{k=1}^\infty c_kf_k\biggr\|_{L_N}\asymp \|c\|_{\ell_\psi}.
\end{equation*}
\notag
$$
As a result, to complete the proof it suffices to apply the Vallée Poussin criterion (see Theorem 3.2 in [34]). This proves Theorem 1. The next theorem gives simple sufficient conditions under which the unit ball of a strongly embedded subspace of $L_M$ spanned by independent copies of a mean zero function from $L_M$ consists of functions having equicontinuous norms in $L_M$. Theorem 2. Let $M$ be an Orlicz function such that $1<\alpha_M^\infty\leqslant \beta_M^\infty<2$. Suppose that $\{f_k\}$ is a sequence of mean zero independent functions equimeasurable with a function $f\in L_M$ and $[f_k]\approx\ell_\psi$, where $1<\alpha_\psi^0\leqslant \beta_\psi^0<2$. Assume also that the subspace $[f_k]$ is strongly embedded in $L_M$. If there exists a function $\varphi\in C_{\psi,1}^0$ such that, for some $C>0$ and all $s,t\in [0,1]$, then the unit ball of the subspace $[f_k]$ consists of functions with equicontinuous norms in $L_M$. In particular, this holds if at least one of the following conditions is fulfilled:(a) $\psi$ is submultiplicative, that is, there exists $C>0$ such that, for all $s,t\in [0,1]$, (b) $\psi$ is equivalent to some $\alpha_\psi^0$-convex function for small values of the argument; (c) $t^{-1/p}\in L_M$ for some $p\in (0,\alpha_\psi^0)$. Proof. It is obvious that inequality (22) holds if and only if
$$
\begin{equation}
\psi^{-1}\biggl(\frac{t}{s}\biggr){\varphi^{-1}(s)}\leqslant C_1 {\psi^{-1}(t)}
\end{equation}
\tag{23}
$$
for some $C_1>0$ and all $0<t\leqslant s\leqslant 1$. Hence,
$$
\begin{equation*}
\mathcal{M}_{\psi^{-1}}\biggl(\frac1{s}\biggr)= \sup_{0\leqslant t\leqslant s} \frac{\psi^{-1}(t/s)}{\psi^{-1}(t)}\leqslant C_1\cdot \frac{1}{\varphi^{-1}(s)},\qquad 0<s\leqslant 1.
\end{equation*}
\notag
$$
Since the subspace $[f_k]$ is strongly embedded in $L_M$, $[f_k]\approx\ell_\psi$ and $\varphi\in C_{\psi,1}^0$, by Lemma 5, the function $1/\varphi^{-1}$ belongs to the space $L_M$. Therefore, from the latter inequality it follows that $\mathcal{M}_{\psi^{-1}}(1/s)\in L_M$.
On the other hand, by Lemma 8, the distribution functions $n_f(\tau)$ and $n_{1/\psi^{-1}}(\tau)$ are equivalent for large $\tau>0$. Therefore, as above, the functions $f^*(t)$ and $1/\psi^{-1}(t)$ are equivalent for small $t>0$, and, thanks to the equality $\psi^{-1}(1)=1$, we find that, for some $C>0$ and all $0<s\leqslant 1$,
$$
\begin{equation}
\mathcal{M}_{f^*}(s)\leqslant C\mathcal{M}_{1/\psi^{-1}}(s)= C\mathcal{M}_{\psi^{-1}}\biggl(\frac1{s}\biggr).
\end{equation}
\tag{24}
$$
Thus, $\mathcal{M}_{f^*}\in L_M$, and so, to complete the proof of the first assertion of the theorem, it remains to apply Theorem 1.
Let us show that the remaining assertions of the theorem are consequences of the first one. Indeed, assertion $(a)$ follows because $\psi\in C_{\psi,1}^0$. Next, by Lemma 3, the function $\psi$ is equivalent to some $p$-convex function for small values of the argument if and only if, for some $C_2>0$ and all $0<t,s\leqslant 1$, Therefore, if (b) is satisfied, then the desired statement is an immediate consequence of the fact that the function $t^{\alpha_\psi^0}$ belongs to the set $C_{\psi,1}^0$ (see § 2.2).Finally, by the definition of the index $\alpha_\psi^0$, for each $p\in (0,\alpha_\psi^0)$, the function $\psi$ is equivalent to some $p$-convex function for small values of the argument; that is, inequality (25) holds for such $p$. Now the required result follows from condition $(c)$. This proves Theorem 2. Remark 3. In general, Theorem 2 cannot be extended to the whole class of subspaces of an Orlicz space $L_M$ that are isomorphic to some Orlicz sequence spaces. As shown in [8] (see Theorem 2 and its proof), if the function $t^{-1/\beta_M^\infty}\in L_M$, then $L_M$ contains a strongly embedded subspace $H$ of such a type whose unit ball consists of functions with nonequicontinuous norms in $L_M$. 4.3. Subspaces of Orlicz spaces generated by mean zero identically distributed independent functions and Matuszewska–Orlicz indicesIn the case $t^{-1/\beta_M^\infty}\notin L_M$ (this is so, for example, for $L^p$), all subspaces under consideration, which are strongly embedded in the Orlicz space $L_M$, can be characterized by using the Matuszewska–Orlicz indices of the corresponding functions. Moreover, the same condition is equivalent to the fact that the unit ball of such a subspace consists of functions with equicontinuous $L_M$-norms. Theorem 3. Let $M$ be an Orlicz function such that $1<\alpha_M^\infty\leqslant \beta_M^\infty<2$ and $t^{-1/\beta_M^\infty}\notin L_M$. If $f\in L_M$ and $\{f_k\}$ is a sequence of mean zero independent functions equimeasurable with $f$, then the following conditions are equivalent: (a) the unit ball of the subspace $[f_k]$ consists of functions with equicontinuous norms in $L_M$; (b) the subspace $[f_k]$ is strongly embedded in $L_M$; (c) $\alpha_\psi^0>\beta_M^\infty$, where the Orlicz function $\psi$ is such that $[f_k]_{L_M}\approx\ell_\psi$. Proof. As above, we can assume that $f=f^*$.
The implication (a) $\Rightarrow$ (b) follows from Lemma 2. The implication (b) $\Rightarrow$ (c) is obvious if $\alpha_\psi^0\geqslant 2$. In the case $\alpha_\psi^0<2$, this implication follows from Lemma 5 (see also its proof). So, it remains only to show that (c) implies (a). So, let $\alpha_\psi^0>\beta_M^\infty$. Also, assume that $p\in (\beta_M^\infty,\alpha_\psi^0)$. Then, by the definition of the index $\beta_M^\infty$, To prove (a) it suffices to show that the norms of the spaces $L_M$ and $L^p$ are equivalent on $H$, or what is the same, to check that, $f\in L^p$ and $[f_k]_{L^p}\approx\ell_\psi$. Indeed, in this case, the unit ball $B_H$ of the subspace $H:=[f_k]_{L_M}$ is bounded in $L^p$, and, therefore, by (26), according to the Vallée Poussin criterion (see, for example, Theorem 3.2 in [34]), the set $B_H$ consists of functions having equicontinuous norms in $L_M$, which verifies (a). First of all, due to the inequality $\alpha_\psi^0>p$ and Lemma 3, the function $\psi$ is equivalent to some $(p+\varepsilon)$-convex function for small values of the argument whenever $\varepsilon>0$ is sufficiently small. Therefore, ${1}/{\psi^{-1}}\in L^p$ and, applying Proposition 3.1 in [16], we find that
$$
\begin{equation*}
\biggl\|\sigma_{1/t}\biggl(\frac1{\psi^{-1}}\biggr)\biggr\|_{L^p}= \biggl(\frac1t\int_0^t \frac{ds}{(\psi^{-1}(s))^p}\biggr)^{1/p}\preceq \frac{1}{\psi^{-1}(t)},\qquad 0<t\leqslant 1.
\end{equation*}
\notag
$$
Next, since $f(t)$ does not increase, $L_M\subset L^1$ and $[f_k]_{L_M}\approx\ell_\psi$, from (10) it follows
$$
\begin{equation*}
f(t)\leqslant \frac1t\int_0^t f(s)\,ds=\|\sigma_{1/t}f\|_{L^1}\leqslant \|\sigma_{1/t}f\|_{L_M}\preceq \frac{1}{\psi^{-1}(t)},\qquad 0<t\leqslant 1.
\end{equation*}
\notag
$$
Therefore, in particular, $f\in L^p$. In addition, from the last relations and (10) we get
$$
\begin{equation*}
\begin{aligned} \, \|\sigma_nf\|_{L^p}+\biggl(n\int_{1/n}^1f(s)^2\,ds\biggr)^{1/2}&\preceq \biggl\|\sigma_n\biggl(\frac1{\psi^{-1}}\biggr)\biggr\|_{L^p} + \biggl(n\int_{1/n}^1f(s)^2\,ds\biggr)^{1/2} \\ &\preceq \frac{1}{\psi^{-1}(1/n)},\qquad n\in\mathbb{N}. \end{aligned}
\end{equation*}
\notag
$$
In view of the embedding $L^p\subset L_M$ and relation (10), we obtain also the opposite inequality, that is,
$$
\begin{equation*}
\frac{1}{\psi^{-1}(1/n)}\asymp \|\sigma_nf\|_{L^p} + \biggl(n\int_{1/n}^1f(s)^2\,ds\biggr)^{1/2}, \qquad n\in\mathbb{N}.
\end{equation*}
\notag
$$
Thus, $[f_k]_{L^p}\approx \ell_\psi$, which proves Theorem 3. Remark 4. The condition $t^{-1/\beta_M^\infty}\notin L_M$ is used only in the proof of the implication (b) $\Rightarrow$ (c) (when applying Lemma 5). Hence, the implication (c) $\Rightarrow$ (a) holds for any Orlicz space $L_M$ such that $1<\alpha_M^\infty\leqslant \beta_M^\infty<2$. Remark 5. Let us assume that an Orlicz function $M$ satisfies the conditions of Theorem 3. According to Theorem 3 in [8], conditions (a) and (b) are equivalent for all subspaces of $L_M$, which are isomorphic to Orlicz sequence spaces. In particular, for $L^p$-spaces from the last theorem and its proof we get the following supplement to Rosenthal’s theorem (see § 1). Corollary 2. Let $1 < p < 2$, $f \in L^p$, and let $\{f_k\}$ be a sequence of mean zero independent functions equimeasurable with $f$ such that $[f_k]_{L^p}\approx\ell_\psi$. Then the following conditions are equivalent: (a) $[f_k]_{L^p}$ is a $\Lambda(p)$-space; (b) $[f_k]_{L^p}$ is a $\Lambda(q)$-space for some $q>p$; (c) $\alpha_\psi^0>p$. 4.4. Subspaces of $L^2$ spanned by independent copies of a mean zero function $f\in L^2$So far, we have considered subspaces of Orlicz spaces $L_M$ lying “strictly to the left” of the space $L^2$, or, more precisely, such that $1<\alpha_M^\infty\leqslant \beta_M^\infty< 2$. The following result shows that in the case where $M(t)=t^2$ (that is, in $L^2$), the situation is much simpler: the unit ball of any subspace of $L^2$ spanned by mean zero identically distributed independent functions consists of functions with equicontinuous $L^2$-norms. Theorem 4. Let $\{f_k\}_{k=1}^\infty$ be a sequence of mean zero independent functions equimeasurable with some function $f\in L^2$. Then the unit ball $B_H$ of the subspace $H:=[f_k]_{L^2}$ consists of functions having equicontinuous norms in $L^2$. Proof. As usual, we assume that $f^*=f$.
By Lemma 7, we find a function $N$ equivalent to some $2$-convex Orlicz function such that $\widetilde{N}\in\Delta_2^\infty$, $\lim_{u\to\infty}{N(u)}u^{-2}=\infty$, and $N(|f|)\in L^1$. Without loss of generality we can assume that $N$ is itself a $2$-convex Orlicz function on $[0,\infty)$, and, therefore, the Orlicz space $L_N$ is $2$-convex (see § 2.2). In addition, from the above relations it follows that $L_N\stackrel{\ne}{\subset} L^2$ and $f\in L_N$. Let $[f_k]_{L_N}\approx \ell_\psi$ and let $\phi_{\ell_\psi}$ be the fundamental function of the space $\ell_\psi$. By (10) and the definition of the operator $\sigma_n$, we have, for any $n\in\mathbb{N}$,
$$
\begin{equation*}
\begin{aligned} \, \phi_{\ell_\psi}(n)&\asymp \|\sigma_nf\|_{L_N} {+}\, \biggl\|f\biggl(\frac{\cdot}{n}\biggr)\biggr\|_{L^2[1,\infty)} {=}\,\|\sigma_n(f\chi_{[0,1/n]})\|_{L_N} {+}\,\biggl\|f\chi_{[1/n,1]} \biggl(\frac{\cdot}{n}\biggr)\biggr\|_{L^2[1,\infty)} \\ &\leqslant C'n^{1/2}(\|f\chi_{[0,1/n]}\|_{L_N}+\|f\chi_{[1/n,1]}\|_{L^2})\leqslant Cn^{1/2}\|f\|_{L_N}. \end{aligned}
\end{equation*}
\notag
$$
On the other hand, $\{f_k/\|f\|_{L^2}\}_{k=1}^\infty$ is an orthonormal sequence in $L^2$, and hence $[f_k]_{L^2}\approx \ell^2$. Since $\ell_\psi\subset \ell^2$ and $\phi_{\ell^2}(n)=n^{1/2}$, $n=1,2,\dots$, it follows by the above that $\phi_{\ell_\psi}(n)\asymp n^{1/2}$, that is, $[f_k]_{L_N}\approx \ell^2$. Thus, the ball $B_H$ is bounded in $L_N$, and now the desired result follows by another appeal to the Vallée Poussin criterion. Remark 6. The following example shows that the result of Theorem 4 cannot be extended to all subspaces generated by mean zero independent (but, in general, not identically distributed) functions. Let $\{f_k\}_{k=1}^\infty$ be a sequence of independent zero mean functions on $[0,1]$ such that $|f_k(t)|=2^{k/2}$, $t\in E_k$, where $m(E_k)=2^{-k-1}$, and $|f_k(t)|=1$, $t\in [0,1]\setminus E_k$ $(k=1,2,\dots)$. According to Example 2 in [29], the subspace $[f_k]$ is strongly embedded in $L^2$, but there is no symmetric space $X$ such that $X \stackrel{\ne}{\subset} L^2$ and $X\supset [f_k]$. Taking into account the Vallée Poussin criterion, we conclude that the norms of functions of the unit ball of the subspace $[f_k]$ are not equicontinuous in $L^2$. 
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