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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
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     § 1. IntroductionAccording to the classical Khintchine inequality (see, for example, Theorem V.8.4 in [1]), for each 0<p<∞, there exist constants Ap>0 and Bp>0 such that, for any sequence of real numbers (ck)∞k=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 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, 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 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:
\begin{equation*}
M(t)\geqslant\int_{t/2}^t N(u)\, \frac{du}{u}\geqslant N\biggl(\frac{t}2\biggr)\geqslant K^{-1}N(t), \qquad t>0,
\end{equation*}
\notag
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 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
\begin{equation}
\frac1{\psi^{-1}(t)} \asymp \biggl(\frac1t\int_0^tf(s)^p\,ds\biggr)^{1/p} +\biggl(\frac1t\int_t^1 f(s)^2\,ds\biggr)^{1/2},\qquad 0<t\leqslant 1,
\end{equation}
\tag{11}
\begin{equation}
\frac1{\varphi^{-1}(t)} \asymp \frac1t\int_0^tf(s)\,ds + \biggl(\frac1t\int_t^1f(s)^2\,ds\biggr)^{1/2},\qquad 0<t\leqslant 1.
\end{equation}
\tag{12}
Now, a combination of standard estimates with integration by parts leads to the following equivalences (where the constants depend only on p):
\begin{equation*}
\begin{aligned} \, \frac1t\int_0^tf(s)^p\,ds &=\frac1t\int_0^t \ln^{-3/2} \biggl(\frac{e}{s}\biggr)\,\frac{ds}{s} \asymp \frac{1}{t\ln^{1/2}(e/t)}, \qquad 0<t\leqslant 1, \\ \frac1t\int_0^tf(s)\,ds &=\frac1t\int_0^t s^{-1/p}\ln^{-3/(2p)} \biggl(\frac{e}{s}\biggr)\,ds \asymp \frac{1}{t^{1/p}\ln^{3/(2p)}(e/t)}, \qquad 0<t\leqslant 1, \\ \frac1t\int_t^1f(s)^2\,ds &=\frac1t\int_t^1 s^{-2/p}\ln^{-3/p} \biggl(\frac{e}{s}\biggr)\,ds \asymp \frac{1}{t^{2/p}\ln^{3/p}(e/t)}, \qquad 0<t\leqslant \frac{1}{2}. \end{aligned}
\end{equation*}
\notag
An appeal to (11) and (12) shows that
\begin{equation*}
\psi^{-1}(t)\asymp t^{1/p}\ln^{1/(2p)}\biggl(\frac{e}{t}\biggr)\quad\text{and} \quad \varphi^{-1}(t)\asymp t^{1/p}\ln^{3/(2p)}\biggl(\frac{e}{t}\biggr), \qquad 0<t\leqslant 1.
\end{equation*}
\notag
Hence, the functions \psi and \varphi are not equivalent, and hence \ell_\psi\stackrel{\ne}{\subset}\ell_{\varphi}. Thus, [f_k]_{L^p} is not a \Lambda(p)-subspace. 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
\begin{equation*}
\int_0^1M(f(t))\,dt \asymp \int_0^1\ln^{p\alpha-2} \biggl(\frac{e}{t}\biggr)\,\frac{dt}{t}<\infty,
\end{equation*}
\notag
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 fLet 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], 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 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,
\begin{equation*}
\lim_{\delta\to 0}\sup_{x\in B_H,\, m(\operatorname{supp}x)\leqslant\delta}\|x\|_{L_M}=0,
\end{equation*}
\notag
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
\begin{equation}
n_g(\tau) = \min\biggl(\mathcal{M}_\psi\biggl(\frac{1}{\tau}\biggr),1\biggr). \qquad \tau>0,
\end{equation}
\tag{17}
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 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^2So 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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