Mathematical Notes, vol. 75, no. 1, 2004, pp. 107–123. Translated from Matematicheskie Zametki, vol. 75, no. 1, 2004, pp. 115–134. c Original Russian Text Copyright 2004 by S. Ya. Novikov.
A-Systems, Independent Functions, and Sets Bounded in Spaces of Measurable Functions S. Ya. Novikov Received April 1, 2002; in final form, May 28, 2003
Abstract—Let U ⊂ L◦ [0, 1], M, m be a set of Lebesgue measurable functions. Suppose also that two seminormed spaces of real number sequences are given: A and B . We study (A, B)-sets U defined by the classes A and B as follows: ∀a = (an ) ∈ A, ∃E = E(a) ⊂ [0, 1],
∀(fn (t)) ∈ uN mE = 1
(or for sequences similar to (fn (t))
such that
{an fn (t)1E (t)} ∈ B,
t ∈ [0, 1].
We consider three versions of the definition of (A, B)-sets, one of which is based on functions independent in the probability sense. The case B = l∞ is studied in detail. It is shown that (A, l∞ )-independent sets are sets bounded or order bounded in some well-known function spaces ( Lp , Lp,q , etc.) constructed with respect to the Lebesgue measure. A characterization of such sets in terms of seminormed spaces of number sequences is given. The (l1 , c◦ )- and (A, l1 )-sets were studied by E. M. Nikishin. Key words: A-system, Lebesgue measurable function, function space, bounded set, orderbounded set, seminormed space of number sequences.
1. INTRODUCTION First, we describe the structure of the problems studied in this paper. Suppose that U ⊂ L◦ [0, 1], M, m is a set of Lebesgue measurable functions and two classes of real number sequences are given: A ⊂ R∞ and B ⊂ R∞ . We are interested in the (A, B)-sets U determined by the classes A and B as follows: for any sequence a = {an } ∈ A and any sequence of functions {fn } in the set U , there exists a measurable subset E = E(a, {fn }) ⊂ [0, 1] of full measure (mE = 1) such that the number sequence {an fn (t)} belongs to B for any t ∈ E . For specific spaces A and B , some criteria for (A, B)-sets were obtained by E. M. Nikishin at the beginning of the seventies. So, for A = l1 , B = c◦ , and a countable set U , we have an A-system; for B = l1 , we have absolutely converging systems in A (e.g., see [1]). We show that the space c◦ in the definition of an A-system can be replaced by the space l∞ (Corollary 2.6) and precisely this case, i.e., the case B = l∞ , is the key point in this paper. This case is interesting because of the following. It has turned out that (A, l∞ )-sets are sets bounded in some well-known function spaces. Thus, a characterization of such sets in terms of seminormed spaces of number sequences can be obtained. Similar results were also obtained previously. Thus, Nikishin [1] (also see [2]) obtained a criterion for A-systems, which can be stated as follows. 0001-4346/2004/7512-0107
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Nikishin’s criterion for A-systems. Let {fn } ∈ (L◦ (m))N be a sequence of measurable functions on [0, 1] . The following assertions are equivalent: (1) {fn } is an A-system or an (l1 , c◦ )-set; (2) for any ε > 0 , there exists a measurable subset Eε ⊂ [0, 1] such that mEε ≥ 1 − ε and the sequence {fn 1Eε } is bounded in the space L1,∞ (m) . Later, G. Pisier [3] (also see the surveys [4, 5]), developing Nikishin’s results, proved the equivalence of the following assertions for the sequence {fn } ∈ (L◦ (m))N and for p > 0: (1) for any sequence a = {an } ∈ lp , we have supn |an fn (t)| < ∞ almost everywhere on [0, 1] and supn |an fn (t)| ∈ Lr (m) for some r ∈ (0, p) ; (2) there is a function f ∈ L1 (m) , f ≥ 0 , [0,1] f dm ≤ 1 , such that {f = 0} ⊂ n {fn = 0} and the sequence {fn · f −1/r } is bounded in the space Lp,∞ (ν) , where ν(E) = E f dm , E ∈ M. Nikishin’s criterion was proved in [1] as a corollary of general operator-factorization theorems. Later, in [6], this criterion led to the proof of another very general factorization theorem. It should be noted that the (A, B)-sets in Nikishin’s criterion turn out to be bounded in Lp,∞ (m) after multiplication by some function, while these sets in Pisier’s theorem are bounded in the spaces Lp,∞ (ν) obtained by a replacement of the measure m . In this paper, we modify the definition of A-systems so as to obtain a precise description of sets that are bounded in function spaces constructed with respect to the Lebesgue measure m . To this end, we introduce the notions of (A, B)-independent and (A, B)-free sets. We obtain the boundedness conditions not only in the spaces Lp,∞ (m) . For example, we describe symmetrically order-bounded sets in the Lebesgue spaces Lp , p > 0 . To do this, for the coefficient classes A we consider not only the classical spaces lp , but also some other seminormed spaces of sequences. At the same time, we establish the property that (A, B)-sets are stable with respect to the class A (Theorem 4.2 and its corollaries) and, in a natural way, arrive at functions that are independent (in the probability sense). These functions were not mentioned in the previous papers. These results allow us to obtain probability characterizations for operators of weak type, which will be studied in a subsequent paper. In the first part, we prove several general auxiliary assertions concerning (A, B)-sets. In the subsequent sections, we study the case B = l∞ in detail. Our notation is consistent with the notation used in the book [2]. By the symbol 1E we denote the characteristic function of the set E , and the symbol := means equality by definition. A colon in lines of logical symbols stands for the words “such that.” 2. MODIFICATIONS OF (A, B)-SETS AND RELATED OPERATORS In what follows, the sets A and B are quasi-Banach symmetric spaces of real number sequences. This means that each of them is equipped with a seminorm · and the following axioms are satisfied: (1) if a = {an } ∈ A and |b| = {|bn |} ≤ |a| = {|an |} , then b ∈ A and b ≤ a ; (2) the seminorm of each of the spaces is invariant under rearrangements of coordinates of its elements. The seminorm · is a functional that satisfies all axioms of norms except the triangle inequality. The latter is replaced by the weaker inequality a + b ≤ K(a + b),
a, b ∈ A,
(2.1)
with some K ≥ 1 . It is well known that any seminormed space is metrizable [7, Sec. 3.10], and, in what follows, we assume that such spaces are complete. Each seminormed space has an equivalent MATHEMATICAL NOTES
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seminorm which is continuous in the topology induced by it [8, 6.1.9]. Therefore, in what follows, we assume that the seminorm is continuous. If f ∈ L◦ ([0, 1], M, m) , then df (τ ) = m(t : |f (t)| > τ ) , τ ≥ 0 . We shall say that sequences of measurable functions {fn } ∈ L◦ (m) and {gn } ∈ L◦ (m) are similar if, for any n ∈ N and for any τ ≥ 0 , the relation dfn (τ ) = dgn (τ ) holds, i.e., if the functions fn and gn are equimeasurable. It is well known that, for any sequence {fn } ∈ L◦ (m) , there exists a sequence {Fn } ∈ L◦ (m) which is similar to it and consists of independent (in the probability sense) functions (e.g., see [2, Chap. 2] or [9, Sec. 26]). Definition 2.1. (a) A set U ⊂ L◦ (m) is called an (A, B)-set if, for any sequence a ∈ A and for any sequence {fn } ∈ U N , the following inclusion holds: {an fn (t)} ∈ B
t -almost everywhere.
(b) A set U ⊂ L◦ (m) is called an independent (A, B)-set if for any sequence a ∈ A , for any sequence {fn } ∈ U N , and for any sequence {Xn } ∈ (L◦ (m))N of independent functions, which is similar to {fn } , the following inclusion holds: {an Xn (t)} ∈ B
t -almost everywhere.
(c) A set U ⊂ L◦ (m) is called a free (A, B)-set if for any sequence a ∈ A , for any sequence {fn } ∈ U N , and for any function sequence {Fn } ∈ (L◦ (m))N similar to {fn } , the following inclusion holds: t -almost everywhere. {an Fn (t)} ∈ B If by F(A, B) , I(A, B) , and S(A, B) we denote the classes of free, independent, and (A, B)sets, respectively, then the definitions imply the inclusions F(A, B) ⊂ I(A, B),
F(A, B) ⊂ S(A, B).
In this paper, we show that the first inclusion can be replaced by the equality for B = c◦ and for B = l∞ and the second inclusion is strict. Lemma 2.2. Let X be a quasi-Banach space. There exists a sequence of positive numbers {εn } xn converges in X . such that for any sequence {xn } ∈ X N with xn ≤ εn , n ∈ N , the series Proof. We shall prove the inequality l l−1 i x K xi + K l−1 xl , l ≤ i=1
l = 2, 3, . . . ,
i=1
by induction. We set εi = (2K)−i , i = 1, 2, . . . . If xi ≤ εi , then l−1 k+l l−1 i l−1 xi ≤ K xk+i + K xk+l ≤ K i εk+i + K l−1 εk+l i=1
i=k+1
i=1
l−1
1 1 1 + k+l k+1 ≤ . i+k K k k 2 2 K (2K) i=1 Thus, the Cauchy criterion for the convergence of the series xi is satisfied. =
Let U be an (A, B)-set, and let {fn } ∈ U N . We consider the operator T{fn } : a ∈ A → {an fn (t)}B . This operator is positive and quasiconvex. The latter means that (1) T (a1 + a2 ) ≤ K(T (a1 ) + T (a2 )) almost everywhere on [0, 1] ; (2) T (λa) = |λ|T (a) , λ ∈ R , almost everywhere on [0, 1] . MATHEMATICAL NOTES
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Lemma 2.3. For a quasiconvex operator T : A → L◦ (m) , the following assertions are equivalent: (1) the operator T is bounded : it takes any set bounded in A to a set bounded with respect to the measure (in the space L◦ (m) ); (2) the operator T is continuous: if an → 0 in A , then T an → 0 with respect to the measure. Proof. Suppose that the operator T is bounded and an → 0 in A . The space A is metrizable. Therefore, there exists a sequence of positive numbers {γn } such that γn → ∞ : γn an → 0 [10, Theorem 1.28]. Since the set {γn an } is bounded in A , the set (T {γn an }) is bounded in L◦ (m) . Hence we have T xn = γn−1 T {γn an } → 0 [10, Theorem 1.30]. Now we assume that the operator T is continuous, E is bounded in A , and W is a neighborhood of zero in L◦ (m) . The space A is metrizable. Hence there exists a neighborhood V of zero in A such that T (V ) ⊂ W [10, Theorem 1.32]. Since the set E is bounded in A , we have E ⊂ tV for sufficiently large t > 0 . We have T (E) ⊂ T (tV ) = tT (V ) ⊂ tW , i.e., T (E) is bounded in L◦ (m) .
Theorem 2.4. Let U be an (A, B)-set. If the seminorm of the space B is order semicontinuous (xn ↑ x ∈ B ⇒ xn ↑ x), then the operator family T{fn } , {fn } ∈ U N , is uniformly bounded, i.e., the set T = (T{fn } (a))(t) : aA ≤ 1, {fn } ∈ U N is bounded in the space L◦ (m) . Proof. First, we prove that the function t → {an fn (t)}B is measurable. Indeed, for each k ∈ N , the function t → {a1 f1 (t), . . . , ak fk (t), 0, 0, . . . }B is measurable as a composition of finitely many functions that are measurable and continuous. Since the seminorm is order semicontinuous, we have {an fn (t)}B = sup {a1 f1 (t), . . . , ak fk (t), 0, 0, . . . }B , k
and the supremum of a sequence of measurable functions is also measurable. Now let us prove the uniform boundedness of the operator family T{fn } . We assume that the set T is unbounded in L◦ (m) . Then for the sequence of numbers εk in Lemma 2.2, we have k = 1, 2, . . . : ∃{a(k) } ∈ AN , a(k) ≥ 0, a(k) A ≤ 1, ∃{fn(k) }∞ n=1 ,
k (k) = m t : εk a(k) m(Ek ) := m t : (T{f k } (a(k) ))(t) > n fn (t) n B > k ≥ δ. εk
∃δ > 0,
(m)
Let n → (m, k) be a bijection of N on N × N . We consider the sequences an := εm ak and (m) fn := fk , n = 1, 2, . . . . We have {an } ∈ A (this follows from Lemma 2.2) and {fn } ∈ U N . ∞ ∞ Now we note that if t ∈ M := limEk = l=1 k=l Ek , then ∀l ∈ N ∃k ≥ l :
(k) (εk a(k) n fn (t))n B > k,
(k) (k) i.e., for infinitely many k ∈ N , the sequence {an fn (t)} contains a subsequence εk an fn (t) n (perhaps, rearranged) whose seminorm in the space B is larger than k . Since the seminorm is monotone, we have t ∈ M , m(M ) ≥ δ. {an fn (t)}B = +∞, This contradict the definition of the (A, B)-set.
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Corollary 2.5. Suppose that the seminorm in the space B is order semicontinuous. If U is a free (A, B)-set (or, respectively, an independent (A, B)-set), then there exists a function χ : R+ → [0, 1],
lim χ(τ ) = 0,
τ →∞
such that ∀a ∈ A,
∀{fn } ∈ U N
aA ≤ 1,
m t : {an Fn (t)}B > τ ≤ χ(τ ),
τ ≥ 0,
where {Fn } is a sequence of measurable functions similar to {fn } (respectively, a sequence of independent functions similar to {fn } ). Proof. The proof of this corollary repeats that of the theorem, except that the definitions of free and independent sets are used. Corollary 2.6. Suppose that the seminorm in the space B is order semicontinuous and Q is the set of finite sequences. If Q is dense in A , then any (A, B)-set is an (A, BQ )-set, where BQ is the closure of Q in B . In particular, the (l1 , l∞ )-set is an A-system (see the introduction). Proof. For a sequence a ∈ A we define the “cut-off functions” a[k] = (0, . . . , 0, ak+1 , . . . ) and note that a[k] A → 0 as k → ∞ by assumption. The operator T : a ∈ A → {an fn (t)}B is continuous (Lemma 2.3 and Theorem 2.4); hence T (a[k] ) → 0 with respect to the measure, and T (a[k] ) ≥ T (a[k+1] ) . We obtain T (a[k] ) = {0, . . . , 0, ak+1 fk+1 (t), ak+2 fk+2 (t), . . . }B → 0 for almost all t ∈ [0, 1] . This means that {an fn (t)} ∈ BQ for almost all t ∈ [0, 1] . 3. COINCIDENCE OF INDEPENDENT AND FREE (A, l∞ )-SETS For B = l∞ , we have the curious phenomenon that independent and free (A, l∞ )-sets coincide, which is described by the following theorem. Theorem 3.1. For a set U ⊂ L◦ (m) , the following assertions are equivalent: (1) U is an independent (A, l∞ )-set; (2) there exists a positive number τ◦ such that for all a ∈ A with aA ≤ 1 and for all ∞ {fn } ∈ U N , the series n=1 dan fn (τ◦ ) converges; (3) U is a free (A, l∞ )-set. Thus, the relation F(A, l∞ ) = I(A, l∞ ) holds for an arbitrary quasi-Banach space of sequences A . Proof. (1) ⇒ (2) . It follows from Corollary 2.5 with B = l∞ that there exists a positive number τ◦ such that m t : sup |an Fn (t)| > τ◦ < 1 n
for any sequence a with aA ≤ 1 , for any sequence {fn } ∈ U N , and for any sequence {Fn } of independent functions, which is similar to {fn } . We consider the relation m t : sup |an Fn (t)| > τ◦ = 1 − m t : sup |an Fn (t)| ≤ τ◦ = n
n
(by independence) =1−
1 − m(|an Fn | > τ◦ ) . m |an Fn (t)| ≤ τ◦ = 1 −
n
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Thus, we have
n
1 − m(|an Fn | > τ◦ ) > 0 . This implies the convergence of the series
m |an Fn | > τ◦ = dan fn (τ◦ ).
n
n
(2) ⇒ (3) . Suppose that a ∈ A , a ≤ 1 , {fn } ∈ U N , and {Fn } is an arbitrary sequence similar to {fn } . We consider the relation
t : |an Fn (t)| > τ m t : sup |an Fn (t)| > τ = m n
≤
n
m t : |an Fn (t)| > τ = dan fn (τ ) < ∞
n
n
for sufficiently large τ . Now we note that dan fn (τ ) = t : |an fn (t)| > τ n
n
is the distribution function for some function defined on the positive semiaxis. (We can obtain such a function (we denote it by an fn ) as follows: we transfer each of the functions an fn on the half-interval [n − 1, n) and “glue together” the parts thus obtained into a single function.) Thus, the function an fn belongs to L◦ ([0, ∞)) and d an fn (τ ) < ∞ . Hence d an fn (τ ) → 0 as τ → ∞ . Thus, we have m t : supn |an Fn (t)| = ∞ = 0 . (3) ⇒ (1) . This implication holds by definition. It should be noted that the procedure of transferring functions from an interval to the semiaxis, which is described in the proof, has been effectively used by many authors [11, 12]. 4. INDEPENDENT AND FREE (lp,q , l∞ )-SETS, 0 < q ≤ p < ∞ First, we describe several transformations of positive functions, which we then use to state the results. Let ϕ : R+ → R+ be an increasing concave function, lim ϕ(t) = 0,
t→+0
We define the function ϕ(t) ˆ := inf
s>0
lim ϕ(t) = ∞.
t→∞
1 + ts , ϕ(s)
t ∈ R+ .
The function ϕˆ : R+ → R+ is concave as the greatest lower bound of a set of linear functions [13, Chap. 2], 1 + t · 1t 1 2 1 1 = inf s ≤ ϕ(t) 1 = 1 , ˆ ≤ t > 0; s>0 ϕ t ϕ 1+ts ϕ t ϕ t in particular, lim ϕ(t) ˆ = 0,
t→+0
lim ϕ(t) ˆ = ∞.
t→∞
Moreover, ϕˆ is strictly increasing and hence has an inverse ϕˇ : R+ → R+ , which is a convex strictly increasing function such that ˇ = 0, lim ϕ(t)
t→+0
lim ϕ(t) ˇ = ∞.
t→∞
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Besides, we have the inequalities 1 ≤ ϕ(s) ˇ ≤
ϕ−1 2s
1 ϕ−1 ( 1s )
,
s > 0.
Using the function ϕ , we introduce two classes of seminormed spaces, which are usually called Lorentz and Marcinkiewicz spaces [13]. We define them on an arbitrary space (J , ν) with a σ-finite measure ν so that the elements of these spaces can be both functions from L◦ (m) and number sequences. So we assume that (J , ν) is a space with a complete σ-finite positive measure ν , L◦ (J , ν) is the space of all almost everywhere measurable finite functions, ν(J ) ∗ x (t)dϕ(t) < ∞ , Λ(J , ν ; ϕ) = x ∈ L◦ (J , ν) : x = 0 ∗ M(J , ν ; ϕ) = x ∈ L◦ (J , ν) : x = sup x (t)ϕ(t) < ∞ . 0
Here x∗ (t) denotes a nonincreasing rearrangement of the function |x(t)| , i.e., x∗ (t) = inf(τ : dx (τ ) < t) = inf τ : ν(j : |x(j)| > τ ) < t . The functional determining the space Λ(ϕ) for a concave function ϕ is a norm, and the functional determining the space M(ϕ) is a seminorm [13, Chap. 2]. We also recall the definition of the spaces Lp,q , 0 < p < ∞ , 0 < q ≤ ∞ , mentioned in the title of this section: ν(J ) 1/q ∗ q q/p (x (t)) d(t ) <∞ for q < ∞, Lp,q (J , ν) = x ∈ L◦ (J , ν) : x = 0 Lp,∞ (J , ν) = x ∈ L◦ (J , ν) : x = sup x∗ (t)t1/p < ∞ . 0
For p = q , the spaces Lp,q and Lp coincide; if q1 < q2 , then Lp,q1 ⊂ Lp,q2 . Usually, the spaces Lp,q (N) are denoted by lp,q . These spaces are well known (e.g., see [7, 14]). Lemma 4.1. Let Γ be an arbitrary nonempty set; and let Φ : R+ × Γ → R+ be an arbitrary bounded function. The following assertions are equivalent: (1) there exists a positive number δ such that, for all a ∈ Λ(N ; ϕ) with aΛ(N, ; ϕ) ≤ δ and for all {γn } ∈ Γ N , ∞ Φ(|an |, γn ) < ∞ ; n=1
(2) there exist positive numbers / and k such that the inequality Φ(s, γ) ≤ ϕ(ks) ˇ holds for all s ∈ [0, /] and for all γ ∈ Γ . Proof. (1) ⇒ (2) . Let us consider the function ρ(s) := sup sup Φ(t, γ), 0≤t≤s γ∈Γ
s > 0.
The function ρ(s) increases monotonically and lims→+0 ρ(s) = 0 . By assumption, there exists a ρ(|an |) converges for all a with aΛ(ϕ) ≤ δ◦ . We are interested δ◦ > 0 such that the series MATHEMATICAL NOTES
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in the behavior of the function ρ(s) in a neighborhood of zero. Hence we can assume that this function takes only finite values and lims→+∞ ρ(s) = +∞ . Let ρ˜(t) := sup{s ≥ 0: ρ(s) ≤ t} . The function ρ˜(t) increases. We need to prove that
1 ϕ(2n) > 0. inf ρ˜ n∈N 4n Assume the contrary: inf n∈N ρ˜(1/(4n))ϕ(2n) = 0 . We choose an arbitrary δ◦ > 0 and construct a sequence of positive integers n1 < n2 < · · · by induction as follows: (1) nj ≥ Nj−1 , where Nj−1 =
j−1
nj , j ≥ 2 , N◦ := 0 (this implies the inequality Nj ≤ 2nj );
i=1
δ◦ 1 ϕ(2nj ) ≤ j+1 , j = 1, 2, . . . . (2) ρ˜ 4nj 2 We define the number sequence
1 ρ an := 2˜ 4nj
for Nj−1 < n ≤ Nj .
The sequence a = {an } has the following properties: (1) an ≥ 0 , an ↓ ; (2) we have aΛ(ϕ)
∞
1 = an (ϕ(n) − ϕ(n − 1)) = 2 ρ˜ ϕ(Nj ) − ϕ(Nj−1 ) 4nj n j=1
∞ δ◦ 1 ϕ(2nj ) ≤ 2 ρ˜ = δ◦ . ≤2 4nj 2j+1 j=1
On the other hand,
ρ(an ) = ∞ , since Nj
n=Nj−1 +1
1 1 1 > nj ρ(an ) = nj ρ 2˜ ρ = , 4nj 4nj 4
which contradicts the assumptions. Thus, we have
1 ϕ(2n) > 0. κ := inf ρ˜ n 4n
For t ≥ 2 , we have 2n ≤ t < 2n + 2 < 4n , where n := [t/2] . We obtain the inequality
1 1 ϕ(t) ≥ ρ˜ ϕ(2n) ≥ κ , ρ˜ t 4n which implies that
κ 1 ≥ρ , t ϕ(t)
t ≥ 2.
By setting s := κ/ϕ(t) , t ≥ 2 , we obtain ρ(s) ≤
1 ≤ ϕ(ks), ˇ
ϕ−1 κs
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where k = 2/κ and 0 ≤ s ≤ κ/(ϕ(2)) . (2) ⇒ (1) . Without loss of generality, we can assume that Φ(s, γ) ≤ ϕ(ks) ˇ for all γ ∈ Γ and for all s ≥ 0 . We set δ := 1/k and choose a ∈ Λ(N, ϕ) with a ≤ δ . Let hj := a∗j − a∗j+1 , so ∞ that a∗n = j=n hj , n ∈ N . Then aΛ(N,ϕ) =
a∗n
∞ ∞ hj ϕ(n) − ϕ(n − 1) ϕ(n) − ϕ(n − 1) = n=1
=
j ∞
j=n
∞ hj ϕ(n) − ϕ(n − 1) = ϕ(j)hj ≤ δ.
j=1 n=1
j=1
Now, for a fixed n ∈ N , we have ϕ(ka ˇ ∗n )
∞ ∞ ∞ ∞ 1 1 kϕ(j)hj ≤ ≤k , = ϕˇ k hj = ϕˇ kϕ(j)hj kϕ(j)hj ϕˇ ϕ(j) ϕ(j) j j=n j=n j=n j=n
since
∞
kϕ(j)hj ≤
j=n
∞
ϕ(j)hj ≤ kδ = 1.
j=1
Thus, ∞ n=1
Φ(|an , γn ) ≤
∞
ϕ(k|a ˇ n |) =
n=1
∞
ϕ(ka ˇ ∗n )
n=1
∞ ∞ ∞ kϕ(j)hj ≤ ϕ(j)hj ≤ kδ = 1 =k j n=1 j=n j=1
for any sequence of elements {γn } ∈ Γ N . To state the following theorem, we use the Orlicz space lψ of real number sequences. Let us recall this definition. Suppose that ψ : R+ → R+ is a continuously increasing convex function such that ψ(0) = 0 , ∞ lim ∞t→∞ ψ(t) = ∞ . The Orlicz space lψ consists of all sequences a ∈ R satisfying the condition n=1 ψ(|an |/ρ) < ∞ for some ρ > 0 . The space lψ with the norm
∞ |an | ψ a = inf ρ > 0 : ≤1 ρ n=1 is a Banach space [15, Chap. 4].
Theorem 4.2. For a set U ⊂ L◦ [0, 1], M, m , the following assertions are equivalent: (1) U is an independent (Λ(N, ϕ), l∞ )-set; (2) U is bounded in the space M [0, 1], M, m ; ϕˆ ; (3) U is a free (lϕˇ , l∞ )-set. Proof. (1) ⇒ (2) . By Theorem 3.1, there exists a τ◦ > 0 such that the series dan fn (τ◦ ) n
converges for all a ∈ Λ(N, ϕ) with a ≤ 1 and for all {fn } ∈ U N . We apply Lemma 4.1 to the function (s, f ) → dsf (τ◦ ) defined on R+ × U and ranging in [0, 1]: ∃ε > 0,
∃k > 0 :
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ˇ dsf (τ◦ ) ≤ ϕ(ks).
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Since ϕ(u) ˇ ↑ ∞ as u → ∞ , we can assume that k is sufficiently large for the last inequality to hold for any s > 0 . Taking the definitions of the nonincreasing rearrangement and of the function ϕˆ into account, we obtain ˆ ≤ kτ◦ , f ∈ U, sup f ∗ (t) · ϕ(t) t>0
i.e., we see that U is bounded in the space M [0, 1], M, m ; ϕˆ . ˆ (2) ⇒ (3) . Let a ∈ lϕˇ , and let {fn } ∈ U N . The boundedness of U in the space M([0, 1], ϕ) means that there exists a positive number k such that the inequality
k , t > 0, df (t) ≤ ϕˇ t holds for all f ∈ U . Now it is easy to see that the series ∞
dan fn (τ ) ≤
n=1
∞ |an |k ϕˇ τ n=1
converges for sufficiently large τ . It follows from Theorem 3.1 that U is a free (lϕˇ , l∞ )-set. (3) ⇒ (1) . Comparing the fundamental functions {1, 1, . . . , 1, 0, . . . }Λ(ϕ) = ϕ(n)
1 {1, 1, . . . , 1, 0, . . . }lϕˇ = 1 ≤ ϕ(n), ϕˆ n
and
n
n
we establish the inclusion Λ(N, ϕ) ⊂ lϕˇ [13, Chap. 2], after which the implication is obvious. Let r > 0 . We construct an r-power transformation of the quasi-Banach space X , which we denote by X r . Recall that 1/r X r = {y : |y|r ∈ X}, yX r = |y|r X . We note that the spaces Lp,q coincide with the spaces Λq (tq/p ) , and the spaces Lp,∞ coincide with the spaces Mq (tq/p ) = M(t1/p ) , 0 < p < ∞ , 0 < q < ∞ . Applying the theorem to the set {|f |r : f ∈ U } , we obtain the following assertion. Corollary 4.3. Let r > 0 . For a set U ⊂ L◦ ([0, 1], M, m) , the following assertions are equivalent: (1) U is an independent (Λr (N, ϕ), l∞ )-set; ˆ ; (2) U is bounded in the space Mr ([0, 1], M, m ; ϕ) r , l )-set. (3) U is a free (lϕ ˇ ∞ By setting ϕ(t) = tq/p and r = q in Corollary 4.3, we obtain the following assertion. Corollary 4.4. Let 0 < q ≤ p < ∞ . For a set U ⊂ L◦ ([0, 1], M, m) , the following assertions are equivalent: (1) U is an independent (lp,q , l∞ )-set; (2) U is bounded in the space Lp,∞ ([0, 1], M, m) ; (3) U is a free (lp , l∞ )-set. Thus, comparing these results with Nikishin’s criterion stated in the introduction, we note that
=
F(l1 , l∞ ) ⊂ S(l1 , l∞ ). On the other hand, we have I(lp,q , l∞ ) = F(lp,q , l∞ ) = F(lp , l∞ ),
0 < q ≤ p < ∞.
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5. INDEPENDENT AND FREE (lp,q , l∞ )-SETS, 0 < p ≤ q ≤ ∞ Suppose that ϕ : R+ → R+ is a strictly increasing continuous function and ϕ(0) = 0 . By Lϕ ([0, 1], m) we denote the set of all functions x(t) that are measurable on the interval [0, 1] and satisfy the condition 1
0
ϕ(γ|x(t)|) dt < ∞
for some γ > 0 . If the function ϕ satisfies the ∆2 -condition, then the set Lϕ ([0, 1], m) coincides with the class ϕ(L) , which was studied in detail in [16]. If ϕ is an N -function [16, 17], then Lϕ ([0, 1], m) is a Banach space and is called the Orlicz space [17]. We start from the simple “extreme” case. Theorem 5.1. Let ϕ : R+ → R+ be a strictly increasing continuous function, and let ϕ(0) = 0 . For s set U ⊂ L◦ ([0, 1], M, m) , the following assertions are equivalent: (1) U is an independent (M(N, ϕ), l∞ )-set; (2) there exists a τ◦ > 0 such that ∞ d(U , τ◦ ϕ(t)) dt < ∞, 0
where d(U , τ ) := supf ∈U df (τ ) and τ > 0 ; (3) the set U is order bounded in the set Lϕ−1 ([0, 1], m) , i.e., there exists a nonincreasing function h ∈ Lϕ−1 ([0, 1], m) such that the inequality f ∗ (t) ≤ h(t),
0 < t ≤ 1;
holds for all f ∈ U ; (4) U is a free (M(N, ϕ), l∞ )-set. Proof. We immediately note that (1) and (4) are equivalent, which follows from Theorem 3.1. Then we have the following set of equivalent assertions: • assertion (1) ⇐⇒ (by Theorem 3.1) there exists a number τ◦ > 0 such that the series dan fn (τ◦ ) converges for any sequence a with aM(N,ϕ) ≤ 1 and for any sequence {fn } ∈ U N ; dfn (τ◦ ϕ(n)) converges for any sequence • there exists a number τ◦ > 0 such that the series N {fn } ∈ U ; ∞ • there exists a number τ◦ > 0 such that the series n=1 d(U , τ◦ ϕ(n)) converges; ∞ • there exists a number τ◦ > 0 such that the integral 0 d(U , τ◦ ϕ(t)) converges; 1 • there exists a number τ◦ > 0 such that the integral 0 ϕ−1 (h(t)/τ◦ ) dt , where h(t) := supf ∈U f ∗ (t) and 0 < t < 1 , converges; • there is a function h ∈ Lϕ−1 ([0, 1], m) such that the inequality f ∗ (t) ≤ h(t) , 0 < t ≤ 1 , holds for all f ∈ U . Just as in the preceding section, we shall prove two corollaries. Corollary 5.2. Suppose that ϕ : R+ → R+ is a strictly increasing continuous function, ϕ(0) = 0 , and r > 0 . For a set U ⊂ L◦ ([0, 1], M, m) , the following assertions are equivalent: (1) U is an independent (Mr (N, ϕ), l∞ )-set; (2) U is an order-bounded set in r-power transformations of the Orlicz space Lrϕ−1 ([0, 1], m) , i.e., there exists a nonincreasing function h ∈ Lrϕ−1 ([0, 1], m) such that the inequality f ∗ (t) ≤ h(t) , 0 < t ≤ 1 , holds for all f ∈ U ; (3) U is a free (Mr (N, ϕ), l∞ )-set. MATHEMATICAL NOTES
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Corollary 5.3. Let p > 0 . For a set U ⊂ L◦ ([0, 1], M, m) , the following assertions are equivalent: (1) U is an independent (lp,∞ , l∞ )-set; (2) U is a symmetrically order-bounded set in the space Lp ([0, 1], m) , i.e., there exists a nonincreasing function h ∈ Lp ([0, 1], m) such that the inequality f ∗ (t) ≤ h(t) , 0 < t ≤ 1 , holds for all f ∈ U ; (3) U is a free (lp,∞ , l∞ )-set. To state the final result (Theorem 5.9), we note that condition (2) in the last corollary can be reformulates as follows: the set U is bounded in the space M([0, 1] ; 1/h(t)) , where h(t) ∈ Lp ([0, 1], m) . Lemma 5.4. If X is a quasi-Banach space, K is a constant in inequality (2.1), γ ∈ (0, 1/(2K)) n xn converges in the space X . and xn ≤ γ , n = 1, 2, . . . , then the series Proof. The proof is similar to that of Lemma 2.2. Lemma 5.5. The extension operator στ x(t) := x(t/τ ) , τ > 0 , is bounded in any quasi-Banach symmetric space of functions on [0, 1] . Proof. The proof repeats the argument in the normed case [13, Chap. 2]. We obtain the following estimate for the “norm”: στ := supx≤1 στ x ≤ Kτ . Lemma 5.6. Let X be a quasi-Banach symmetric space of functions on [0, 1] . There exists a constant R > 1 such that, for any nonincreasing function x(t) ∈ X , there exists a nonincreasing function x1 (t) ∈ X , and moreover, the inequalities x(t) ≤ x1 (t) and x1 (t/2) ≤ Rx1 (t) hold almost everywhere. Proof. We choose α ∈ (0, (2K)−1 σ2 −1 ) and set x1 (t) =
t αn x n , 2 n=0 ∞
t ∈ (0, 1].
The function x1 (t) does not increase, and x(t) ≤ x1 (t) . Since n αn σ2n x ≤ ασ2 X→X x,
ασ2 X→X ∈ (0, (2K)−1 ),
we have x1 (t) ∈ X (Lemma 5.4). Moreover, we have
∞ ∞ t 1 n 1 t t n α x n+1 ≤ α x n = x1 (t), = x1 2 2 α n=0 2 α n=0
t ∈ (0, 1].
It remains to set R := 1/α . In the preceding section, we defined the spaces Λ(J , ν ; ϕ) for the concave function ϕ . Similarly, we define the seminormed spaces with respect to a convex N -function (for polynomial functions, see [14, Chap. 5] and [18]). The notion of a convex N -function and of its complementary function were studied in detail in [17]. The Orlicz spaces constructed with respect to the N -function ϕ and with respect to its complementary N -function ϕ∗ are in duality. Following [17], we say that an N -function ϕ(t) satisfies the ∆2 -condition for t ≥ t◦ if there exists a constant K > 2 such that ϕ(2t) ≤ Kϕ(t),
t ≥ t◦ .
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Lemma 5.7. Let ϕ(t) be an N -function satisfying the ∆2 -condition for t ≥ 1 with the complementary N -function ϕ∗ (t) . For an arbitrary increasing function ρ(t) : R+ → R+ , the following assertions are equivalent: (1) the series ρ(an ) converges for all a ∈ Λ(N, ϕ) ; (2) the function κ(t) := ρ(t)/t , where t > 0 , belongs to the Orlicz space Lϕ∗ ([0, 1], m) . Proof. (1) ⇒ (2) . First, by induction, we prove that the inequality ϕ(t1 + · · · + tm ) ≤
m−i+1 m
K i=1
2
ϕ(ti )
(∗)
holds for arbitrary t1 , . . . , tm ≥ 1 . This inequality is obvious for m = 1 , since we have K ≥ 2 for the constant. If ( ∗ ) holds for m ∈ N , then, for t1 , . . . , tm , tm+1 ≥ 1 , we have
t1 + · · · + tm + tm+1 K K ≤ ϕ(t1 + · · · + tm ) + ϕ(tm+1 ) ϕ(t1 + · · · + tm + tm+1 ) ≤ Kϕ 2 2 2
m m+1 m−i+1 K m−i+2 K K K ϕ(ti ) + ϕ(tm+1 ) = ϕ(ti ). ≤ 2 i=1 2 2 2 i=1 We choose a γ ∈ (0, 2/K) , and by {an } we denote the sequence 1, . . . , 1 , γ , . . . , γ , . . . , γ j−1 , . . . , γ j−1 , . . . , n1
n2
nj
where the numbers n1 , n2 , . . . will be chosen later. Since {an }Λ(N,ϕ) =
∞
ϕ(n)(an − an+1 ) = (1 − γ)
n=1
ϕ(n1 + · · · + nj ) ≤
j
i=1
K 2
j−i+1
∞
ϕ(n1 + · · · + nj )γ j−1 ,
j=1
ϕ(ni ),
j ∈ N,
we have {an }Λ(N,ϕ) ≤ (1 − γ)
∞
γ j−1
j−i+1 j
K
ϕ(ni ) 2 j=1 i=1 j−i+1 ∞ ∞
∞ K K(1 − γ) j−1 = (1 − γ) ϕ(ni ) γ = ϕ(ni )γ i−1 . 2 2 − Kγ i=1 j=1 i=1
( ∗∗ )
We shall write the sum of the converging number series as follows: ∞
ρ(an ) =
n=1
∞
nj ρ(γ
j−1
j=1
We shall also show that
∞ ρ(γ j−1 ) j=1
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γj
∞ 1 0 j=1
nj 1(γ j ,γ j−1 ] (t) ·
1(γ j ,γ j−1 ] (t) ∈ Lϕ∗ ([0,1]).
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γj
1(γ j ,γ j−1 ] (t) dt.
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Indeed, if we assume the contrary, then there is a sequence of nonnegative numbers (sj ) such that 1 ∞ ∞ ∞ ρ(γ j−1 ) j j−1 j j−1 s 1 ≤ 1, s 1 (t) · 1(γ j ,γ j−1 ] (t) dt = ∞. j (γ ,γ j (γ ,γ ] ] γj 0 j=1
Lϕ ([0,1])
j=1
j=1
We set nj = [sj ] + 1 , j ∈ N ; then
∞ 1
0 j=1
∞ ρ(γ j−1 )
nj 1(γ j ,γ j−1 ] (t) ·
γj
j=1
1(γ j ,γ j−1 ] (t) dt = ∞
and hence ρ(an ) = ∞ . At the same time, we have (1 − γ)
∞
ϕ(nj )γ
j−1
≤ (1 − γ)
j=1
∞
ϕ(sj + 1)γ
j−1
≤ (1 − γ)
j=1
∞
[Kϕ(sj ) + ϕ(2)]γ j−1
j=1
= ϕ(2) + (1 − γ)K
∞
ϕ(sj )γ j−1 ≤ ϕ(2) + K ,
j=1
which, together with ( ∗∗ ), implies {an }Λ(N,ϕ) ≤
K(ϕ(2) + K) . 2 − Kγ
The contradiction obtained means that ∞ ρ(γ j−1 ) j=1
And since κ(t) ≤
γj
1(γ j ,γ j−1 ] (t) ∈ Lϕ∗ ([0,1]).
∞ ρ(γ j−1 ) j=1
γj
0 < t ≤ 1,
1(γ j ,γ j−1 ] (t),
we also have κ(t) ∈ Lϕ∗ ([0, 1]) . (2) ⇒ (1) . We can assume that the sequence {an } decreases and {an }Λ(N,ϕ) ≤ 1/4 . We define a sequence of positive integers (Nj ) as follows: a1 j ∈ N. N0 := 0, Nj := sup n ∈ N : an ≥ j , 2 Then we have
∞ a1 a1 ρ(an ) ≤ (Nj − Nj−1 )ρ j−1 ≤ Nj ρ j−1 2 2 n=1 j=1 j=1 a1 2a1 ∞ ∞ ρ 2j−1 a a a1 Nj 1( j−1 (t) · 1( j−1 =2 1 , 1 , a1 j−2 ] ∞
∞
0
≤2
j=1 ∞ 2a1
0
j=1
2
N j 1(
a1 2j−1
2
,
j=1
a1 2j−2
∞ a a Nj 1( j−1 ≤ 4 1 , 1 ] 2 2j−2 j=1
] (t)
·
2j−2
2
a1 2j−2
] (t) dt
ρ(t) dt t
Lϕ ([0,2a1 ])
· κLϕ∗ ([0,2a1 ]) .
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Taking into account the relation ∞ a1 a1 a1 a1 a1 a1 a1 a1 ϕ(Nj ) j−1 = 4 ϕ(Nj ) j+1 = 4 , . . . , , 2 , ... , 2 , ... , j , ... , j , ... 2 2 Λ(N,ϕ) 2 2 2 2 2 2 j=1 j=1
∞
N1
N2 −N1
Nj −Nj−1
≤ 4(an )Λ(N,ϕ) ≤ 1, we obtain
∞ a1 a1 Nj 1( j−1 , j−2 ] 2 2
Lϕ ([0,2a1 ])
j=1
and hence
∞
≤ 1,
ρ(an ) ≤ 4κLϕ∗ ([0,2a1 ]) .
n=1
Corollary 5.8. Let 0 < p < q < ∞ . For an arbitrary increasing function ρ(t) : R+ → [0, 1] , the following assertions are equivalent: (1) the series ρ(an ) converges for all a ∈ lp,q ; (2) the function 1/λ , where λ(s) := sup{t ≥ 0 : ρ(t) ≤ s} , s ∈ [0, 1] , belongs to the space Lp,r ([0, 1], m) ; the number r is determined by the equation 1/r = 1/p − 1/q . Proof. We have the following set of equivalent assertions: (1) ⇐⇒ for any sequence (bn ) ∈ lq/p,1 , 1 1/q q/(q−p) ρ(t ) 1/q dt converges; the series ρ(bn ) converges (by Lemma 5.7); • the integral t 0 ∞ 1/q q/(q−p) ρ(t ) dt converges (the change t = θ−q/r ); • the integral t 0 ∞ • the integral [ρ(θ−1/r )]r/p dθ converges; 0 1 inf θ ≥ 0 : [ρ(θ−1/r )]r/p ≤ τ dτ converges; • the integral 0 1 −r dτ converges; • the integral sup{t ≥ 0 : ρ(t) ≤ τ p/r } 0 1 • the integral [λ(τ p/r )]−r dτ converges (the change s := τ p/r ); 0 r 1 1 d(sr/p ) converges; the function λ1 belongs to Λr (sr/p ) = Lp,r . • the integral λ(s) 0 In conclusion, we state the theorem that combines the results obtained in Sections 4 and 5 for independent sets. Theorem 5.9. Let 0 < p < ∞ , and let 0 < q ≤ ∞ . For a set U ⊂ L◦ ([0, 1], M, m) , the following assertions are equivalent: (1) U is an (lp,q , l∞ )-independent set; (2) U is an (lp,q , l∞ )-free set; (3) there exists a function ψ(t) such that U is bounded in the space M([0, 1], M, m ; ψ) and, moreover, 1 1 1 1 ∈L % p,r ([0, 1], M, m), = max 0, − . ψ r p q
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Proof. The case 0 < q ≤ p < ∞ was studied in Corollary 4.4. Indeed, in this corollary it was shown that condition (1) is equivalent to the condition that a set is bounded in the spaces Lp,∞ and Lp,∞ ([0, 1]) = M(t1/p ) . Thus, condition (3) with r = ∞ was obtained. Now let 0 < p < q < ∞ . We construct the following sequence of equivalent assertions: ∞ (1) ⇐⇒ (2) there exists a number τ0 > 0 such that the series n=1 dan fn (τ0 ) converges for any sequence {an } ∈ lp,q with {an }lp,q ≤ 1 and for any sequence {fn } ∈ U N ; ∞ (3) the series n=1 dfn (1/an ) converges for any sequence {an } ∈ lp,q and for any sequence {fn } ∈ U N ; ∞ (4) for any sequence {an } ∈ lp,q , the series n=1 dU (1/an ) converges (by Corollary 5.8); (5) the function 1 ∈ Lp,r ([0, 1]) ; sup t ≥ 0 : dU 1t ≤ s (6) the function supf ∈U f ∗ ∈ Lp,r ([0, 1]) . We note that the implication (1) ⇐⇒ (6) holds for q = ∞ (Corollary 5.3). To complete the proof, we show that ((2)) ⇐⇒ (6) . (The double two-sided parentheses show that we mean the condition in the statement of the theorem.) The implication ((2)) ⇒ (6) is obvious. Let h := supU f ∗ . By Lemma 5.6, there exists a nonincreasing function h1 (t) ∈ Lp,r ([0, 1], m) such that, for some constant R > 1 , we have h(t) ≤ h1 (t) and h1 (t/2) ≤ Rh1 (t) almost everywhere on [0, 1] . We write ψ := 1/h1 . The function thus defined does not decrease and satisfies the ∆2 -condition: ψ(t) ≤ Kψ(t/2), 0 < t ≤ 1 . It follows from (6) that the set U lies in the “unit ball” in the space M([0, 1], m ; ψ) and 1/ψ ∈ Lp,r ([0, 1], m) , i.e., (6) =⇒ ((2)) . We note that in the last theorem with the assumptions 1 < p < ∞ and 0 < q ≤ ∞ , the seminormed space M([0, 1], m ; ψ) can be replaced by the traditional normed Marcinkiewicz space M(n) ([0, 1], m ; ψ) := x ∈ L◦ ([0, 1], m) : x = sup x∗∗ (t)ψ(t) < ∞ , 0
where 1 x (t) := t ∗∗
t
x∗ (s) ds,
0 < t ≤ 1,
0
constructed with respect to the quasiconcave function ψ : [0, 1] → R+ . In this case, the Hardy–Littlewood operator 1 Hx(t) := t
t
acts in the space Lp,r (see [13, Chap. 2]). We set 1 ψ := , Hh we then have
0 < t ≤ 1,
x(s) ds, 0
h = sup f ∗ ; U
sup ψ(t)f ∗∗ (t),
f ∈ U,
0
the function ψ is quasiconcave, and 1 = Hh ∈ Lp,r ψ
for
h ∈ Lp,r .
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Thus, we see that (6) implies the following statement: (7) there exists a quasiconcave function ψ : [0, 1] → R+ satisfying the conditions 1/ψ ∈ Lp,r and supt ψ(t)f ∗∗ (t) ≤ 1, f ∈ U . On the other hand, we have (7) =⇒ supU f ∗∗ ∈ Lp,r =⇒ (6) . A particular case of Theorem 5.9 for independent and identically distributed random variables was proved earlier in [19]. REFERENCES 1. E. M. Nikishin, “Resonance theorems and hyperlinear operators,” Uspekhi Mat. Nauk [Russian Math. Surveys], 25 (1970), no. 6, 129–191. 2. B. S. Kashin and A. A. Saakyan, Orthogonal Series [in Russian], AFTs, Moscow, 1999. 3. G. Pisier, “Factorization of operators through Lp,∞ and Lp,1 ,” Math. Ann., 276 (1986), 105–136. 4. B. M. Makarov, “ p-absolutely summing operators and some of their applications,” Algebra i Analiz [St. Petersburg Math. J.], 3 (1991), no. 2, 1–76. 5. S. V. Kislyakov, “Absolutely summing operators on the disk-algebra,” Algebra i Analiz [St. Petersburg Math. J.], 3 (1991), no. 4, 1–77. 6. E. M. Nikishin, “The resonance theorem and the series in eigenfunctions of the Laplace operator,” Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.], 36 (1972), no. 4, 795–813. 7. J. Bergh and J. L¨ ofstrom, Interpolation Spaces, Springer-Verlag, Heidelberg, 1976. 8. A. Pietsh, Operator Ideals, Deutscher Verlag der Wissenschaften, Berlin, 1980. 9. K. Parthasarathy, Introduction to Probability and Measure, McMillan Company of India, Deli, 1980. 10. W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973. 11. W. B. Johnson and G. Schechtman, “Sums of independent random variables in rearrangement invariant function spaces,” Ann. of Probability, 17 (1989), no. 2, 789–808. 12. N. L. Carothers and S. J. Dilworth, “Inequalities for sums of independent random variables,” Proc. of Amer. Math. Soc., 104 (1988), no. 1, 221–226. 13. S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators [in Russian], Nauka, Moscow, 1978. 14. E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, 1971. 15. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, I, Springer-Verlag, Berlin, 1977. 16. P. L. Ulyanov, “Representation of functions by series and the ϕ(L) classes,” Uspekhi Mat. Nauk [Russian Math. Surveys], 27 (1972), no. 2, 3–52. 17. M. A. Krasnoselskii and Ya. B. Rutitskii, Convex Functions and Orlicz Spaces [in Russian], Fizmatgiz, Moscow, 1958. 18. R. A. Hunt, “On L(p, q) spaces,” L’Enseignement Math., 12 (1966), no. 4, 249–276. 19. C. Ya. Novikov and A. M. Shteinberg, “Lorentz spaces and boundedness almost surely . . . ,” Sibirsk. Mat. Zh. [Siberian Math. J.], 30 (1989), no. 2, 138–144. Samara State University E-mail :
[email protected]
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