and J. Diestel, [1], [2], the following vector measure was considered: A â M([0 ... tion theory of Bartle, Dunford, and Schwartz, [8] (see also [24]), the space L. 1.
Operator Theory: Advances and Applications, Vol. 201, 135–148 c 2009 Birkh¨ auser Verlag Basel/Switzerland
How Summable are Rademacher Series? Guillermo P. Curbera Abstract. Khintchin inequalities show that a.e. convergent Rademacher series belong to all spaces Lp ([0, 1]), for finite p. In 1975 Rodin and Semenov considered the extension of this result to the setting of rearrangement invariant spaces. The space LN of functions having square exponential integrability plays an prominent role in this problem. Another way of gauging the summability of Rademacher series is considering the multiplicator space of the Rademacher series in a rearrangement invariant space X, that is, an rn ∈ X . Λ(R, X) := f : [0, 1] → R : f · an rn ∈ X, for all The properties of the space Λ(R, X) are determined by its relation with some classical function spaces (as LN and L∞ ([0, 1])) and by the behavior of the logarithm in the function space X. In this paper we present an overview of the topic and the results recently obtained (together with Sergey V. Astashkin, from the University of Samara, Russia, and Vladimir A. Rodin, from the State University of Voronezh, Russia.) Mathematics Subject Classification (2000). Primary 46E35, 46E30; Secondary 47G10. Keywords. Rademacher functions, rearrangement invariant spaces.
1. Introduction: a problem on vector measures The problem which originated the research that we are going to present arises from the study of vector measures and the space of scalar functions which are integrable with respect to them.
Partially supported D.G.I. #MTM2006-13000-C03-01 (Spain).
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In the note Sequences in the range of a vector measure, by R. Anantharaman and J. Diestel, [1], [2], the following vector measure was considered: rn (t) dt ∈ 2 , A ∈ M([0, 1]) −→ ν(A) := A
where M([0, 1]) is the σ-algebra of Lebesgue measurable sets of the interval [0,1] and (rn ) are the Rademacher functions (see Section 2). According to the integration theory of Bartle, Dunford, and Schwartz, [8] (see also [24]), the space L1 (ν) of functions which are integrable with respect to ν is the set of all f : [0, 1] → R such that, for every A ∈ M([0, 1]), the sequence of Rademacher–Fourier coefficients of f χA belongs to 2 , that is, f (t)rn (t) dt ∈ 2 . (1.1) A
The problem we were interested in was identifying the functions in L1 (ν). A similar problem, related to the Hausdorff-Young inequality, is to describe the space Fp (T), for 1 < p < 2, of all functions f ∈ L1 (T) such that for every A ∈ B(T) 1 −int f (t)e dt ∈ 1/p . 2π A Recently, Mockenhaupt and Ricker have shown that Lp (T) Fp (T), [22]. Thus answering a problem posed by R.E. Edwards in the 1960s. The underlying measure in this case is 1 −int A ∈ B(T) −→ ν(A) := e dt ∈ 1/p . 2π A
2. The Rademacher system We briefly recall the main properties of the system. It was defined by Hans Rademacher in 1922 in Section VI, Ein spezielles Orthogonalsystem, of [26]. The functions of the system are rn (t) := sign sin(2n πt),
t ∈ [0, 1], n ∈ N.
The system is uniformly bounded and has a strongly orthogonality property: 1 rn1 (t)p1 rn2 (t)p2 . . . rnk (t)pk dt = 0, ni = nj , 1 ≤ i < j ≤ k, 0
unless all pj are even, in which case the integral is equal to 1. It follows that the closed linear subspace generated by (rn ) in L2 ([0, 1]) is isometric to the 2 , ∞ ∞ 1/2 an rn = a2n , n=1
2
n=1
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which write as Rad (L2 ) = 2 . It also follows that the system is not complete. In fact, the set of all finite products of different Rademacher functions constitute the Walsh system, which is complete. An important property of the Rademacher system is related to the almost everywhere convergence of Rademacher series. Namely, ∞
an rn (t) converges a.e. ⇐⇒
n=1
∞
a2n < ∞.
n=1
The reverse implication was proved by Rademacher in 1922, [26], and the direct implication by Khintchin and Kolmogorov in 1925, [15]. Regarding the closed linear subspace generated by (rn ) in other Lp spaces, it is easy to see that Rad (L∞ ) = 1 , since n n n ai ri = sup ai ri (t) = |ai |, i=1
∞
t∈I(a1 ,...,an ) i=1
i=1
where I(a1 , . . . , an ) is the dyadic interval where ri = sign ai for 1 ≤ i ≤ n. For other values of p, Khintchin proved in 1923, [14], that there exists constants Ap , Bp such that ∞ ∞ ∞ 1/2 1/2 Ap · a2n ≤ an rn ≤ Bp · a2n . n=1
n=1
p
n=1
p
(This formulation in terms of L -convergence and square summability was given by Paley and Zygmund in 1930, [25].) It follows that the closed linear subspace generated by (rn ) in Lp ([0, 1]), p = ∞, is isomorphic to 2 ; we write this as √ Rad (Lp ) ≈ 2 . Regarding the constants, Bp ≤ p. The best constants for these inequalities where found by Szarek in 1976, for p = 1, and for general p by Haagerup √ in 1982. Asymptotically, we have Bp ∼ p. Concerning best constants, it is worth mentioning [18], where they are discussed for Kahane’s inequalities, i.e., the vector version of Khintchin inequalities. The power series expansion of the exponential function together with Khintchin inequalities allow to prove that 1 a r 2 n n 2 exp < ∞, for some λ > 0. an < ∞ =⇒ λ 0 It follows that Rad (LN ) ≈ 2 , where LN (= Lψ2 ) is the Orlicz space associated to 2 the function ψ2 (t) = et − 1 and consisting of all functions f such that 2 1 |f | exp < ∞, for some λ > 0. λ 0 The space LN is ‘close’ to L∞ in the sense that L∞ LN ⊂ Lp , for all 1 ≤ p < ∞. Are there any other function spaces on [0, 1] where the Rademacher functions generate a subspace isomorphic to 2 ? A precise answer was given by Rodin and Semenov in 1975, [27], in the context of rearrangement invariant (r.i.) spaces. These
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are Banach function spaces where the norm of a function f depends only on its distribution function, λ → m({t : |f (t)| > λ}). For the definition and properties of these spaces, see [10], [17], [19]. Rodin and Semenov proved that, if X is an r.i. space over [0,1], then Rad (X) ≈ 2 ⇐⇒ (LN )0 ⊂ X, where (LN )0 is the closure of L∞ in LN . The proof is based on the Central Limit Theorem and the role of the function log1/2 (2/t) in the space LN : f ∈ LN ⇐⇒ f ∗ (t) ≤ M · log1/2 (2/t), where f ∗ is the decreasing rearrangement of f (i.e., the right continuous inverse of its distribution function). The above equivalence is due to the fact that, as the 2 function ψ2 (t) = et − 1 increases very rapidly, the Orlicz space LN coincides with the Marcinkiewicz space (see Section 4.3) associated to log−1/2 (2/t), [21]. The situation when Rad (X) is complemented in X was characterized, in terms of (LN )0 , by Rodin and Semenov, [28], and independently by Lindenstrauss and Tzafriri, [19, Theorem 2.b.4].
3. A problem on function spaces The problem of identifying the space L1 (ν) of functions satisfying (1.1) can reformulated as follows. Describe the space of all functions f : [0, 1] → R such that 1 an rn (t) dt < ∞, (3.1) f (t) · 0
for every (an ) ∈ 2 . At this stage, it is reasonable to abandon the original vector measure. Thus, we change notation and label this space by Λ(R). The space Λ(R) is Banach function space for the norm an rn . (3.2)
f Λ(R) := sup f · (an )∈B2
1
Note that, due to Khintchin inequalities, the space Λ(R) satisfies Lp ⊂ Λ(R) L1 p>1
The attempts to identify the space Λ(R) with any of the classical Banach function spaces (Lp , Orlicz, Lorentz, Marcinkiewicz, Zygmund, Lorentz–Zygmund, . . . ) fail. The reason is revealed by the following result. Theorem 3.1 ([11, Theorem]). Λ(R) is not a rearrangement invariant space. The strategy for proving the result is building sequences of sets (Bn ) and (Dn ) with m(Bn ) = m(Dn ) = 2nn and such that
χBn Λ(R) −→ 0.
χDn Λ(R)
(3.3)
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This contradicts Λ(R) being r.i., since χBn and χDn have the same distribution function, so their norms in any r.i. space should be the same (or equivalent). Let us sketch how these sets are built. Note that, according to (3.2), n n ∞ ai ri ≤ χE Λ(R) ≤ sup χE · ai ri + χE · ai ri . sup χE ·
(ai )∈B2
i=1
1
(ai )∈B2
1
i=1
n+1
1
n
Let ∆1n , . . . , ∆2n be the dyadic intervals of order n. Let aij be the value of the Rademacher function ri on the interval ∆jn . The values of r1 , r2 , . . . , rn over the n intervals ∆1n , . . . , ∆2n are shown in the following matrix: ∆1 n r1 1 .. .. . . ri ai1 .. . . .. 1 rn
∆2n 1 .. .
... ... .. .
n
∆jn a1j .. .
. . . ∆2n . . . −1 .. .. . . . . . ai2n .. .. . .
ai2 .. .
. . . aij .. .. . . −1 . . . anj . . . −1 Choose columns J1 so that, for Bn = j∈J1 ∆jn , we have that sup (ai )∈B2
χBn
n
ai ri Λ
i=1
is small; and choose columns J2 so that, for Dn = sup (ai )∈B2
χDn
n
j∈J2
∆jn , we have that
ai ri Λ
i=1
is large. From this, together with an adequate control of the norm of the tails of the series, we deduce (3.3).
4. The Rademacher multiplicator space It was a suggestion of S. Kwapie´ n to the author to consider the above result substituting the role played by the space L1 ([0, 1]) in the definition of the space Λ(R), namely: an rn ∈ L1 , Λ(R) = f : f · an rn ∈ L1 , for all by Lp ([0, 1]). We went a step further and considered an arbitrary r.i. space. 4.1. The space Λ(R, X) Definition 4.1. Let X be an r.i. space on [0,1]. The Rademacher multiplicator space for X is Λ(R, X) := f : f · an rn ∈ X, for all an rn ∈ X .
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Note that when X = L1 , or X = Lp for finite p, the condition an rn ∈ X corresponds to (an ) ∈ 2 , due to Khintchin inequalities. However, this is not the case for a general r.i. space. Some examples are in order: 2 , for X = LN , due to Rodin and Semenov’s result. Rad (X) ≈ q ,∞ , for X = Lψq with ψq (t) = exp(tq ) − 1, q > 2. 1 , for X = L∞ . In general, for X an arbitrary r.i. space Rad (X) is (isomorphic to) a sequence space which is an interpolation space between 2 and 1 (that is, for every bounded linear operator T satisfying T : i → i , i = 1, 2, we have T : Rad (X) → Rad (X)). The converse to this result is also true. Theorem 4.2 ([3]). Every interpolation space between 2 and 1 is a space Rad (X) for some r.i. space X. Theorem 3.1 can be extended to this more general setting. Note that the space Λ(R, X) is a Banach function space for the norm an rn :
f Λ(R,X) := sup f · an rn ≤ 1 . X
X
Theorem 4.3 ([11, Theorem], [5, Theorem 2.1]). If X is an r.i. space such that the lower dilation index of its fundamental function ϕX satisfies γϕX > 0, then Λ(R, X) is not rearrangement invariant. Recall that the fundamental function of an r.i. space X is defined by ϕX (t) :=
χ[0,t] X for 0 ≤ t ≤ 1. In particular, for X = Lp , 1 ≤ p ≤ ∞, we have ϕX (t) = t1/p . The lower dilation index γϕ of a positive function ϕ is γϕ := lim
t→0+
log sup 0 0, [10, III.5.12]. p,q
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4.2. The symmetric kernel of Λ(R, X) Theorem 4.3 implies that for ‘most’ of the classical r.i. spaces the multiplicator space is not r.i. Thus, it becomes relevant to identify the largest r.i. space contained in Λ(R, X), which we call the symmetric kernel of Λ(R, X). Let us illustrate the concept of symmetric kernel with a simple example. The function space 1 Z= f: |f (t)|t1/p−1 dt < ∞ 0
is not r.i. (to see this, consider the functions f (t) = (1 − t)−1/p and g(t) = t−1/p , they have the same distribution function but, f ∈ Z and g ∈ Z). The largest r.i. space inside Z is easy to identify: it is the Lorentz space 1 Lp,1 = f : f ∗ (t)t1/p−1 dt < ∞ . 0
Note that the inclusion Lp,1 ⊂ Z follows from a result of Hardy and Littlewood on rearrangements of functions; see [10, II.2.2]. The definition of the symmetric kernel of the multiplicator space follows. Definition 4.4. The symmetric kernel of Λ(R, X) is the space Sym (R, X) := f ∈ Λ(R, X) : if g f, then g ∈ Λ(R, X) , where g f means that g and f have the same distribution function. The norm in Sym (R, X) is
f Sym (R,X) := sup g Λ(R,X) : g f For the identification of Sym (R, X) we need to recall the associate space of a Banach function space X. It is the space X of all measurable functions g such that g·f is integrable, for every f ∈ X. If X is r.i., then also X is r.i. The biassociate of X is the space defined by X := (X ) . Theorem 4.5 ([5, Theorem 2.8], [7, Proposition 3.1]). Let X be an r.i. space with LN ⊂ X, then Sym (R, X) = f : f ∗ log1/2 (2/t)X < ∞ . Recall that LN is the space functions with square exponential integrability. As could be expected, the proof relies on the Central Limit Theorem. Indeed, the proof is based on the inequalities: ∗ f ∗ (t) an rn (t) ≤ K (an ) 2 f ∗ (t) log1/2 (2/t), n ∗ ri √ (t). f ∗ (t) log1/2 (2/t) ≤ C f ∗ (t)· lim n n 1 Some examples following from Theorem 4.5 are in order. In many problems in classical analysis the class of Lorentz–Zygmund spaces plays a prominent role;
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see [9], [10, IV.6.13]. For 0 < p, q ≤ ∞, α ∈ R, the Lorentz–Zygmund space Lp,q (log L)α consists of all measurable functions f : [0, 1] → R for which 1 q dt 1/q t1/p logα (2/t)f ∗ (t)
f p,q;α = < ∞, t 0 with the usual modification in the case q = ∞. Note that the Lp spaces are Lorentz–Zygmund spaces for parameters p = q and α = 0. (Below, we denote by A B the existence of constants C, c > 0 such that c·A ≤ B ≤ C·A.) • For X = Lp with 1 ≤ p < ∞, we have 1 p 1/p f ∗ (t) log1/2 (2/t) dt .
f Sym (R,X) 0
p
Hence, Sym (R, L ) is the Zygmund space Lp (logL)1/2 (these are Lorentz– Zygmund spaces with p = q, see [10, IV.6.11]). • For X = Lp,q (log L)α with either 1 < p < ∞, 1 ≤ q < ∞ and α ∈ R, or p = q = 1 and α ≥ 0, we have 1 q dt 1/q t1/p log1/2+α (2/t)f ∗ (t) .
f Sym (R,X) t 0 Hence, Sym(R,Lp,q (log L)α ) is the Lorentz–Zygmund space Lp,q (log L)1/2+α. • For X = Lp,∞ (log L)α with 1 < p < ∞ and α ∈ R we have
f Sym (R,X) sup t1/p log1/2+α (2/t)f ∗ (t). 0 2, we have Λ(R, Lψq ) = L∞ , [11, Example 3]. Observe that if Λ(R, X) = L∞ then, necessarily, Sym (R, X) = L∞ . The next results shows that, unexpectedly, both conditions are, in view of Theorem 4.7, equivalent. Partial results, still of interest, giving conditions for Λ(R, X) = L∞ were given in [13, Theorem 2] and [4, Theorem 1]. Theorem 4.8 ([6, Theorem 1]). Let X be an r.i. space and X0 denote the closure of L∞ in X. Then: / X0 Λ(R, X) = L∞ ⇐⇒ log1/2 (2/t) ∈ The proof uses a remarkable formula, due to Montgomery-Smith, [23], for the distribution function of a Rademacher series: 2 ≤ et /2 , m t: an rn > K(a, t; 1 , 2 ) 2 an rn > c−1 ·K(a, t; 1 , 2 ) ≥ c−1 ·ect , m t: where K(a, t; 1 , 2 ) is the K-functional of Petree for the sequence a = (an ) ∈ 2 with respect to the spaces 1 and 2 ; see [10, V.1.1]. Note that when X is separable we have X0 = X and so the condition in Theorems 4.7 and 4.8 becomes LN ⊂ X. Note also that if L∞ Λ(R, X) then Rad (X) ≈ 2 . Another consequence of Theorem 4.8 is that X ⊂ Y implies Λ(R, X) ⊂ Λ(R, Y ), [6, Corollary 3], which is far from obvious in view of Definition 4.1. It could be thought that LN is the largest r.i. space with Λ(R, X) = L∞ . This is not the case, as shown by Theorem 4.8 and Example 3.5 in [6].
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Can the multiplicator space Λ(R, X) be a r.i. space different from L∞ ? The answer is yes, and a first example is X = Lexp , the space of functions f of expo 1 nential integrability: 0 exp(|f |/λ) < ∞, for some λ > 0. In this case we have, [11, Example 3], Λ(R, Lexp ) = LN . Other similar cases are X = Lψq , the Orlicz space associated to the function ψq (t) = exp(tq ) − 1 with 0 < q < 2, where, [13, Theorem 3], Λ(R, Lψq ) = Lψα ,
for α := 2q/(2 − q);
and X an exponential Orlicz space ExpLφ , that is, the Orlicz space associated to the function eφ(t) − 1. In this case we have, [5, Theorem 4.7], Λ(R, ExpLφ ) = ExpLΨ , √ where Ψ is the Orlicz function satisfying Ψ−1 (t) := φ−1 (t)/ t (whenever this last function is increasing). These examples have a common feature. Namely, the norm of the space X can be obtained via an extrapolation formula. For example, equivalent expressions for the norm in LN and Lexp are the following
f LN = sup
1≤p
