Tangent Sequences in Orlicz and Rearrangement Invariant Spaces

BY PAWEŁ HITCZENKO

Department of Mathematics, Box 8205, North Carolina State University, Raleigh, NC 27695 – 8205, USA

AND STEPHEN J. MONTGOMERY-SMITH

Department of Mathematics, University of Columbia – Missouri, Columbia, MO 65211, USA

Abstract Let (f_{n}) and (g_{n}) be two sequences of random variables adapted to an increasing sequence of σ-algebras (F_{n}) such that the conditional distributions of f_{n} and g_{n} given F_{n−1} coincide. Suppose further that the sequence (g_{n}) is conditionally independent. Then it is known that ∥∑ f_{k}∥_{p} ≤ C ∥∑ g_{k}∥_{p}, 1 ≤ p ≤ ∞, where the number C is a universal constant. The aim of this paper is to extend this result to certain classes of Orlicz and rearrangement invariant spaces. This paper includes fairly general techniques for obtaining rearrangement invariant inequalities from Orlicz norm inequalities.

1. Introduction Let (F_{n}) be an increasing sequence of σ-algebras on some probability space (Ω, F, P). We will assume that F_{0} = {∅, Ω}. A sequence (f_{n}) of random variables is called (F_{n})- adapted if f_{n} is F_{n}-measurable for each n ≥ 1. In the sequel we will simply write ‘adapted’ if there is no risk of confusion. For any sequence (f_{n}) of random variables, we will write f^{∗} = sup_{n} |f_{n}| and f_{n}^{∗} = max_{1≤k≤n} |f_{k}|. Throughout the paper all equalities or inequalities between random variables are assumed to hold almost surely.

Given a σ-algebra A ⊂ F and an integrable random variable f, we will denote the conditional expectation of f given A by E_{A}f. If A = F_{k} then we will simply write E_{k}f for E_{F_{k}}f.

The conditional distribution of a random variable f given A is denoted by L(f|A). Thus, L(f|A) = L(g|A) means that for each real number t we have that P (f > t|A) = P (g > t|A).

The following definition was introduced by Kwapień and Woyczyński in a preprint of their paper [10], which was distributed as early as 1986. We refer the reader to their book [11] for more information on tangent sequences.

Definition. Let (F_{n}) be an increasing sequence of σ-algebras on (Ω, F, P).

(a) Two adapted sequences (f_{n}) and (g_{n}) of random variables are tangent if for each n ≥ 1

we have

L(f_{n}|F_{n−1}) = L(g_{n}|F_{n−1}).

(b) An adapted sequence (g_{n}) of random variables satisfies condition (CI) if there exists a σ-algebra G ⊂ F such that for each n ≥ 1

L(g_{n}|F_{n−1}) = L(g_{n}|G),

and (g_{n}) is a sequence of G-conditionally independent random variables.

A sequence (f_{n}) is conditionally symmetric if (f_{n}) and (−f_{n}) are tangent sequences of random variables.

Every sequence of random variables (f_{n}) admits (possibly on an enlarged probability space) a tangent sequence which satisfies condition (CI) (cf. e.g. [11 p. 104]). Throughout this paper, such a sequence will be denoted by (\overline{f}_{n}), and will be called a decoupled version of (f_{n}). It is useful to note that the σ-algebra G can be chosen so that the random variables f_{n}, n ≥ 1, are G-measurable.

In this paper, we will be interested in comparing Orlicz and rearrangement invariant norms of sums of tangent sequences. A rearrangement invariant space X is a space of random variables f equipped with a complete quasi-norm ∥ · ∥_{X} such that either the conditions (i), (ii) and (iii) or (i), (ii) and (iii′) below hold:

(i) if g^{#} ≤ f^{#} and f ∈ X, then g ∈ X with ∥g∥_{X} ≤ ∥f∥_{X}; (ii) if f is simple with finite support then f ∈ X; (iii) f_{n} ∈ X and f_{n} ↘ 0 implies ∥f_{n}∥_{X} ↘ 0 (iii′) f_{n} ∈ X and 0 ≤ f_{n} ↗ f and sup_{n} ∥f_{n}∥_{X} < ∞ imply f ∈ X with ∥f∥_{X} = sup_{n} ∥f_{n}∥_{X}. Here f^{#} denotes the decreasing rearrangement of |f|, that is, f^{#}(s) = sup{ t : P (|f| > t) > s }.

Examples of rearrangement invariant spaces include Orlicz spaces and Lorentz spaces. Let Φ : [0, ∞) → [0, ∞) be an increasing function such that Φ(0) = 0, and such that there is a constant c > 0 such that Φ(ct) ≥ 2Φ(t) for all t ≥ 0. (Functions satisfying the latter condition have been called dilatory in [14]; let us note that if Φ is convex, and Φ(0) = 0 then this condition is satisfied with c = 2.) Given such a function Φ we define the Orlicz norm of a random variable f to be

∥f∥_{Φ} = inf {λ > 0 : E(Φ(|f| /λ)) ≤ 1} .

We let L_{Φ} = {f : ∥f∥_{Φ} < ∞}. Note that if Φ is convex, then L_{Φ} is a normed space. However, we do not wish to restrict ourselves to normed spaces.

Other examples are the Lorentz spaces. Given 0 < p, q ≤ ∞, we define the space L_{p,q} to be those random variables f for which the following quantity is finite:

∥f∥_{p,q} = \{ \begin{matrix}(\frac{q}{p} ∫_{0}^{1} s^{(q/p)−1}f^{#}(s)^{q} ds)^{1/q} & if q < ∞ \\ sup_{s>0} s^{1/p}f^{#}(s) & if q = ∞ . \end{matrix}

Note that L_{p,q} is not a normed space unless 1 ≤ q ≤ p ≤ ∞.

Note that the L_{p} spaces are special cases: L_{p} = L_{Φ} = L_{p,p}, where Φ(t) = t^{p}. We refer the reader to [13] for more details about these spaces.

In the present paper we will be interested in the domination of a rearrangement invariant norm of a sum of an arbitrary sequence of adapted random variables by the rearrangement invariant norm of a sum of its decoupled version. It is already known (see [4]) that if Φ satisfies the Δ_{2}-condition, that is, there is a constant c > 0 such that Φ(2t) ≤ c Φ(t) for all t ≥ 0, then there is a constant C_{Φ} such that for every adapted sequence (f_{n}) of random variables one has:

∥ ∑ f_{i}∥_{Φ} ≤ C_{Φ}∥ ∑ \overline{f}_{i}∥_{Φ}.  (1.1)

Building on some special situations considered by Klass [4, Theorem 3.1] and Kwapień [9], Hitczenko [6] began to investigate how the constant C_{Φ} depends upon Φ. He showed that there is a universal constant C > 0 such that

∥ ∑ f_{i}∥_{p} ≤ C ∥ ∑ \overline{f}_{i}∥_{p}  1 ≤ p ≤ ∞.  (1.2)

In this present paper, we will show among other things, that inequality (1.1) holds with C_{Φ} uniformly bounded, at least for certain classes of Orlicz functions. Our first theorem extends a result of Klass who proved (1.1) for randomly stopped sums of independent random variables.

Let us define classes of Orlicz functions. Following Klass [8], for q > 0 we define the class F_{q} as the class of all functions Φ such that:

(i) Φ : [0, ∞) → [0, ∞), Φ(0) = 0, (ii) Φ is nondecreasing and continuous, and (iii) Φ satisfies the growth condition: Φ(cx) ≤ c^{q}Φ(x) for all x ≥ 0, c ≥ 2. For p > 0, we define the class G_{p} as the class of all functions Φ such that

(i) Φ : [0, ∞) → [0, ∞), Φ(0) = 0, (ii) Φ is nondecreasing and continuous, and (iii) Φ satisfies the growth condition: Φ(cx) ≥ c^{p}Φ(x) for all x ≥ 0, c ≥ 2.

Then we obtain the following results.

Theorem 1.1. There is a universal constant C > 0 such that if Φ ∈ F_{q} for some q > 0, then for every adapted sequence (f_{n}) of random variables one has:

EΦ (|∑ f_{i}|) ≤ C^{1+q}EΦ (|∑ \overline{f}_{i}|) .

This inequality had already been obtained by Klass [8] in the special case that f_{k} = I(τ ≥ k)ξ_{k}, where (ξ_{k}) is a sequence of independent random variables and τ is a stopping time. More precisely, Klass proved his result for Banach space valued random variables (ξ_{k}) (with absolute value replaced by norm). To discuss Banach space valued random variables one needs to adjust notation; for a random variable Y and a σ-algebra A, we use L(Y |A) to denote the regular version of the conditional distribution of Y given A, that is, L(Y |A) = L(Z|A) means that for every Borel subset A of the Banach space, we have P (Y ∈ A|A) = P (Z ∈ A|A). Recall that the existence of the regular versions of the conditional distributions is guaranteed, as long as our random variables take values in a separable Banach space. As it turns out, in our generality, the inequality of Theorem 1.1 need not hold (with any constant), unless some extra conditions are imposed on the geometry of the underlying Banach space (see e.g. [3]). Since it is unclear at this time for which Banach spaces the inequality

(E∥ ∑ f_{k}∥^{p})^{1/p} ≤ c_{p}(E∥ ∑ \overline{f}_{k}∥^{p})^{1/p}

holds (even if the constant c_{p} is allowed to depend on p), we confine our discussion to real valued random variables.

Corollary 1.2. Given numbers p_{0} > 0 and r ≥ 1, there is a constant c_{p_{0},r} such that if p ≥ p_{0}, and if Φ ∈ G_{p} ∩ F_{rp}, then

∥ ∑ f_{i}∥_{Φ} ≤ c_{p_{0},r}∥ ∑ \overline{f}_{i}∥_{Φ}.

The next step is to extend these results to rearrangement invariant spaces. This will be accomplished through a rather general method of obtaining rearrangement invariant norm inequalities from Orlicz norm inequalities. We believe that this technique will prove useful in other contexts as well. In particular, we would like to mention that this method could be used to deduce martingale inequalities obtained by Johnson and Schechtman [7] from the corresponding inequalities for Orlicz functions.

Corresponding to the notions of Orlicz spaces lying in G_{p} ∩ F_{q}, we have the following notion. We say that a rearrangement invariant space is an interpolation space for (L_{p}, L_{q}) (in short, a (p, q)-interpolation space) if there is a constant c > 0 such that for every operator T : L_{p} ∩ L_{q} → L_{p} ∩ L_{q} for which ∥T ∥_{L_{p}→L_{p}} ≤ 1 and ∥T ∥_{L_{q}→L_{q}} ≤ 1 we have that ∥T ∥_{X→X} ≤ c.

However, this notion is not quite what we need. Define

K_{p,q}(f, t) = inf{∥f′∥_{p} + t ∥f′′∥_{q} : f′ + f′′ = f^{#}}.

We will say that a rearrangement invariant space X is a (p, q)-K-interpolation space if there is a constant c such that whenever f and g are such that K_{p,q}(f, t) ≤ K_{p,q}(g, t) (t > 0), and g ∈ X, then f ∈ X and ∥f∥_{X} ≤ c ∥g∥_{X}. The (p, q)-K-interpolation constant of X, denoted by C_{p,q}(X), is the infimum of c that work for all functions f and g.

It is quite easily seen that every (p, q)-K-interpolation space is a (p, q)-interpolation space. It is also known that if 1 ≤ p, q ≤ ∞, then every normed (p, q)-interpolation space is a (p, q)-K-interpolation space (see [1]). We are able to establish the following method for obtaining rearrangement invariant inequalities from Orlicz inequalities.

Theorem 1.3. Suppose that ∥f∥_{Φ} ≤ ∥g∥_{Φ} for all Φ ∈ F_{q} ∩ G_{p}, where 0 < p < q < ∞. If X is a (p, q)-K-interpolation space, then ∥f∥_{X} ≤ 2^{2+1/p}C_{p,q}(X) ∥g∥_{X}.

Corollary 1.4. Given numbers p_{0} > 0 and r ≥ 1, there is a constant c_{p_{0},r} such that if p ≥ p_{0}, and if X is a (p, pr)-K-interpolation space, then

∥ ∑ f_{i}∥_{X} ≤ c_{p_{0},r}C_{p,pr}(X)∥ ∑ \overline{f}_{i}∥_{X}.

In particular, we are able to obtain the following result for Lorentz spaces. Note that if one is interested in normed Lorentz spaces, then p_{0} below can be taken to be 1, and the resulting inequality extends (1.2).

Corollary 1.5. Given a number p_{0} > 0, there is a constant c_{p_{0}} such that if p, q ≥ p_{0}, then

∥ ∑ f_{i}∥_{p,q} ≤ c_{p_{0}}∥ ∑ \overline{f}_{i}∥_{p,q}.

2. Inequalities for Orlicz functions We begin with a proof of Theorem 1.1. Since our proof is based on well understood techniques we will be somewhat sketchy and we refer the reader to [6] for details that are not explained here. Throughout this section we let (M_{n}) be a martingale with difference sequence (Δ_{k}). Since F_{q_{1}} ⊂ F_{q_{2}} whenever q_{1} ≤ q_{2}, we can assume without loss of generality that q ≥ 1. Our departing point is the following result which can be found in the just mentioned paper (Theorem 5.1 and the beginning of the proof of Lemma 2.3).

Lemma 2.1. Let 1 ≤ q < ∞, and let (Δ_{k}) be a conditionally symmetric martingale difference sequence, and (\overline{Δ}_{k}) its decoupled version. Set T_{n,q}(M) = (E| ∑_{k=1}^{n} \overline{Δ}_{k}|^{q}|G)^{1/q}. Then there exist δ_{1} > 0, β > 1 + δ_{1} and ϵ with 0 < ϵ ≤ 1/2 such that for every λ > 0 we have

P (M^{∗} ≥ βλ, (T_{q}^{∗}(M) ∨ Δ^{∗}) < δ_{1}λ) ≤ ϵ^{q}P (M^{∗} ≥ λ).

From this, we obtain

Lemma 2.2. Let (Δ_{k}) be as above, and assume that w_{n} is a F_{n−1}-measurable random variable such that |Δ_{n}| ≤ w_{n} for each n ≥ 1. Set N_{n} = ∑_{k=1}^{n} \overline{Δ}_{k} (n ≥ 1). Suppose that δ_{1}, β, and ϵ are as in Lemma 2.1. Then, there exist δ > 0, δ_{2} > 0 and 0 < α ≤ 1/2 such that for every λ > 0 we have

P (M^{∗} ≥ βλ, N^{∗} < δλ) ≤ ϵ^{q}P (M^{∗} ≥ λ) + P (w^{∗} ≥ δ_{2}λ) + (1 − α^{q})P (M^{∗} ≥ βλ).

Proof: We have that

\begin{matrix}P (M^{∗} ≥ βλ, N^{∗} < δλ) ≤ & P (M^{∗} ≥ βλ, T_{q}^{∗}(M) < δ_{1}λ, w^{∗} < δ_{2}λ) + P (w^{∗} ≥ δ_{2}λ) \\ + & P (M^{∗} ≥ βλ, T_{q}^{∗}(M) ≥ δ_{1}λ, w^{∗} < δ_{2}λ, N^{∗} < δλ). \end{matrix} (2.1)

Suppose δ_{2} ≤ δ_{1}. Then, in view of Lemma 2.1, for the first probability on the right-hand side of (2.1) we have that

\begin{matrix}P (M^{∗} ≥ βλ, T_{q}^{∗}(M) < δ_{1}λ, w^{∗} < δ_{2}λ) ≤ & P (M^{∗} ≥ βλ, T_{q}^{∗}(M) ∨ w^{∗} < δ_{1}λ) \\ ≤ & ϵ^{q}P (M^{∗} ≥ λ). \end{matrix}

It remains to estimate the last probability in (2.1). Since M^{∗}, w^{∗} and T_{q}^{∗}(M) are Gmeasurable, by conditioning on G, we see that the last probability in (2.1) is equal to:

E{I(M^{∗} ≥ βλ, T_{q}^{∗}(M) ≥ δ_{1}λ, w^{∗} < δ_{2}λ)P (N^{∗} < δλ|G)}.

By Kolmogorov’s converse inequality (see e.g. [12, Remark 6.15, p. 161]) for all sequences of independent and symmetric random variables (ξ_{k}) and for all t > 0 we have that

P (S^{∗} ≥ t) ≥ \frac{1}{2^{q}} (1 − \frac{2^{2q}(t^{q} + E(ξ^{∗})^{q})}{E(S^{∗})^{q}}),

where S_{n} = ∑_{k=1}^{n} ξ_{k}. Applying this result conditionally on G, we obtain that

P (N^{∗} < δλ|G) ≤ 1 − \frac{1}{2^{q}} (1 − \frac{2^{2q}((δλ)^{q} + E_{G}(\overline{Δ}^{∗})^{q})}{E_{G}(N^{∗})^{q}}).  (2.2)

Also, if w_{n} < δ_{2}λ, then |Δ_{n}| < δ_{2}λ, and since w_{n} is F_{n−1}-measurable, and the conditional distributions of Δ_{n} and \overline{Δ}_{n} coincide, it follows that |\overline{Δ}_{n}| < δ_{2}λ. Therefore, on the set

{T_{q}^{∗}(M) ≥ δ_{1}λ, w^{∗} < δ_{2}λ},

we have

\frac{(δλ)^{q} + E_{G}(\overline{Δ}^{∗})^{q}}{E_{G}(N^{∗})^{q}} ≤ \frac{(δλ)^{q} + (δ_{2}λ)^{q}}{(δ_{1}λ)^{q}} ,

so that the conditional probability in (2.2) does not exceed

1 − \frac{1}{2^{q}} (1 − \frac{4^{q}(δ^{q} + δ_{2}^{q})}{δ_{1}^{q}}).

Choosing δ = δ_{2} = δ_{1}/12, we obtain that

1 − \frac{1}{2^{q}} (1 − \frac{4^{q}(δ^{q} + δ_{2}^{q})}{δ_{1}^{q}}) = 1 − \frac{1}{2^{q}} (1 − \frac{2}{3^{q}} ) ≤ 1 − α^{q},

whenever α ≤ 1/6. Therefore,

\begin{matrix}E\{I(M^{∗} ≥ βλ & , T_{q}(M) ≥ δ_{1}λ, w^{∗} < δ_{2}λ)P (N^{∗} < δλ|G)\} \\ ≤ & (1 − α^{q})EI(M^{∗} ≥ βλ, T_{q}(M) ≥ δ_{1}λ, w^{∗} < δ_{2}λ) ≤ (1 − α^{q})P (M^{∗} ≥ βλ). \end{matrix}

This completes the proof of Lemma 2.2.

Now we are ready to complete the proof of Theorem 1.1. By an argument similar to one used in [6, proof of Lemma 2.1], it follows that in order to prove

EΦ(| ∑ f_{k}|) ≤ c^{q}EΦ(| ∑ \overline{f}_{k}|),  (2.3)

it suffices to establish (2.3) for (f_{k}) = (Δ_{k}), a conditionally symmetric martingale difference sequence. By a routine application of Davis’ decomposition (cf. e.g. [2] and references therein), we may also assume that |Δ_{n}| ≤ w_{n}, where w_{n} is a F_{n−1}-measurable random variable, and that w^{∗} ≤ 2Δ^{∗}. The latter inequality, together with the inequality P (f^{∗} ≥ t) ≤ 2P (g^{∗} ≥ t) valid for all tangent sequences (f_{k}) and (g_{k}) (cf. [4] or [11, Theorem 5.2.1 (i)]), implies that

P (w^{∗} ≥ t) ≤ P (Δ^{∗} ≥ t/2) ≤ 2P (\overline{Δ}^{∗} ≥ t/2) ≤ 2P (N^{∗} ≥ t/4).  (2.4)

By Lemma 2.2, we have that

P (M^{∗} ≥ βλ) ≤ P (N^{∗} ≥ δλ) + P (w^{∗} ≥ δ_{2}λ) + (1 − α^{q})P (M^{∗} ≥ βλ),

so that

α^{q}P (M^{∗} ≥ βλ) ≤ P (N^{∗} ≥ δλ) + P (w^{∗} ≥ δ_{2}λ).

Consequently,

α^{q}EΦ(M^{∗}/β) ≤ EΦ(N^{∗}/δ) + EΦ(w^{∗}/δ_{2}).

Therefore,

\begin{matrix}(\frac{α}{β})^{q}EΦ(M^{∗}) = & (\frac{α}{β})^{q}EΦ(βM^{∗}/β) \\ ≤ & (\frac{α}{β})^{q}β^{p}EΦ(M^{∗}/β) ≤ EΦ(N^{∗}/δ) + EΦ(w^{∗}/δ_{2}) \\ ≤ & δ^{−q}EΦ(N^{∗}) + δ_{2}^{−q}EΦ(w^{∗}). \end{matrix}

By (2.4) we have that

EΦ(w^{∗}) ≤ 2EΦ(4N^{∗}) ≤ 2 · 4^{q}EΦ(N^{∗}),

so that

EΦ(M^{∗}) ≤ (\frac{β}{α})^{q}\{\frac{1}{δ^{q}} + \frac{2 · 4^{q}}{δ_{2}^{q}} \}EΦ(N^{∗}).

Conditionally given G, (\overline{Δ}_{k}) is a sequence of independent and symmetric random variables. Therefore, by Levy’s inequality,

P (N^{∗} ≥ t|G) ≤ 2P (N ≥ t|G).

This implies (see e.g. [11, Proposition 0.2.1]) that for every increasing function ϕ : R^{+} → R^{+} we have Eϕ(N^{∗}) ≤ 2Eϕ(N). This completes the proof of Theorem 1.1.

As we mentioned in the introduction, inequality (2.3) extends a result of Klass, who considered sequences (f_{k}) of special form. On the other hand, it follows from a result of Kwapień [9] that if f_{k} = (∑_{j=1}^{k−1} a_{j,k}ξ_{j})ξ_{k}, where (ξ_{j}) is a sequence of independent zero mean random variables, then (2.3) holds in the stronger form:

EΦ(| ∑ f_{k}|) ≤ EΦ(c| ∑ \overline{f}_{k}|),

for some absolute constant c and for every convex function Φ. Thus one may wonder whether our restrictions on Φ can be relaxed. We wish to close this section with a negative result showing that (2.3) does not hold for all convex functions Φ. (However, it is still possible that (2.3) holds under weaker assumption than ours.) Our example is an easy adaptation of an example due to Talagrand concerning comparison of tail behavior for sums of tangent sequences. This example was included in [5], and we refer the reader to the latter paper for details that are not included here.

Proposition 2.3. For every constant c > 0, there exists a convex function Φ : R → R and a sequence (f_{k}) such that

EΦ(| ∑ f_{k}|) ≥ cEΦ(c| ∑ \overline{f}_{k}|).

Proof: We will show that for every k ∈ N there exists a convex function Φ and a sequence (f_{k}) for which

EΦ(| ∑ f_{k}|) ≥ \frac{2^{2^{k+2}}}{k^{2}2^{2k}} EΦ(\frac{k}{4} | ∑ \overline{f}_{k}|).

Let (r_{n}) denote the Rademacher random variables, that is, a sequence of independent random variables such that P (r_{n} = ±1) = 1/2. Fix k ∈ N. Given an integer N_{1} to be specified in a moment, define N_{2}, … N_{k} as follows:

N_{i} − N_{i−1} = 2^{−(i−1)}N_{1}, i = 2, …, k.

Put

Ω_{1} = {r_{1} = … = r_{N_{1}} = 1},

and then

Ω_{i} = Ω_{i−1} ∩ {r_{N_{i−1}+1} = … = r_{N_{i}}}, i = 2, …, k.

Define a sequence of random variables (v_{i}) by the formulas:

v_{1} = … = v_{N_{1}} = 1
v_{N_{1}+1} = … = v_{N_{2}} = 2I_{Ω_{1}}
. . . . . . . . . . . .
v_{N_{k−1}+1} = … = v_{N_{k}} = 2^{k−1}I_{Ω_{k−1}}.

We let f_{j} = v_{j}r_{j} for j = 1, … N_{k}. Then \overline{f}_{j} = v_{j}r′_{j}, where (r′_{j}) is an independent copy of (r_{j}) (cf. [10, Example 4.3.1]). For 0 < δ < 1, let Φ_{δ} be a convex function defined by Φ_{δ}(x) = (x − δkN_{1})^{+}. Note that ∑ |v_{j}| = kN_{1}, and therefore

EΦ_{δ}(| ∑ v_{j}r_{j}|) ≥ (1 − δ)kN_{1}P (| ∑ v_{j}r_{j}| ≥ kN_{1}).

On the other hand, if | ∑ v_{j}r′_{j}| < 4N_{1}, then | ∑ v_{j}r′_{j}| ≤ 4N_{1} − 1, so that with δ = 1 − 1/(4N_{1}) we get

Φ_{δ}(\frac{k}{4} | ∑ v_{j}r′_{j}|) ≤ (kN_{1} − \frac{k}{4} − δkN_{1})^{+} = 0.

Since

P (| ∑ v_{j}r′_{j}| ≥ 4N_{1}) ≤ k2^{−N_{1}/2^{k−2}}P (| ∑ v_{j}r_{j}| ≥ kN_{1}),

(cf. [5, top half of page 176]) we obtain

\begin{matrix}EΦ_{δ}(\frac{k}{4} | ∑ v_{j}r′_{j}|) ≤ & Φ_{δ}(\frac{k^{2}N_{1}}{4} )P (| ∑ v_{j}r′_{j}| ≥ 4N_{1}) \\ ≤ & kN_{1} \frac{k}{4} k2^{−N_{1}/2^{k−2}}P (| ∑ v_{j}r_{j}| ≥ kN_{1}), \end{matrix}

and it follows that

\frac{EΦ_{δ}(| ∑ v_{j}r_{j}|)}{EΦ_{δ}((k/4)| ∑ v_{j}r′_{j}|)} ≥ \frac{4(1 − δ)kN_{1}2^{N_{1}/2^{k−2}}}{k^{3}N_{1}} = \frac{2^{N_{1}/2^{k−2}}}{k^{2}N_{1}} ≥ \frac{2^{2^{k+2}}}{k^{2}2^{2k}} ,

for N_{1} ≥ 2^{k}. This completes the proof.

In the above example the sequence (f_{k}) may be constructed so that ∑ f_{k} is a randomly stopped sum of independent random variables (see Remark on p. 176 of [5]). Thus, the conclusion of this Remark applies here as well.

3. Rearrangement invariant norm inequalities

Lemma 3.1. Suppose that Φ and Ψ are two Orlicz functions. Let Θ = Φ ∧ Ψ, and Θ_{1}(x) = \frac{1}{2}Θ(x). Then

\frac{1}{2} ∥f∥_{Θ_{1}} ≤ inf\{∥f′∥_{Φ} + ∥f′′∥_{Ψ} : f′ + f′′ = f^{#}\} ≤ 2 ∥f∥_{Θ} .

Proof: To show the left hand side, suppose that f^{#} = f′ + f′′, and ∥f′∥_{Φ} + ∥f′′∥_{Ψ} ≤ 1. Then EΦ(|f′|) ≤ 1 and EΨ(|f′′|) ≤ 1, and so

EΘ_{1}(\frac{1}{2}|f|) ≤\frac{1}{2}EΘ(max\{|f′| , |f′′|\}) = \frac{1}{2}E max\{Θ(|f′|), Θ(|f′′|)\}
≤\frac{1}{2}(EΦ(|f′|) + EΨ(|f′′|)) ≤ 1.

To show the right hand side, suppose that ∥f∥_{Θ} ≤ 1, that is, EΘ(|f|) ≤ 1. Let

f′(t) = \{ \begin{matrix}f^{#}(t) & if Φ(|f(t)|) ≤ Ψ(|f(t)|) \\ 0 & otherwise, \end{matrix}

and f′′ = f^{#} − f′. Then we see that EΦ(|f′|) ≤ EΘ(|f|) ≤ 1 and that EΨ(|f′′|) ≤ EΘ(|f|) ≤ 1, and the result follows.

The next lemma follows immediately.

Lemma 3.2. Let Φ_{t}(x) = x^{p} ∧ (tx)^{q}, where 0 < p < q < ∞. Then

2^{−1−1/p} ∥f∥_{Φ_{t}} ≤ K_{p,q}(f, t) ≤ 2 ∥f∥_{Φ_{t}} .

Now we will prove Theorem 1.3, using the above Lemma. From the hypothesis of Theorem 3.1, it follows that ∥f∥_{Φ_{t}} ≤ ∥g∥_{Φ_{t}}. Hence by Lemma 3.2, it follows that K_{p,q}(f, t) ≤ 2^{2+1/p}K_{p,q}(g, t). Now the result follows by the definition of (p, q)-K-interpolation space.

Corollary 1.4 follows easily from Theorems 1.1 and 1.3. To show Corollary 1.5, we only need the following result. The methods below are all fairly standard in interpolation theory, and indeed if one is not concerned about uniform estimates, may be taken directly from the literature.

Lemma 3.3. Given p_{0} > 0, there is a constant c_{p_{0}} > 0 such that if p, q ≥ p_{0}, then L_{p,q} is a (p/2, 2p)-K-interpolation space with constant bounded by c_{p_{0}}.

Proof: First let us define some norms. For p ≤ q, let

∥f∥_{a(t)} = inf{t^{−2/p} ∥f′∥_{p/2} + t^{−1/2p} ∥f′′∥_{2p} : f′ + f′′ = f^{#}},

∥f∥_{b(t)} = (\frac{1}{t} ∫_{0}^{t} f^{#}(s)^{p/2} ds)^{2/p} + (\frac{1}{t} ∫_{t}^{∞} f^{#}(s)^{2p} ds)^{1/2p} .

Clearly ∥f∥_{a(t)} ≤ ∥f∥_{b(t)}. Also ∥f∥_{a(t)} ≥ min{1, 2^{1−2/p}}f^{#}(t). This is because if f^{#} = f′ + f′′, then

t^{−2/p} ∥f′∥_{p/2} + t^{−1/2p} ∥f′′∥_{2p} ≥ (\frac{1}{t} ∫_{0}^{t} |f′(s)|^{p/2} ds)^{2/p} + (\frac{1}{t} ∫_{0}^{t} |f′′(s)|^{2p} ds)^{1/2p}

≥ (\frac{1}{t} ∫_{0}^{t} |f′(s)|^{p/2} ds)^{2/p} + (\frac{1}{t} ∫_{0}^{t} |f′′(s)|^{p/2} ds)^{2/p}

≥ min\{1, 2^{1−2/p}\} (\frac{1}{t} ∫_{0}^{t} f^{#}(s)^{p/2} ds)^{2/p}

≥ min{1, 2^{1−2/p}}f^{#}(t).

Next, given a function f, let us define the function Hf(t) = ∥f∥_{b(t)}. Then it follows that ∥Hf∥_{p,q} ≤ 32^{1/ min{p,q}} ∥f∥_{p,q}. To see this, first note that ∥Hf∥_{p,q} ≤ (∥H_{1}f∥_{p,q}^{q} + ∥H_{2}f∥_{p,q}^{q})^{1/q}, where

H_{1}f(t) = (\frac{1}{t} ∫_{0}^{t} f^{#}(s)^{p/2} ds, )^{2/p}

H_{2}f(t) = (\frac{1}{t} ∫_{t}^{∞} f^{#}(s)^{2p} ds)^{1/2p}

We will use two properties of L_{p,q}. First, if v ≤ q, and if f_{1}, f_{2}, …, f_{n} are functions, then

∥(∑_{i=1}^{n}(f_{i}^{#})^{v})^{1/v}∥_{p,q} ≤ (∑_{i=1}^{n} ∥f_{i}∥_{p,q}^{v})^{1/v} .

Second, if we define the operators D_{a}f(t) = f^{#}(at) for 0 < a < ∞, then ∥D_{a}f∥_{p,q} = a^{−1/p} ∥f∥_{p,q}. (The first property follows from Minkowski’s inequality for L_{q/v}, the second is a simple change of variables argument.) Let u = max{p/q, 2}, v = min{q, p/2}. Then

∥H_{1}f∥_{p,q} ≤ ∥(∫_{0}^{1}(D_{a}f^{#})^{p/2} da)^{2/p}∥_{p,q}

≤ ∥(∑_{n=0}^{∞} ∫_{2^{−u(n+1)}}^{2^{−un}}(D_{a}f^{#})^{p/2} da)^{2/p}∥_{p,q}

≤ ∥(∑_{n=0}^{∞} 2^{−un}(D_{2^{−u(n+1)}}f^{#})^{p/2})^{2/p}∥_{p,q}

≤ ∥(∑_{n=0}^{∞} 4^{−n}(D_{2^{−u(n+1)}}f^{#})^{v})^{1/v}∥_{p,q}

≤ (∑_{n=0}^{∞} 4^{−n} ∥D_{2^{−u(n+1)}}f^{#}∥_{p,q}^{v})^{1/v}

≤ (∑_{n=0}^{∞} 4^{−n}2^{u(n+1)v/p}))^{1/v} ∥f∥_{p,q}

≤ (∑_{n=0}^{∞} 2^{1−n})^{1/v} ∥f∥_{p,q}

≤ 4^{1/v} ∥f∥_{p,q} .

Now let u = max{2p/q, 1}, v = min{q, 2p}.

∥H_{2}f∥_{p,q} ≤ ∥(∫_{1}^{∞}(D_{a}f^{#})^{2p} da)^{1/2p}∥_{p,q}

≤ ∥(∑_{n=0}^{∞} ∫_{2^{un}}^{2^{u(n+1)}}(D_{a}f^{#})^{2p} da)^{1/2p}∥_{p,q}

≤ ∥(∑_{n=0}^{∞} 2^{u(n+1)}(D_{2^{un}}f^{#})^{2p})^{1/2p}∥_{p,q}

≤ ∥(∑_{n=0}^{∞} 2^{n+1}(D_{2^{un}}f^{#})^{v})^{1/v}∥_{p,q}

≤ (∑_{n=0}^{∞} 2^{n+1} ∥D_{2^{un}}f^{#}∥_{p,q}^{v})^{1/v}

≤ (∑_{n=0}^{∞} 2^{n+1}2^{−unv/p}))^{1/v} ∥f∥_{p,q}

≤ (∑_{n=0}^{∞} 2^{1−n})^{1/v} ∥f∥_{p,q}

≤ 4^{1/v} ∥f∥_{p,q} .

Finally, to finish, suppose that K_{p/2,2p}(f, t) ≤ K_{p/2,2p}(g, t) for all t > 0. Then it follows that

f^{#}(t) ≤ max{1, 2^{2/p−1}} ∥f∥_{a(t)} ≤ max{1, 2^{2/p−1}} ∥g∥_{a(t)} ≤ max{1, 2^{2/p−1}} ∥g∥_{b(t)}
=2^{2/p}Hg(t).

Hence

∥f∥_{p,q} ≤ 2^{2/p} ∥Hg∥_{p,q} ≤ 128^{1/ min{p,q}} ∥g∥_{p,q} .

Acknowledgments The second named author would like to express warm gratitude to Victor de la Peña for introducing him to this problem. Research of both authors was partially supported by separate NSF grants. The second named author was also supported by the Research Board of the University of Missouri.

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