Georgian Mathematical Journal Volume 8 (2001), Number 2, 245–274

ENTROPY NUMBERS OF CERTAIN SUMMATION OPERATORS

J. CREUTZIG AND W. LINDE

Abstract. Given nonnegative real sequences a = (α_{k})_{k∈ℤ} and b = (β_{k})_{k∈ℤ} we study the generated summation operator

S_{a,b}(x) := (α_{k}[∑_{l<k} β_{l}x_{l}])_{k∈ℤ} , x = (x_{k})_{k∈ℤ} ,

regarded as a mapping from ℓ_{p}(ℤ) to ℓ_{q}(ℤ). We give necessary and sufficient conditions for the boundedness of S_{a,b} and prove optimal estimates for its entropy numbers relative to the summation properties of a and b. Our results are applied to the investigation of the behaviour of

ℙ(∑_{k∈ℤ} α_{k}^{q}|W (t_{k})|^{q} < ε^{q})  and  ℙ(sup_{k∈ℤ} α_{k}|W (t_{k})| < ε)

as ε → 0, where (t_{k})_{k∈ℤ} is some nondecreasing sequence in [0, ∞) and (W (t))_{t≥0} denotes the Wiener process.

2000 Mathematics Subject Classification: Primary: 47B06. Secondary: 47B37, 60G15. Key words and phrases: Summation operator, entropy numbers, small ball probabilities.

1. Introduction

Volterra integral operators are known to play an important role within Functional Analysis as well as Probability Theory. A special class of interest is that of so-called weighted Volterra integral operators T_{ρ,ψ} mapping L_{p}(0, ∞) into L_{q}(0, ∞) and defined by

T_{ρ,ψ} : f ↦ ρ(s) ∫_{0}^{s} ψ(t)f(t) dt ,  (1.1)

where ρ, ψ ≥ 0 are suitable functions on (0, ∞). The main task is to describe properties of T_{ρ,ψ} (boundedness, degree of compactness, etc.) in terms of certain properties of the weight functions ρ and ψ. Although several results have been proved in this direction ([12], [5], [6], [10]), interesting problems remain open. For example, we do not know of a complete characterization of ρ’s and ψ’s such that the approximation or entropy numbers of T_{ρ,ψ} behave exactly as those of T_{1} on [0, 1]. (T_{1} = T_{1,1} denotes the ordinary integral operator).

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

Given sequences a = (α_{k})_{k∈ℤ} and b = (β_{k})_{k∈ℤ}, the summation operator S_{a,b} : ℓ_{p}(ℤ) → ℓ_{q}(ℤ) defined by

S_{a,b} : x = (x_{k})_{k∈ℤ} ↦ (α_{k} ∑_{j<k} β_{j}x_{j})_{k∈ℤ})  (1.2)

can be viewed as a discrete counterpart to weighted Volterra operators. Thus one may expect for S_{a,b} similar compactness properties (relative to a and b) as for T_{ρ,ψ} (relative to ρ and ψ). This is indeed so as long as one considers upper estimates for the entropy numbers e_{n}(S_{a,b}). Here the most surprising result (Theorem 2.3) asserts that summation operators can behave as badly and irregularly as weighted integral operators. Yet the situation becomes completely different when considering lower estimates for e_{n}(S_{a,b}). The deeper reason is as follows: If ρ ψ > 0 on some interval (some set of positive measure suffices), then the entropy numbers of T_{ρ,ψ} cannot tend to zero faster than those of T_{1} considered on this interval. Since e_{n}(T_{1}) ≈ n^{−1}, this order is a natural lower bound for e_{n}(T_{ρ,ψ}). In the case of summation operators such a“canonical” operator (as T_{1} for T_{ρ,ψ}) does not exist. Hence it is not surprising that e_{n}(S_{a,b}) can tend to zero much faster than n^{−1}.

The aim of this paper is to investigate these phenomena more precisely. We state optimal conditions for a and b in order that sup_{n} n e_{n}(S_{a,b}) < ∞. More precise statements can be formulated under some additional regularity assumptions on a and b, e.g., monotonicity or an exponential decay of the β_{k}’s.

As is well-known, the ordinary Volterra operator T_{1} is closely related to Brownian motion. More precisely, let (ξ_{k})_{k=1}^{∞} be an i.i.d. sequence of standard normal distributed random variables and let (f_{k})_{k=1}^{∞} be some orthonormal basis in L_{2}(0, ∞). Then

W (t) = ∑_{k=1}^{∞} ξ_{k}(T_{1}f_{k})(t),  t ≥ 0 ,  (1.3)

is a standard Wiener process over (0, ∞). In this sense, summation operators S_{a,b} : ℓ_{2} → ℓ_{q} generate Gaussian sequences X = (X_{k})_{k∈ℤ} ∈ ℓ_{q}(ℤ) with

X_{k} = α_{k}W (t_{k}),  k ∈ ℤ,  (1.4)

where t_{k}’s are defined via β_{k}^{2} = t_{k+1} − t_{k}.

Recent results ([9], [11]) relate the entropy behaviour of an operator with estimates for the probability of small balls of the generated Gaussian random variable. We transform the entropy results proved for S_{a,b} into the small ball estimates for X = (X_{k})_{k∈ℤ} defined above. Hence we find sufficient conditions for the α_{k}’s and t_{k}’s such that

lim_{ε→0} ε^{2} log ℙ(∥X∥_{q} < ε) = 0.  (1.5)

These conditions turn out to be (in this language) the best possible ones.

The paper is organized as follows. In Section 2, the main results about the entropy of S_{a,b} are stated. Section 3 provides basic tools and well-known facts, which are used in Section 4 to prove our results. Section 5 is devoted to the study of some examples. Finally, in Section 6 we establish some small ball estimates for Gaussian vectors as in (1.4), using our entropy results for S_{a,b} proved before.

2. Notation and Main Results

For a given sequence x = (x_{k})_{k∈ℤ} ⊆ ℝ and p ∈ [1, ∞] set

∥x∥_{p} := ( ∑_{k=−∞}^{∞} |x_{k}|^{p})^{1/p} if p < ∞, ∥x∥_{∞} := sup_{k∈ℤ}|x_{k}|,

and let ℓ_{p}(ℤ) := {x : ∥x∥_{p} < ∞}. As usual, for p ∈ [1, ∞] the adjoint p′ is given by 1/p + 1/p′ = 1. Now, let p, q ∈ [1, ∞] be arbitrary. For nonnegative sequences a = (α_{k})_{k∈ℤ} and b = (β_{k})_{k∈ℤ} satisfying

A_{k} := (∑_{l≥k} α_{l}^{q})^{1/q} < ∞ and B_{k} := (∑_{l<k} β_{l}^{p′})^{1/p′} < ∞  (2.1)

for all k ∈ ℤ (with obvious modifications for q = ∞ or p = 1), the expression

S_{a,b}(x) := (α_{k}[∑_{l<k} β_{l}x_{l}])_{k∈ℤ}  (2.2)

is well-defined for any x = (x_{k})_{k∈ℤ} ∈ ℓ_{p}(ℤ). Our first result is a version of the well-known Maz’ja-Rosin Theorem (see, e.g., [12], pp. 39-51), characterizing the boundedness of S_{a,b} in terms of A_{k} and B_{k} given in (2.1):

Theorem 2.1. Under the above assumptions, the operator S_{a,b} is well-de-fined and bounded from ℓ_{p}(ℤ) to ℓ_{q}(ℤ) iff D(a, b) < ∞, where

D(a, b) := \begin{cases}sup_{k∈ℤ}[A_{k} · B_{k}] & for p ≤ q , \\ (∑_{k∈ℤ}[A_{k} · Bk^{\frac{p′}{q′}}]^{\frac{pq}{p−q}} β_{k−1}^{p′})^{\frac{p−q}{pq}} & for p > q . \end{cases}  (2.3)

Moreover, there is a universal C_{p,q} > 0 with

\frac{1}{C_{p,q}} D(a, b) ≤ ∥S_{a,b} : ℓ_{p}(ℤ) → ℓ_{q}(ℤ)∥ ≤ C_{p,q}D(a, b).  (2.4)

Our next aim is to describe the compactness of S_{a,b} in terms of entropy numbers. For the introduction of entropy numbers, let E, F be Banach spaces with unit balls B_{E}, B_{F}. If T : E → F is a linear bounded operator, we set the (dyadic) entropy numbers of T to be

e_{n}(T) := inf{ε > 0 : ∃ y_{1}, …, y_{2^{n−1}} ∈ F s. t. T (B_{E}) ⊆ ⋃_{k=1}^{2^{n−1}}(y_{k} + εB_{F} )},

where “+” means the Minkowski sum. It is well-known that T is compact iff e_{n}(T) → 0, so that it makes sense to consider the speed of decay of e_{n}(T) as a measure for the compactness of T.

To avoid some technical and notational problems, from now on it is assumed that p > 1, hence p′ < ∞. In the following, the number r > 0 defined by

1/r := 1/q + 1/p′  (2.5)

will play a crucial role. Since p > 1, we always have r < ∞.

Later on, we will use the monotone rearrangement δ_{k}^{∗} of a sequence (δ_{k})_{k∈ℤ}. By this we mean that the sequence (δ_{k}^{∗})_{k≥0} is the nonincreasing rearrangement of (δ_{k})_{k≥0}, and that (δ_{k}^{∗})_{k≤−1} is the nondecreasing rearrangement of (δ_{k})_{k≤−1} (i.e., (δ_{−k}^{∗})_{k≥1} is the nonincreasing rearrangement of (δ_{−k})_{k≥1}).

For the nonnegative sequences a and b satisfying (2.1), we define

v_{k} := inf{m ∈ ℤ : ∑_{l≤m} β_{l}^{p′} ≥ 2^{k}}  (2.6)

for any k ∈ ℤ, where inf ∅ := ∞, and with these v_{k}’s we set

δ_{k}(a, b) := \begin{cases}2^{k/p′}( ∑_{l=v_{k}+1}^{v_{k+1}} α_{l}^{q})^{1/q} & if v_{k} < ∞ , \\ 0 & for v_{k} = ∞ . \end{cases}  (2.7)

Here, as well as in the following, the empty sum has to be read as 0. With r given by (2.5) we define

|(a, b)|_{r} := ∥(δ_{k}(a, b))_{k∈ℤ}∥_{r} = (∑_{k∈ℤ}δ_{k}(a, b)^{r})^{1/r}  (2.8)

and

|(a, b)|_{r,∞} := sup_{k≥1} k^{1/r}(δ_{k−1}^{∗}(a, b) + δ_{−k}^{∗}(a, b)).  (2.9)

It is easy to see that

∥(α_{l} β_{l−1})_{l∈ℤ}∥_{r} ≤ |(a, b)|_{r} and |(a, b)|_{r,∞} ≤ c |(a, b)|_{r} .

On the other hand, there are easy examples such that |(a, b)|_{r,∞} < ∞ and ∥(α_{l} β_{l−1})_{l∈ℤ}∥_{r} = ∞, and vice versa, i.e., these two expressions are not comparable.

Let us formulate now the main results of this paper. We have the following general upper estimate for the entropy numbers of S_{a,b}:

Theorem 2.2. Let S_{a,b} map ℓ_{p}(ℤ) into ℓ_{q}(ℤ).

(1) There is a numerical constant c > 0 such that

sup_{n∈ℕ}n e_{n}(S_{a,b}) ≤ c |(a, b)|_{r}.  (2.10)

(2) Whenever the right hand side of (2.10) is finite, we have

lim_{n→∞} n e_{n}(S_{a,b}) = 0.  (2.11)

The next result shows that estimate (2.10) cannot in general be improved to an estimate neither against |(a, b)|_{r,∞} nor against ∥(α_{l} β_{l−1})_{l∈ℤ}∥_{r}:

Theorem 2.3. For any sequence d_{k} > 0, k = 1, 2, … satisfying ∑_{k} d_{k}^{q} < ∞ and ∑_{k} d_{k}^{r} = ∞ there are sequences a = (α_{k})_{k≥1} and b = (β_{k})_{k≥1} such that:

(1) The operator S_{a,b} : ℓ_{p} → ℓ_{q} is bounded.

(2) ∑_{k=1}^{∞}[α_{k+1}β_{k}]^{r} < ∞.

(3) d_{k} = δ_{k}(a, b) = 2^{k/p′}( ∑_{l=v_{k}+1}^{v_{k+1}} α_{l}^{q})^{1/q} for all k ≥ 1

(4) sup_{n∈ℕ}n e_{n}(S_{a,b}) = ∞.

On the other hand, under some additional regularity assumptions about a and b assertion (2.11) holds even under weaker conditions. More precisely, the following is valid.

Proposition 2.4. For a, b given, assume that α_{k}^{q}/β_{k}^{p′} is monotone near ±∞ if q < ∞, or that α_{k} is monotone near ±∞ if q = ∞. Then the condition ∥(α_{l} β_{l−1})_{l∈ℤ}∥_{r} < ∞ implies

lim_{n→∞} n e_{n}(S_{a,b}) = 0.

For sequences b which do not increase too fast (e.g., not superexponentially) at +∞, and do not decrease too fast at −∞, we have the following lower estimate for the entropy numbers of S_{a,b}.

Theorem 2.5. Assume that there is m ≥ 1 such that

|{k′ ∈ ℤ : v_{k′} = v_{k}}| ≤ m  (2.12)

for all k ∈ ℤ. Then, for ρ > 0 with 1/s := ρ − 1/p + 1/q > 0 we have

sup_{n∈ℕ}n^{ρ} e_{n}(S_{a,b} : ℓ_{p} → ℓ_{q}) ≥ c · sup_{n∈ℕ}n^{1/s}(δ_{n−1}^{∗}(a, b) + δ_{−n}^{∗}(a, b)) (2.13)

with c > 0 only depending on p, q, m and ρ. In particular, if ρ = 1, then

sup_{n∈ℕ}n e_{n}(S_{a,b} : ℓ_{p} → ℓ_{q}) ≥ c |(a, b)|_{r,∞} .  (2.14)

Moreover, for ρ and s in (2.13) we have

\underline{lim}_{n→∞} n^{ρ} e_{n}(S_{a,b} : ℓ_{p} → ℓ_{q}) ≥ c \underline{lim}_{n→∞} n^{1/s}(δ_{n−1}^{∗}(a, b) + δ_{−n}^{∗}(a, b)). (2.15)

Remark. Note that for v_{k} < ∞, k ∈ ℤ, i.e., for ∑_{l∈ℤ} β_{l}^{p′} = ∞, condition (2.12) is equivalent to

β_{k} ≤ γ( ∑_{l<k} β_{l}^{p′})^{1/p′} , k ∈ ℤ ,

for some γ > 0.

Of course, condition (2.12) is violated provided that

(a) β_{l} = 0 if l ≤ l_{0} for some l_{0} ∈ ℤ or

(b) ∑_{l∈ℤ}β_{l}^{p′} < ∞ .

Here we have the following weaker variant of Theorem 2.5.

Proposition 2.6. Suppose that there are some k_{0}, k_{1} ∈ ℤ such that in case (a) condition (2.12) holds for all k ≥ k_{0} or in case (b) for all k ≤ k_{1}. Then if 1/s = ρ − 1/p + 1/q > 0, from

sup_{n∈ℕ}n^{ρ}e_{n}(S_{a,b} : ℓ_{p} → ℓ_{q}) < ∞

we conclude that

sup_{n∈ℕ}n^{1/s}(δ_{n−1}^{∗}(a, b) + δ_{−n}^{∗}(a, b)) < ∞

and (2.15) holds for S_{a,b} as well.

Note that in case (a), δ_{−n}(a, b) = 0 for n ≥ n_{0}, and in case (b), δ_{n−1}(a, b) = 0 for n ≥ n_{1} with certain n_{0}, n_{1} ∈ ℤ. Examples (see Proposition 5.3) show that Theorem 2.5 and Proposition 2.6 become false without an additional regularity assumption.

3. Basic Tools

3.1. Connection to Integral Operators. Let ρ, ψ : (0, ∞) → [0, ∞) be measurable functions with

ρ ∈ L_{q}(x, ∞) and ψ ∈ L_{p′}(0, x)  (3.1)

for any x ∈ (0, ∞). Then for f ∈ L_{p}(0, ∞) the function

(T_{ρ,ψ}(f))(s) := ρ(s) ∫_{0}^{s} ψ(t)f(t) dt  (3.2)

is well-defined. The operator T_{ρ,ψ} is called the Volterra integral operator induced by ρ and ψ. There is a close connection between summation and integral operators. Generally speaking, one can always consider a summation operator as “part” of a special Volterra integral operator. This allows one to use, for these operators, results established in [10].

Let us denote Δ_{k} := [2^{k}, 2^{k+1}) for k ∈ ℤ, and set

I_{k} := Δ_{2k} = [2^{2k}, 2^{2k+1}) and J_{k} := Δ_{2k+1} = [2^{2k+1}, 2^{2k+2}). (3.3)

For any set X ⊆ (0, ∞), let 1_{X} be the indicator function of X. We denote by ℓ_{0}(ℤ) := ℝ^{ℤ} the space of all real sequences and set L_{0}(0, ∞) to be the space of all equivalence classes of measurable functions. It is easily checked that for any p ∈ [1, ∞] the mappings

Φ_{I}^{p} : ℓ_{0}(ℤ) → L_{0}(0, ∞), (x_{k})_{k∈ℤ} ↦ ∑_{k∈ℤ}x_{k}1_{I_{k}}|I_{k}|^{−1/p}  (3.4)

and

Φ_{J}^{p} : ℓ_{0}(ℤ) → L_{0}(0, ∞), (x_{k})_{k∈ℤ} ↦ ∑_{k∈ℤ}x_{k}1_{J_{k}}|J_{k}|^{−1/p}  (3.5)

induce isometric embeddings of ℓ_{p}(ℤ) into L_{p}(0, ∞) (for p = ∞ let, as usual, 1/p = 0). Given now sequences a = (α_{k})_{k∈ℤ} and b = (β_{k})_{k∈ℤ}, we set

ρ := Φ_{I}^{q}(a) and ψ := Φ_{J}^{p′}(b) ,  (3.6)

i.e., we have

ρ = ∑_{k∈ℤ}α_{k}1_{I_{k}}|I_{k}|^{−1/q} and ψ = ∑_{k∈ℤ}β_{k}1_{J_{k}}|J_{k}|^{−1/p′} .  (3.7)

The connection between summation and integral operators reads as follows:

Proposition 3.1. In the above setting, ρ and ψ defined in (3.6) satisfy (3.1) if and only if a and b satisfy (2.1), and it holds

T_{ρ,ψ} ∘ Φ_{J}^{p} = Φ_{I}^{q} ∘ S_{a,b}.  (3.8)

Moreover, T_{ρ,ψ} is bounded from L_{p} to L_{q} iff S_{a,b} is so from ℓ_{p} to ℓ_{q}. In this case there is an operator Q : L_{p}(0, ∞) → ℓ_{p}(ℤ) with ∥Q∥ ≤ 1 such that for all f ∈ L_{p}(0, ∞) we have

∥T_{ρ,ψ}(f)∥_{q} = ∥S_{a,b} ∘ Q(f)∥_{q} .  (3.9)

In particular,

∥T_{ρ,ψ} : L_{p} → L_{q}∥ = ∥S_{a,b} : ℓ_{p} → ℓ_{q}∥ .  (3.10)

Proof. The equivalence of (3.1) and (2.1) is clear by the definition of ρ and ψ and of Φ_{I}^{q} and Φ_{J}^{p′}, respectively. We easily compute

T_{ρ,ψ}(Φ_{J}^{p}(x_{k})) = ∑_{k∈ℤ}α_{k}|I_{k}|^{−1/q}[∑_{l<k} β_{l}x_{l}]1_{I_{k}} = Φ_{I}^{q}(S_{a,b}(x_{k})) .

This verifies that (3.8) holds.

If T_{ρ,ψ} is bounded, then, due to (3.8), so is the operator S_{a,b}, and ∥S_{a,b}∥ ≤ ∥T_{ρ,ψ}∥. Conversely, if S_{a,b} is bounded and f ∈ L_{p}((0, ∞)), set

Q(f) := (|J_{k}|^{−1/p′} ∫_{J_{k}} f(t) dt)_{k∈ℤ} .  (3.11)

Due to Hölder’s inequality, Q(f) ∈ ℓ_{p} with ∥Q(f)∥_{p} ≤ ∥f∥_{p}, and, additionally, we find

∥T_{ρ,ψ}(f)∥_{q}^{q} = ∑_{k∈ℤ}α_{k}^{q}|∑_{l<k} β_{l}|J_{l}|^{−1/p′} ∫_{J_{l}} f(t) dt|^{q} = ∥S_{a,b}(Q(f))∥_{q}^{q}. (3.12)

This proves (3.9), and implies, in particular, that ∥T_{ρ,ψ}∥ ≤ ∥S_{a,b}∥ which completes the proof.

As a simple corollary we have

Corollary 3.2. For a and b satisfying (2.1), and ρ and ψ defined as in (3.7) we have

(1/2) e_{n}(T_{ρ,ψ}) ≤ e_{n}(S_{a,b}) ≤ 2 e_{n}(T_{ρ,ψ}).

Proof. First note that

e_{n}(T) ≤ 2 e_{n}(J ∘ T)

for any operator T : E → F and any isometric embedding J : F → F_{0}. Hence, by (3.8) we conclude

e_{n}(S_{a,b}) ≤ 2 e_{n}(Φ_{I}^{q} ∘ S_{a,b}) = 2 e_{n}(T_{ρ,ψ} ∘ Φ_{J}^{p}) ≤ 2 e_{n}(T_{ρ,ψ}) ∥Φ_{J}^{p}∥ ≤ 2 e_{n}(T_{ρ,ψ}) .

On the other hand, by (3.9) we have (use Lemma 4.2 in [10], yet observe that there the entropy numbers were defined slightly different so that additional factor 2 had to be added)

e_{n}(T_{ρ,ψ}) ≤ 2 e_{n}(S_{a,b} ∘ Q) ≤ 2 e_{n}(S_{a,b}) ∥Q∥ ≤ 2 e_{n}(S_{a,b}),

which completes the proof.

For later use we cite now some of the main results in [10] about weighted Volterra integral operators. Let ρ and ψ now be arbitrary nonnegative measurable functions on (0, ∞) satisfying (3.1). For s > 0, set

R(s) := ∥ρ∥_{L_{q}(s,∞)}  and  Ψ(s) := ∥ψ∥_{L_{p′}(0,s)}.  (3.13)

The following version of the Maz’ja–Rosin Theorem can be found in [10], Theorem 6.1:

Theorem 3.3. Let 1 ≤ p, q ≤ ∞ and ρ, ψ ≥ 0 on (0, ∞). Then T_{ρ,ψ} is bounded from L_{p}(0, ∞) into L_{q}(0, ∞) iff D(ρ, ψ) < ∞, where

D(ρ, ψ) := \begin{cases}sup_{s>0} R(s)Ψ(s) & if p ≤ q, \\ (∫_{0}^{∞}[R(s) · Ψ(s)^{p′/q′}]^{\frac{pq}{p−q}} ψ(s)^{p′} ds)^{\frac{p−q}{pq}} & for p > q. \end{cases} (3.14)

Moreover, there are universal constants c_{p,q}, C_{p,q} > 0 such that

c_{p,q}D(ρ, ψ) ≤ ∥T_{ρ,ψ} : L_{p} → L_{q}∥ ≤ C_{p,q}D(ρ, ψ) .  (3.15)

Now we turn to upper estimates for the entropy numbers of T_{ρ,ψ}. Therefore set

u_{k} := inf{s > 0 : ∫_{0}^{s} ψ(t)^{p′} dt ≥ 2^{k}},  k ∈ ℤ ,  (3.16)

and define

δ_{k}(ρ, ψ) := 2^{k/p′}( ∫_{u_{k}}^{u_{k+1}}ψ(t)^{q} dt)^{1/q}.  (3.17)

Similarly to the setting for a and b, with r given by (2.5) define

|(ρ, ψ)|_{r} := ∥(δ_{k}(ρ, ψ))_{k∈ℤ}∥_{r} = (∑_{k∈ℤ}δ_{k}(a, b)^{r})^{1/r}  (3.18)

and

|(ρ, ψ)|_{r,∞} := sup_{k≥1} k^{1/r}(δ_{k}^{∗}(a, b) + δ_{−k+1}^{∗}(a, b)).  (3.19)

With this notation we have ([10], Theorem 4.6):

Theorem 3.4. Let p > 1 and 1 ≤ q ≤ ∞. Then for all ρ, ψ as above, the following statements are valid.

(1) For T_{ρ,ψ} as in (3.2), we have sup_{n} n e_{n}(T_{ρ,ψ}) ≤ C |(ρ, ψ)|_{r} .

(2) Whenever |(ρ, ψ)|_{r} < ∞, we even have

\overline{lim}_{n→∞} n e_{n}(T_{ρ,ψ}) ≤ C ∥ρ ψ∥_{r} .  (3.20)

Remarks. (1) There exist functions ρ and ψ (cf. [10], Theorem 2.3) with ∥ρ ψ∥_{r} < ∞ such that \overline{lim}_{n→∞} n e_{n}(T_{ρ,ψ}) = ∞ . But note that since these ρ’s and ψ’s are not of form (3.7) with suitable sequences a and b, hence they cannot be used to construct similar examples for summation operators. However to prove Theorem 2.3 the general ideas may be taken over in order.

(2) In [10] lower estimates for e_{n}(T_{ρ,ψ}) were also proved. But unfortunately all these lower bounds turn out to be zero in the case of disjointly supported functions ρ and ψ. So they do not provide any information about lower bounds for e_{n}(S_{a,b}).

3.2. Entropy Numbers of Diagonal Operators. For a given sequence σ = (σ_{k})_{k≥1} of nonnegative numbers, one defines the diagonal operator

D_{σ}(x) := (σ_{k}x_{k})_{k∈ℕ} , x = (x_{k})_{k∈ℕ} .  (3.21)

Considered as a mapping from ℓ_{p}(ℕ) to ℓ_{q}(ℕ), the entropy of these operators can be estimated by means of σ_{k}’s. As an example, we have the following special case of general results in [1] or [13].

Theorem 3.5. Let p, q ∈ [1, ∞] be arbitrary, and let σ = (σ_{k})_{k≥0} be a nonnegative bounded sequence of real numbers. Let ρ > 0 satisfy 1/s := ρ − 1/p + 1/q > 0. Then there are constants c_{1}, c_{2} > 0 depending on p, q and ρ only such that

c_{1} sup_{n∈ℕ}n^{1/s}σ_{n}^{∗} ≤ sup_{n∈ℕ}n^{ρ} e_{n}(D_{σ} : ℓ_{p}(ℕ) → ℓ_{q}(ℕ)) ≤ c_{2} sup_{n∈ℕ}n^{1/s}σ_{n}^{∗}. (3.22)

For the lower bound, one even has

n^{1/q−1/p}σ_{n}^{∗} ≤ 6 e_{n}(D_{σ}).  (3.23)

Let us draw an easy conclusion out of this:

Corollary 3.6. In the above setting,

c_{1} \underline{lim}_{n→∞} n^{1/s}σ_{n}^{∗} ≤ \underline{lim}_{n→∞} n^{ρ} e_{n}(D_{σ})  (3.24)

as well as

\overline{lim}_{n→∞} n^{ρ} e_{n}(D_{σ}) ≤ 2 c_{2} \overline{lim}_{n→∞} n^{1/s}σ_{n}^{∗}  (3.25)

holds.

Proof. Without loss of generality, assume that σ_{n}’s are already nonincreasing, e.g., σ_{n} = σ_{n}^{∗}. Then estimate (3.24) is an immediate consequence of (3.23).

To prove (3.25), fix a number N ∈ ℕ and set

D^{N}((x_{i})_{i=1}^{∞}) := (σ_{1}x_{1}, …, σ_{N}x_{N}, 0, …), D_{0}^{N} := D_{σ} − D^{N}.

Because of e_{2n−1}(D_{σ}) ≤ e_{n}(D^{N})+e_{n}(D_{0}^{N}) and the exponential decay of e_{n}(D^{N}) as n → ∞ (cf. [2], 1.3.36), we get

\overline{lim}_{n→∞} n^{ρ}e_{n}(D_{σ}) ≤ 2 \overline{lim}_{n→∞} n^{ρ}e_{n}(D_{0}^{N}) ≤ 2 sup_{n∈ℕ}n^{ρ}e_{n}(D_{0}^{N}).  (3.26)

Now D_{0}^{N} is not a diagonal operator with nonincreasing entries, but this difficulty can be easily overcome. For x = (x_{i})_{i∈ℕ} let

S_{N}^{−}(x) := (x_{i+N})_{i∈ℕ} and S_{N}^{+}(x) := (0, …, 0, x_{1}, …)

be the operators of shifting N times to the right or to the left in ℓ_{p}(ℕ) or ℓ_{q}(ℕ), respectively. If

D_{1}^{N}((x_{i})_{i=1}^{∞}) := (x_{1}σ_{N+1}, x_{2}σ_{N+2}, …)

is the diagonal operator defined by (σ_{N+i})_{i≥1}, we clearly have

D_{0}^{N} = S_{N}^{+} ∘ D_{1}^{N} ∘ S_{N}^{−} and D_{1}^{N} = S_{N}^{−} ∘ D_{0}^{N} ∘ S_{N}^{+},

and therefore (note that both shift operators have norm one) it holds by the multiplicity of the entropy numbers

e_{n}(D_{0}^{N}) = e_{n}(D_{1}^{N}).

But for D_{1}^{N} Theorem 3.5 applies and leads to

sup_{n∈ℕ}n^{ρ}e_{n}(D_{1}^{N}) ≤ c_{2} sup_{n∈ℕ}n^{1/s}σ_{n+N}

with s > 0 defined in this theorem. Since n^{1/s} ≤ (n + N)^{1/s}, this yields

sup_{n∈ℕ}n^{ρ}e_{n}(D_{0}^{N}) ≤ c_{2} sup_{n∈ℕ}(n + N)^{1/s}σ_{n+N} = c_{2} sup_{m>N} m^{1/s}σ_{m}.  (3.27)

Now we combine (3.26) and (3.27) to find that

\overline{lim}_{n→∞} n^{ρ}e_{n}(D_{σ}) ≤ 2c_{2} sup_{n>N} n^{1/s}σ_{n}  (3.28)

holds for all N ∈ ℕ. Taking the infimum over the right side of (3.28), we end up with

\overline{lim}_{n→∞} n^{ρ}e_{n}(D_{σ}) ≤ 2c_{2} inf_{N∈ℕ}(sup_{n>N} n^{1/s}σ_{n}) = 2c_{2} \overline{lim}_{n→∞} n^{1/s}σ_{n},

which proves (3.25).

It is of more use for us to consider diagonal operators mapping ℓ_{p}(ℤ) into ℓ_{q}(ℤ), generated, say, by a sequence (σ_{k})_{k∈ℤ}. However these operators are isomorphic to the product of two diagonal operators generated by (σ_{k−1})_{k≥1} and (σ_{−k})_{k≥1}. It is then quite easy to establish

Corollary 3.7. Let p, q ∈ [1, ∞] be arbitrary, and let σ = (σ_{k})_{k∈ℤ} be a nonnegative bounded sequence of real numbers. For given ρ > 0 assume 1/s := ρ − 1/p + 1/q > 0 . Then there are constants c_{1}, c_{2} > 0 depending only on p, q and ρ such that

c_{1} sup_{n∈ℕ}n^{1/s}(σ_{n−1}^{∗} + σ_{−n}^{∗}) ≤ sup_{n∈ℕ}n^{ρ}e_{n}(D_{σ}) ≤ c_{2} sup_{n∈ℕ}n^{1/s}(σ_{n−1}^{∗} + σ_{−n}^{∗}) (3.29)

and, moreover,

\underline{lim}_{n→∞} n^{ρ}e_{n}(D_{σ}) ≥ c_{1} \underline{lim}_{n→∞} n^{1/s}(σ_{n−1}^{∗} + σ_{−n}^{∗})

as well as

\overline{lim}_{n→∞} n^{ρ}e_{n}(D_{σ}) ≤ c_{2} \overline{lim}_{n→∞} n^{1/s}(σ_{n−1}^{∗} + σ_{−n}^{∗}).

4. Proof of the Results

Let us start with the proof of Theorem 2.1. Proof of Theorem 2.1. Let a = (α_{k})_{k∈ℤ} and b = (β_{k})_{k∈ℤ} be given, satisfying (2.1), and define ρ and ψ by means of (3.7). Then R(s) and Ψ(s) are given by (3.13). Because of Proposition 3.1 and Theorem 3.3 we have the estimate

\frac{1}{C_{p,q}} D(ρ, ψ) ≤ ∥S_{a,b} : ℓ_{p}(ℤ) → ℓ_{q}(ℤ)∥ ≤ C_{pq,}D(ρ, ψ)

with D(ρ, ψ) given in (3.14). So it remains to show that with D(a, b) from (2.3) we have

c_{1}D(ρ, ψ) ≤ D(a, b) ≤ c_{2}D(ρ, ψ) .  (4.1)

Let us treat the case p ≤ q first. Take any s ∈ I_{k} for some k ∈ ℤ. Then on the one hand we have R(s) ≤ A_{k}, while, on the other hand, Ψ(s) = B_{k}. Now assume s ∈ J_{k} for some k ∈ ℤ. Then R(s) = A_{k+1}, while Ψ(s) ≤ B_{k+1}. Combining both cases, we find

sup_{s>0} R(s) · Ψ(s) ≤ sup_{k∈ℤ} [A_{k} · B_{k}] .  (4.2)

Since A_{k} · B_{k} = R(2^{2k}) · Ψ(2^{2k}), we even have equality in (4.2), which proves (4.1) for p ≤ q.

Let us now assume p > q. First note that the integral in the definition of D(ρ, ψ) has only to be taken over ⋃_{k∈ℤ} J_{k} . But if s ∈ J_{k−1}, then R(s) = A_{k} while Ψ(s) ≤ B_{k}, hence (recall ψ(s) = β_{k−1}|J_{k−1}|^{−1/p′})

D(ρ, ψ) ≤ D(a, b) .

Conversely, whenever s ∈ [2^{2k+3/2}, 2^{2k+2}] ⊆ J_{k}, we have R(s) = A_{k+1}, while

Ψ(s) ≥ (B_{k}^{p′} + 2^{−1}β_{k}^{p′})^{1/p′} ≥ 2^{−1/p′}B_{k+1} .

Consequently, we obtain

(∫_{J_{k}} [R(s) · Ψ(s)^{p′/q′}]^{\frac{pq}{p−q}} ψ(s)^{p′} ds)^{\frac{p−q}{pq}} ≥ 2^{−1/q′} [(A_{k+1} · B_{k+1}^{p′/q})^{\frac{pq}{p−q}} β_{k}^{p′}]^{\frac{p−q}{pq}} ,

yielding

D(a, b) ≤ 2^{1/q′} D(ρ, ψ) ,

which proves (4.2) in this case as well. Proof of Theorem 2.2. Let a, b be given, and ρ, ψ be defined via (3.7). In view of Proposition 3.1 and Theorem 3.4 we have to show only that

δ_{k}(a, b) = δ_{k}(ρ, ψ),  (4.3)

where δ_{k}’s are defined in (2.7) and (3.17), respectively. But for v_{k} as in (2.6) and u_{k} as in (3.16), we have u_{k} ∈ J̄_{v_{k}} = [2^{2k+1}, 2^{2k+2}]. Inserting this into the definition of δ_{k}’s yields (4.3), and hence Theorem 2.2. Proof of Theorem 2.3. We treat the case q < ∞ first. Let d_{k} be given with

∑_{k∈ℕ}d_{k}^{q} < ∞ and ∑_{k∈ℕ}d_{k}^{r} = ∞.  (4.4)

We assume 0 < d_{k} ≤ 1. For any k ∈ ℕ, we can find s_{k} ∈ ℕ such that

s_{k} ≤ \frac{1}{d_{k}^{q}} ≤ 2s_{k}.  (4.5)

According to our assumptions, it is possible to find a partition (K_{m})_{m∈ℕ} of ℕ with inf K_{m+1} = sup K_{m} + 1, and

∑_{k∈K_{m}} d_{k}^{r} ≥ m^{r}.  (4.6)

Set ν_{m} := max_{k∈K_{m}} d_{k}^{−1}. Then for any k ∈ K_{m} there are n_{k} ∈ ℕ such that

(ν_{m}d_{k})^{r} ≤ n_{k} ≤ 2(ν_{m}d_{k})^{r}.  (4.7)

Further we define

γ_{k,j} := ∑_{l<k}(1 + 2s_{l}) n_{k} + (j − 1)(1 + 2s_{k})  (4.8)

for k ∈ ℕ and j = 1, …, n_{k} + 1. In particular, γ_{1,1} = 0 and γ_{k,n_{k}+1} = γ_{k+1,1}. Now we are ready to define a partition of ℕ suitable for our purposes: For any k ∈ ℕ and j = 1, …, n_{k} set

A_{k,j}^{+} := {γ_{k,j} + 1, …, γ_{k,j} + s_{k}},  (4.9)

as well as

I_{k,j} := {γ_{k,j} + s_{k} + 1}  (4.10)

and

A_{k,j}^{−} := {γ_{k,j} + s_{k} + 2, …, γ_{k,j+1}}.  (4.11)

Clearly, |A_{k,j}^{+}| = |A_{k,j}^{−}| = s_{k}, and

A_{k,1}^{+} ≺ I_{k,1} ≺ A_{k,1}^{−} ≺ A_{k,2}^{+} ≺ ⋯ ≺ A_{k,n_{k}}^{−} ≺ A_{k+1,1}^{+} ≺ ⋯,  (4.12)

where ≺ denotes the natural ordering of intervals in ℕ. Denoting

U_{k} := ⋃_{j=1}^{n_{k}}(A_{k,j}^{+} ∪ I_{k,j} ∪ A_{k,j}^{−}) = {γ_{k,1} + 1, …, γ_{k+1,1}} ,  (4.13)

it is clear that ℕ = ⋃_{k≥1} U_{k}. Let us define β_{0} := 1, and

β_{l} := \frac{2^{(k−1)/p′}}{n_{k}^{1/p′}(1 + 2s_{k})^{1/p′}}  for l ∈ U_{k} .  (4.14)

This way we ensure (note that |U_{k}| = n_{k}(1 + 2s_{k})) that for v_{k}’s defined in (2.6) we have v_{k} = γ_{k,1} for k ≥ 1 and v_{k} = 0 if k ≤ 0 so that

U_{k} = {v_{k} + 1, …, v_{k+1}},  k ≥ 1.  (4.15)

Now, let us set

α_{l} := \begin{cases}\frac{2^{−k/p′}d_{k}}{n_{k}^{1/q}} & if l ∈ I_{k,j} for some k ∈ ℕ, j = 1, …, n_{k}, \\ 0 & otherwise. \end{cases} (4.16)

We can easily compute that

∑_{l=v_{k}+1}^{v_{k+1}} a_{l}^{q} = ∑_{l=1}^{n_{k}} \frac{2^{−kq/p′}d_{k}^{q}}{n_{k}} = 2^{−kq/p′}d_{k}^{q}.  (4.17)

In particular, we know a = (α_{l})_{l∈ℤ} ∈ ℓ_{q}(ℤ), and δ_{k}(a, b) = d_{k} and hence (3) holds.

If l ∈ I_{k,j}, then l − 1 ∈ U_{k}, which implies

∑_{l∈ℕ}(α_{l}β_{l−1})^{r} = ∑_{k∈ℕ} ∑_{j=1}^{n_{k}}(\frac{2^{−k/p′}d_{k}}{n_{k}^{1/q}} · \frac{2^{(k−1)/p′}}{n_{k}^{1/p′}(1 + 2s_{k})^{1/p′}} )^{r}

≤ ∑_{k∈ℕ}n_{k} \frac{d_{k}^{r}}{n_{k}(1 + 2s_{k})^{r/p′}} ≤ ∑_{k∈ℕ}(d_{k}/(2s_{k})^{1/p′})^{r}.

Because of the definition of s_{k} we assured 2s_{k} ≥ d_{k}^{−q}, and hence we can estimate further

∑_{l∈ℕ}(α_{l}β_{l−1})^{r} ≤ ∑_{k∈ℕ}d_{k}^{(1+q/p′)r} = ∑_{k∈ℕ}d_{k}^{q} < ∞,  (4.18)

i.e., (2) is valid. Using Theorem 2.1 one easily checks that S_{a,b} is bounded, i.e.,(1) is satisfied as well.

So it remains to verify (4). To do so, let us fix k ∈ ℕ and j ∈ {1, …, n_{k}}. Then consider the vector x^{k,j} = (x_{l}^{k,j})_{l∈ℤ} given by

x_{l}^{k,j} := \begin{cases}s_{k}^{−1/p} & if l ∈ A_{k,j}^{+}, \\ −s_{k}^{−1/p} & for l ∈ A_{k,j}^{−}, \\ 0 & otherwise . \end{cases}  (4.19)

Recall that |A_{k,j}^{+}| = |A_{k,j}^{−}| = s_{k}, so ∥x^{k,j}∥_{p} ≤ 2. Setting y^{k,j} := S_{a,b}(x^{k,j}), we have

y_{l}^{k,j} = \begin{cases}s_{k}^{1/p′} \frac{d_{k}}{n_{k}^{1/r}(1 + 2s_{k})^{1/p′}} & if l ∈ I_{k,j} \\ 0 & otherwise . \end{cases}  (4.20)

For m ∈ ℕ, set N_{m} := ∑_{k∈K_{m}} n_{k}. Let us denote by e_{k,j} the standard basis of ℓ_{p}^{N_{m}} or ℓ_{q}^{N_{m}}, respectively, where k runs through K_{m} and j through 1, …, n_{k}. Then we define two mappings, an injection

X_{m} : ℓ_{p}^{N_{m}} → ℓ_{p}(ℤ), e_{k,j} ↦ x^{k,j},  (4.21)

and a quasi-projection

Y_{m} : ℓ_{q}(ℤ) → ℓ_{q}^{N_{m}}, e_{l} ↦ \begin{cases}e_{k,j} & if l ∈ I_{k,j}, \\ 0 & otherwise . \end{cases}  (4.22)

We have ∥X_{m}∥ ≤ 2 and ∥Y_{m}∥ ≤ 1. Additionally,

Y_{m} ∘ S_{a,b} ∘ X_{m} = D_{σ^{m}},  (4.23)

where D_{σ^{m}} is the diagonal operator generated by the sequence

σ_{k,j}^{m} = s_{k}^{1/p′} \frac{d_{k}}{n_{k}^{1/r}(1 + 2s_{k})^{1/p′}} ≥ 3^{−1/p′} \frac{d_{k}}{n_{k}^{1/r}} ≥ 1/6 ν_{m}^{−1}.  (4.24)

(Here we have used the definition of n_{k}.) It is well-known (cf. [13] or (3.23)) that

Ne_{N} (id : ℓ_{p}^{N} → ℓ_{q}^{N} ) ≥ (1/6) N^{1/r},  (4.25)

so, putting things together and using (4.23), (4.24) and (4.25) we get

N_{m}e_{N_{m}}(S_{a,b}) ≥ 1/6 ν_{m}^{−1}N_{m}e_{N_{m}}(id : ℓ_{p}^{N_{m}} → ℓ_{q}^{N_{m}}) ≥ c ν_{m}^{−1}N_{m}^{1/r}. (4.26)

But due to the definition of n_{k} we have

N_{m} ≥ ν_{m}^{r} ∑_{k∈K_{m}} d_{k}^{r},  (4.27)

which, inserted in (4.26), yields, with regard to (4.6),

N_{m}e_{N_{m}}(S_{a,b}) ≥ c( ∑_{k∈K_{m}} d_{k}^{r})^{1/r} ≥ c m,  (4.28)

so that \overline{lim}_{n→∞} n e_{n}(S_{a,b}) = ∞.

Let now q = ∞, so r = p′. This means we have sup_{n} d_{n} < ∞ while ∑_{n} d_{n}^{p′} = ∞. We adapt the construction above, changing only the definition of variables where q is involved. So define s_{k} := 2^{k}, and set α_{l} := 2^{−k/p′}d_{k} for l ∈ I_{k,j}, while all other variables are defined as above (they have different values now, though). It is trivial to see that α_{l} ∈ ℓ_{∞} and that δ_{k}(a, b) = d_{k}. Further, we estimate

∑_{l∈ℤ}[α_{l}β_{l−1}]^{p′} = ∑_{k∈ℕ} ∑_{j=1}^{n_{k}} \frac{d_{k}^{p′}}{n_{k}(1 + 2s_{k})^{p′}} ≤ (sup_{n∈ℕ}d_{n}^{p′}) ∑_{k∈ℕ}2^{−kp′} < ∞ .

As above, Theorem 2.1 shows the boundedness of S_{a,b}. The construction of X_{m} and Y_{m} remains exactly the same, and since (4.25) remains valid for q = ∞, the arguments given above apply also here, so (4.28) holds in this case, too.

Next we want to prove Proposition 2.4. For this it clearly suffices to show the following:

Lemma 4.1. Assume α_{l}^{q}/β_{l−1}^{p′} to be monotone near ±∞ if q < ∞, or α_{k} to be monotone near ±∞ for q = ∞. Then (α_{l} β_{l−1})_{l∈ℤ} ∈ ℓ_{r} always implies |(a, b)|_{r} < ∞.

Remark. It is notable that in the setting of Lemma 4.1, one cannot establish the estimate |(a, b)|_{r} ≤ c ∥(α_{l} β_{l−1})_{l}∥_{r}, as simple examples show.

Proof of Lemma 4.1. Let us first assume q < ∞. Since all possible situations can be treated similarly, we treat, e.g., the case where α_{l}^{q}/β_{l−1}^{p′} is monotonically increasing near ∞, say, for k > k_{0}. So let k > k_{0} be given with v_{k+2} < ∞ and v_{k} ≠ v_{k+1}. Then, α_{l}^{q} ≤ β_{l−1}^{p′}α_{v_{k+1}}^{q}(β_{v_{k+1}−1})^{−p′} holds for any l ∈ {v_{k} +1, …, v_{k+1}} and hence

δ_{k}(a, b)^{r} = 2^{kr/p′}( ∑_{l=v_{k}+1}^{v_{k+1}} α_{l}^{q})^{r/q} ≤ 2^{kr/p′}( ∑_{l=v_{k}}^{v_{k+1}−1} β_{l}^{p′})^{r/q} \frac{α_{v_{k+1}}^{r}}{β_{v_{k+1}−1}^{rp′/q}}

≤ 2^{k+1} \frac{α_{v_{k+1}}^{r}}{β_{v_{k+1}−1}^{rp′/q}} .  (4.29)

On the other hand, for any l ∈ {v_{k+1} + 1, …, v_{k+2} + 1} we have of course α_{l}^{q} ≥ β_{l−1}^{p′}α_{v_{k+1}+1}^{q}(β_{v_{k+1}})^{−p′}, which implies

∑_{l=v_{k+1}+1}^{v_{k+2}+1} α_{l}^{r}β_{l−1}^{r} ≥ ( ∑_{l=v_{k+1}}^{v_{k+2}} β_{l}^{p′})\frac{α_{v_{k+1}+1}^{r}}{β_{v_{k+1}}^{rp′/q}} ≥ 2^{k+1} \frac{α_{v_{k+1}}^{r}}{β_{v_{k+1}−1}^{rp′/q}} .  (4.30)

We combine (4.29) and (4.30) to see that

δ_{k}(a, b)^{r} ≤ ∑_{l=v_{k+1}+1}^{v_{k+2}+1} α_{l}^{r}β_{l−1}^{r}  (4.31)

holds for all k > k_{0} with v_{k+2} < ∞ and v_{k} < v_{k+1}. In order to prove that ∑_{k>k_{0}} δ_{k}(a, b)^{r} < ∞, in view of (4.31) it suffices to show that with K := {k ∈ ℤ : k > k_{0} and v_{k} < v_{k+1} ≤ v_{k+2} < ∞} we have

∑_{k∈K} ∑_{l=v_{k+1}+1}^{v_{k+2}+1} α_{l}^{r}β_{l−1}^{r} ≤ 2 ∑_{l∈ℤ}α_{l}^{r}β_{l−1}^{r}.  (4.32)

This takes place if every summand appearing in the right-hand sum appears at most two times in the sums on the left-hand side. Thus we have to show only that for any l ∈ ℤ the set

M_{l} := {k ∈ ℤ : v_{k} ≠ v_{k+1}, v_{k+1} + 1 ≤ l ≤ v_{k+2} + 1}

consists of at most two elements. So assume that k_{1}, k_{2}, k_{3} ∈ M_{l} with k_{1} < k_{2} < k_{3}. Since v_{k_{3}} < v_{k_{3}+1}, we get

v_{k_{1}+2} ≤ v_{k_{2}+1} ≤ v_{k_{3}} < v_{k_{3}+1} .  (4.33)

But by the definition of M_{l} we must have v_{k_{3}+1} ≤ l − 1 ≤ v_{k_{1}+2}, which is a contradiction. So we know |M_{l}| ≤ 2, and (4.32) is shown.

Let us now consider the case q = ∞, so r = p′. Again, we demonstrate only the summability in +∞, provided that α_{k} is monotonically increasing for k > k_{0}. The arguments above have only to be modified slightly: For any k > k_{0} with v_{k} < v_{k+1} we have

δ_{k}(a, b)^{p′} = 2^{k} sup_{v_{k}<l≤v_{k+1}} α_{l}^{p′} = 2^{k}α_{v_{k+1}}^{p′},

while

∑_{l=v_{k+1}+1}^{v_{k+2}+1} α_{l}^{p′}β_{l−1}^{p′} ≥ α_{v_{k+1}+1}^{p′}( ∑_{l=v_{k+1}}^{v_{k+2}} β_{l}^{p′}) ≥ α_{v_{k+1}}^{p′}2^{k+1}.

So we conclude that it suffices to verify (4.32) for the case q = ∞, which is proved completely analogously as above for the case q < ∞.

Next we prove Theorem 2.5. To do so we construct suitable “embeddings” of diagonal operators. Let m ∈ ℕ be a number for which the regularity condition (2.12) holds. For l ∈ ℤ we set

l + (2m + 4) ℤ := {l + (2m + 4)k : k ∈ ℤ}  (4.34)

as usual. If l ∈ {0, …, 2m + 3}, then we define

π_{l} : ℤ → l + (2m + 4) ℤ,  π_{l}(k) := l + (2m + 4)k .  (4.35)

Now let D^{l} be a diagonal operator generated by the sequence

σ_{k}^{l} := δ_{π_{l}(k)}(a, b) = 2^{π_{l}(k)/p′}( ∑_{j=v_{π_{l}(k)}+1}^{v_{π_{l}(k)+1}} α_{j}^{q})^{1/q}.  (4.36)

With these operators, we have

Proposition 4.2. For any l ∈ {0, …, 2m + 3}, there is a bounded linear operator X^{l} : ℓ_{p}(ℤ) → ℓ_{p}(ℤ) with ∥X^{l}∥ ≤ C_{m} and

∥D^{l}(x)∥ ≤ 2∥S_{a,b} ∘ X^{l}(x)∥  (4.37)

for all x ∈ ℓ_{p}(ℤ).

Before we prove Proposition 4.2, let us mention a useful lemma:

Lemma 4.3. Assume v_{k} ≠ v_{k+1}. Then

2^{k−1} ≤ ∑_{j=v_{k−1}}^{v_{k+1}−1} β_{l}^{p′} ≤ 2^{k+1}.  (4.38)

Proof. The upper estimate is trivial. For the lower estimate, combine

∑_{l<v_{k+1}} β_{l}^{p′} ≥ ∑_{l≤v_{k}} β_{l}^{p′} ≥ 2^{k} with ∑_{l<v_{k−1}} β_{l}^{p′} < 2^{k−1}.

Proof of Proposition 4.2. We prove the proposition for the case l = 0, the other cases can be treated analogously. Let (e_{k})_{k∈ℤ} be the standard base of ℓ_{p}(ℤ). We define the mapping X^{0} by setting X^{0}(e_{k}) := x^{k}, where x^{k}’s are defined below. The construction guarantees that the vectors y^{k} := S_{a,b}(x^{k}) are disjointly supported, that ∥x^{k}∥_{p} ≤ c, and moreover ∥y^{k}∥ ≥ 1/2σ_{k}^{0} with σ_{k}^{0} defined in (4.36) for l = 0. From this, one can directly conclude the assertion. So let k ∈ ℤ be given, and set k_{0} := (2m + 4)k = π_{0}(k). If v_{k_{0}} = v_{k_{0}+1}, then σ_{k}^{0} = δ_{k_{0}}(a, b) = 0, and we set x^{k} := 0. Otherwise, we distinguish two cases:

Case 1: v_{k_{0}−1} ≠ v_{k_{0}}.

Then we fix k_{1} ∈ {k_{0} + m + 2, k_{0} + 2m + 2} such that v_{k_{1}} ≠ v_{k_{1}+1} (this is possible due to our assumption (2.12) ). Since k_{1} − 1 > k_{0} + 1 + m, (2.12) also shows that v_{k_{1}−1} > v_{k_{0}+1}. Now we define x^{k} = (x_{j}^{k})_{j∈ℤ} via

x_{j}^{k} := 2^{−k_{0}/p} · \begin{cases}β_{j}^{p′−1} & if j ∈ \{v_{k_{0}−2}, …, v_{k_{0}} − 1\}, \\ −C_{k} β_{j}^{p′−1} & for j ∈ \{v_{k_{1}−1}, …, v_{k_{1}+1} − 1\}, \\ 0 & otherwise, \end{cases}  (4.39)

where

C_{k} := ( ∑_{j=v_{k_{0}−2}}^{v_{k_{0}}−1} β_{l}^{p′})( ∑_{j=v_{k_{1}−1}}^{v_{k_{1}+1}−1} β_{l}^{p′})^{−1}.  (4.40)

Lemma 4.3 tells us that C_{k} ∈ [2^{−2m−4}, 2^{−m+1}]. It is easy to obtain

∥x^{k}∥_{p}^{p} = 2^{−k_{0}}[ ∑_{j=v_{k_{0}−2}}^{v_{k_{0}}−1} β_{j}^{p′} + C_{k}^{p} ∑_{j=v_{k_{1}−1}}^{v_{k_{1}+1}−1} β_{j}^{p′}] ≤ 1 + 2^{−k_{0}+1}2^{k_{0}+2m+1} ≤ 2^{2m+2}.

Let y^{k} = (y_{l}^{k})_{l∈ℤ} := S_{a,b}(x^{k}). We will not calculate all values of y_{l}^{k}’s, since it is only important to note that y_{l}^{k} ≥ 0, and

y_{l}^{k} = \begin{cases}0 & if l < v_{k_{0}−2}, \\ 2^{−k_{0}/p}α_{l} ∑_{j=v_{k_{0}−2}}^{v_{k_{0}}−1} β_{j}^{p′} & when l ∈ \{v_{k_{0}} + 1, …, v_{k_{0}+1}\}, \\ 0 & for l ≥ v_{k_{1}+1}. \end{cases}  (4.41)

Case 2: v_{k_{0}} = v_{k_{0}−1}.

In this case, because of v_{k_{0}} < v_{k_{0}+1}, we know that

2^{k_{0}−1} ≤ β_{v_{k_{0}}}^{p′} ≤ 2^{k_{0}+1}.  (4.42)

Let k_{1} be fixed like in case 1, and set

x_{j}^{k} := 2^{−k_{0}/p} \begin{cases}β_{v_{k_{0}}}^{p′−1} & for j = v_{k_{0}}, \\ −C_{k} β_{j}^{p′−1} & for j ∈ \{v_{k_{1}−1}, …, v_{k_{1}+1} − 1\}, \\ 0 & otherwise, \end{cases}  (4.43)

where this time

C_{k} := β_{v_{k_{0}}}^{p′} ( ∑_{j=v_{k_{1}−1}}^{v_{k_{1}+1}−1} β_{l}^{p′})^{−1}.  (4.44)

Because of (4.42) and Lemma 4.3, we know that C_{k} ∈ [2^{−2m−4}, 2^{−m+1}]. Again, it is easy to obtain ∥x^{k}∥_{p} ≤ 2^{(2m+2)/p}. Setting y^{k} := S_{a,b}(x^{k}), we conclude that

y_{l}^{k} = \begin{cases}0 & if l < v_{k_{0}}, \\ 2^{−k_{0}/p}α_{l}β_{v_{k_{0}}} & when l ∈ \{v_{k_{0}} + 1, …, v_{k_{0}+1}\}, \\ 0 & for l ≥ v_{k_{1}+1}. \end{cases}  (4.45)

So in both cases we have defined our vectors x^{k}. Since k_{1} ≤ k_{0} + 2m + 2 ≤ π_{0}(k + 1) − 2, we easily see that the y^{k} ’s are as disjointly supported as the x^{k} ’s are with

∥y^{k}∥_{q} ≥ 2^{k_{0}/p′−1}( ∑_{j=v_{k_{0}}+1}^{v_{k_{0}+1}} α_{j}^{q})^{1/q} = 1/2σ_{k}^{0}.  (4.46)

Taking into account y^{k} = S_{a,b} ∘ X^{0}(e_{k}), we conclude that (4.37) holds.

Before we can finally prove Theorem 2.5, a technical difficulty has to be fixed. Note that for a subsequence σ_{π(n)} of a sequence σ_{n}, the sequence (σ_{π(n)})_{n}^{∗} cannot, in general, be estimated from below by σ_{n}^{∗}, e.g. (σ_{π(n)})_{n}^{∗} could be constantly 0 and σ_{n}^{∗} be 1. In particular, for the sequences σ^{l} defined in (4.36), (σ^{l})_{n}^{∗} cannot be estimated from below by δ_{n}^{∗}(a, b), but fortunately this is possible for the maximum:

Lemma 4.4. Let (σ_{n})_{n≥0} be a nonnegative sequence tending to zero, and let (σ_{π_{1}(n)})_{n∈ℕ}, …, (σ_{π_{N} (n)})_{n∈ℕ} be partial sequences of (σ_{n})_{n∈ℕ}, where the π_{j}’s satisfy ⋃_{j=1}^{N} π_{j}(ℕ) = ℕ. Then for any s > 0 we have

sup_{n∈ℕ}n^{1/s}σ_{n}^{∗} ≤ 2N^{1/s} max_{j=1,... ,N} sup_{n∈ℕ}n^{1/s}(σ_{π_{j}(n)})_{n}^{∗},  (4.47)

and for all k ∈ ℕ

(Nk)^{1/s}σ_{kN}^{∗} ≤ 2 N^{1/s}k^{1/s} max_{j=1,... ,N}(σ_{π_{j}(n)})_{k}^{∗}  (4.48)

holds.

Proof. For x ≥ 0 denote ⌊x⌋ := sup{n ∈ ℕ : n ≤ x}. Given n ∈ ℕ, there are k_{1}, …, k_{n} ∈ ℕ with σ_{k_{i}} ≥ σ_{n}^{∗}/2. For l := ⌊n/N⌋ + 1 we conclude that there is j ∈ ℕ such that |π_{j}(ℕ) ∩ {k_{1}, …, k_{n}}| ≥ l. Consequently, for this j we have (σ_{π_{j}(n)})_{l}^{∗} ≥ σ_{n}^{∗}/2, and since n/l ≤ N, we find

n^{1/s}σ_{n}^{∗} ≤ 2 (n/l)^{1/s}l^{1/s}(σ_{π_{j}(n)})_{l}^{∗} ≤ 2N^{1/s}l^{1/s}(σ_{π_{j}(n)})_{n}^{∗} ,

which implies (4.47). The second assertion is analogous.

Proof of Theorem 2.5. Using Lemma 4.2 from [10], it is easy to deduce from Proposition 4.2 and Corollary 3.7 that

sup_{n∈ℕ}n^{ρ}e_{n}(S_{a,b}) ≥ 1/4 max_{l≤2m+3} sup_{n∈ℕ}n^{ρ}e_{n}(D^{l})
≥ 1/4 max_{l≤2m+3} sup_{n∈ℕ}n^{1/s}((σ^{l})_{n}^{∗} + (σ^{l})_{−n+1}^{∗})  (4.49)

with D^{l} and σ^{l} defined in (4.36). On the other hand, by Lemma 4.4 with δ_{n} := δ_{n}(a, b) we get

sup_{n∈ℕ}n^{1/s}(δ_{n−1}^{∗} + δ_{−n}^{∗}) ≤ sup_{n∈ℕ}n^{1/s}δ_{n−1}^{∗} + sup_{n∈ℕ}n^{1/s}δ_{−n}^{∗}
≤ C max_{l=0,... ,2m+3}(sup_{n∈ℕ}n^{1/s}(σ^{l})_{n−1}^{∗})
+ C max_{l=0,... ,2m+3}(sup_{n∈ℕ}n^{1/s}(σ^{l})_{−n}^{∗})
≤ 2 C max_{l=0,... ,2m+3}(sup_{n∈ℕ}n^{1/s}((σ^{l})_{n−1}^{∗} + (σ^{l})_{−n}^{∗})) .

Combined with (4.49), this shows the first assertion.

The last estimate is based on the fact that for a_{n}, b_{n} ≥ 0 we have

sup_{n}(a_{n} + b_{n}) ≤ sup_{n} a_{n} + sup_{n} b_{n} ≤ 2 sup_{n}(a_{n} + b_{n}).

264 J. CREUTZIG AND W. LINDE

Such estimates cannot be established for the infimum. So for estimate (2.15) we have to be a little more careful. First we define two subspaces of ℓ_{p}(ℤ), namely

ℓ_{p}^{+} := {(x_{k})_{k∈ℤ} ∈ ℓ_{p}(ℤ) : x_{k} = 0 for k < 0} ,  (4.50)

as well as

ℓ_{p}^{−} := {(x_{k})_{k∈ℤ} ∈ ℓ_{p}(ℤ) : x_{k} = 0 if k ≥ 0, ∑_{j<0} β_{j}x_{j} = 0} .  (4.51)

The corresponding restrictions of S_{a,b} are denoted analogously, e.g.,

S_{a,b}^{+} := (S_{a,b})_{|ℓ_{p}^{+}}, S_{a,b}^{−} := (S_{a,b})_{|ℓ_{p}^{−}}. (4.52)

We observe that Im (S_{a,b}^{+}) ∩ Im (S_{a,b}^{−}) = {0}. It is easy to deduce that

e_{n}(S_{a,b}^{+}) + e_{n}(S_{a,b}^{−}) ≤ C_{p,q} e_{n}(S_{a,b}).  (4.53)

A careful inspection of the proof of Proposition 4.2 reveals that X^{l}(e_{k}) ∈ ℓ_{p}^{+} if k ≥ 0, and that X^{l}(e_{k}) ∈ ℓ_{p}^{−} if k < −1. As above, this yields

e_{n}(S_{a,b}^{+}) ≥ C max_{0≤l≤2m+3} e_{n}(D_{(σ_{k}^{l})_{k≥0}})

and

e_{n}(S_{a,b}^{−}) ≥ C′ max_{0≤l≤2m+3} e_{n}(D_{(σ_{−k}^{l})_{k≥2}}).

From this we easily derive with the help of (3.23) and (4.48) that

n^{ρ}e_{n}(S_{a,b}^{+}) ≥ C n^{1/s}δ_{(2m+4)(n−1)}^{∗}  (4.54)

and

n^{ρ}e_{n}(S_{a,b}^{−}) ≥ C′ n^{1/s}((δ_{−k})_{k≥2m+4})_{(2m+4)n}^{∗} .  (4.55)

But it is not hard to see that ((δ_{−k})_{k≥2m+4})_{(2m+4)n}^{∗} ≥ δ_{−2(2m+4)n}^{∗}, which allows to continue the estimate (4.55) into

n^{ρ}e_{n}(S_{a,b}^{−}) ≥ C′ n^{1/s}δ_{−2(2m+4)n}^{∗}.  (4.56)

Combined with (4.54) and (4.53), this reveals

n^{ρ}e_{n}(S_{a,b}) ≥ C′′ n^{1/s}(δ_{2(2m+4)n−1}^{∗} + δ_{−2(2m+4)n}^{∗}).

From this we are directly led to (2.15).

5. Examples and Special Cases

At first we will study polynomial growth of the α_{k}’s and β_{k}’s: Let us assume that for α, β ∈ ℝ we have α_{k} = k^{−α} and β_{k} = k^{−β} if k ≥ 1, and α_{k} = 0 = β_{k} otherwise. First of all, (2.1) is satisfied iff α > 1/q for q < ∞, and iff α ≥ 0 when q = ∞. For A_{k} and B_{k} given by (2.1) we find that then A_{k} ≈ k^{−α+1/q} while

B_{k} ≈ \begin{cases}k^{−β+1/p′}, & β < 1/p′, \\ log k, & β = 1/p′, \\ c, & β > 1/p′. \end{cases}  (5.1)

Assume first p ≤ q. Then Theorem 2.1 tells us that S_{a,b} is bounded iff A_{k} · B_{k} is so; by treating the three cases given in (5.1) separately and regarding 1/p′ ≥ 1/r − α one can easily check that S_{a,b} is bounded from ℓ_{p} to ℓ_{q} iff α + β ≥ 1/r. For p > q, the condition in Theorem 2.1 seems to be more complicated; however, using (5.1) one can verify that in this case we can estimate

[A_{k} · B_{k}^{p′/q′}]^{\frac{pq}{p−q}} β_{k−1}^{p′} ≈ \begin{cases}k^{\frac{pq}{p−q} (1−α−β)}, & β < 1/p′, \\ k^{\frac{pq}{p−q} (1/q−α)−1}(log k)^{\frac{pq}{q′(p−1)}} , & β = 1/p′, \\ k^{\frac{pq}{p−q} (1/q−α)−βp′}, & β > 1/p′. \end{cases}  (5.2)

For β ≥ 1/p′ this is always summable, while for β < 1/p′ we infer that S_{a,b} is bounded from ℓ_{p} to ℓ_{q} iff \frac{pq}{p−q}(1−α−β) < −1, which is equivalent to α+β > 1/r. Since 1/p′ > 1/r − α (we have q < ∞ in this case), we find that for p > q the operator S_{a,b} is bounded iff α + β > 1/r. Next we wish to calculate δ_{k}(a, b). For this purpose we assume from now on that β < 1/p′. Then it is easy to see that for v_{k} as in (2.6) we have v_{k} ≈ 2^{k/(1−βp′)}, and consequently,

δ_{k}(a, b) ≈ 2^{k/p′} ( ∑_{j=2^{k/(1−βp′)}}^{2^{(k+1)/(1−βp′)}} j^{−αq})^{1/q} ≈ 2^{k((1/q−α)/(1−βp′)+1/p′)}.

Since it is easily verified that (1/q − α)/(1 − βp′) < −1/p′ if and only if β > 1/r − α, we derive that S_{a,b} is compact iff β > 1/r − α, and in this case, we have an exponential decrease of δ_{k}(a, b). Through easy direct calculations one verifies that δ_{k}(a, b) decrease even more rapidly for β ≥ 1/p′. Summing up our results and using Theorem 2.2, we verify the following proposition:

Proposition 5.1. For α, β ∈ ℝ, define S_{α,β} : ℓ_{p}(ℕ) → ℓ_{q}(ℕ) by

S_{α,β} : x = (x_{k})_{k∈ℕ} ↦ (k^{−α} ∑_{l=1}^{k−1} l^{−β}x_{l})_{k∈ℕ}.

Then (i) S_{α,β} is bounded from ℓ_{p} to ℓ_{q} iff

α ≥ 0 and α + β ≥ 1/r  if q = ∞,
α > 1/q and α + β ≥ 1/r if p ≤ q < ∞,
α > 1/q and α + β > 1/r for p > q;

and (ii) S_{α,β} is compact iff α > 1/q ( α ≥ 0, if q = ∞ ), and α + β > 1/r. In

this case, we have lim_{n→∞} ne_{n}(S_{α,β}) = 0.

As a second class of interest we investigate the case that β_{k}’s grow exponentially:

Proposition 5.2. Let a = (α_{k})_{k∈ℤ} be arbitrary, and let β_{k} = 2^{k/p′}, k ∈ ℤ. Then we have

c_{1}|(a, b)|_{r,∞} ≤ sup_{n∈ℕ}n e_{n}(S_{a,b}) ≤ c_{2}|(a, b)|_{r,∞}  (5.3)

and

c_{1} \underline{lim}_{n→∞} n^{1/r}τ_{n} ≤ \underline{lim}_{n→∞} n e_{n}(S_{a,b}) ≤ \overline{lim}_{n→∞} n e_{n}(S_{a,b}) ≤ c_{2} \overline{lim}_{n→∞} n^{1/r}τ_{n}, (5.4)

with τ_{n} := δ_{n−1}^{∗}(a, b) + δ_{−n}^{∗}(a, b).

Proof. Note first that δ_{k}(a, b) = 2^{k/p′}α_{k} for k ∈ ℤ. Let V : ℓ_{p}(ℤ) → ℓ_{p}(ℤ) denote an auxiliary summation operator defined by

V ((x_{k})_{k∈ℤ}) := (2^{−k/p′} ∑_{j<k} x_{j}2^{j/p′})_{k∈ℤ}.

Using Theorem 2.1 one easily checks that V is bounded. We set now D : ℓ_{p}(ℤ) → ℓ_{q}(ℤ) to be the diagonal operator generated by the sequence δ_{k}(a, b). Then it is trivial to obtain S_{a,b} = D ∘ V, yielding e_{n}(S_{a,b}) ≤ Ce_{n}(D). On the other hand, let R : ℓ_{p}(ℤ) → ℓ_{p}(ℤ) be given by R(e_{k}) := 2e_{k−1} −e_{k}. We find that D = S_{a,b} ∘ R, and thus e_{n}(D) ≤ C′e_{n}(S_{a,b}). Now we have to estimate e_{n}(D) only, which has already been done in Corollary 3.7.

Next we study the case of a rapid (i.e., superexponential) increase of β_{k}:

Proposition 5.3. Set β_{k} = 2^{2^{k}} for k ≥ 0 and β_{k} = 0 for k < 0. Then for any sequence a = (α_{k})_{k∈ℤ} we have

|(a, b)|_{r,∞} ≥ c sup_{l∈ℕ}l^{1/r}(α_{l} β_{l})_{l}^{∗} ,  (5.5)

while

sup_{l∈ℕ} l e_{l}(S_{a,b}) ≤ C sup_{l∈ℕ}l^{1/r}(α_{l} β_{l−1})_{l}^{∗} .  (5.6)

In particular, if we set α_{l} := 2^{−l} β_{l−1}^{−1} whenever l ≥ 0, and α_{l} = 0 otherwise, then |(a, b)|_{r,∞} = ∞ while sup_{n} n e_{n}(S_{a,b}) < ∞.

Proof. It is easy to see that v_{k} = ⌊log_{2} k⌋ for k ≥ k_{0}, where as usual

⌊x⌋ := inf{n ∈ ℕ : n ≥ x} .

In particular, v_{k} ≠ v_{k+1} iff k = 2^{l} − 1 for some l ≥ 1. So for k = 2^{l} − 1 we derive

δ_{k}(a, b) = 2^{k/p′}α_{v_{k+1}} = 2^{(2^{l}−1)/p′}α_{l} ≥ c_{p} β_{l} α_{l}.  (5.7)

Since k^{1/r} ≥ l^{1/r}, this implies (5.5).

On the other hand, let V : ℓ_{p}(ℤ) → ℓ_{p}(ℤ) be defined by

V ((x_{l})_{l∈ℤ}) := (β_{l−1}^{−1} ∑_{j<l} β_{j}x_{j})_{l∈ℤ} .  (5.8)

Then we have by Theorem 2.1 that ∥V ∥ ≤ C_{p} and, additionally,

S_{a,b} = D_{α_{l}·β_{l−1}} ∘ V.  (5.9)

Thus e_{n}(S_{a,b}) ≤ C_{p} e_{n}(D_{(α_{l}·β_{l−1})}) and so (5.6) follows by virtue of Theorem 3.5.

Finally, we will estimate the first entropy numbers of finite-dimensional standard summation operators. Estimates for the approximation, Gelfand and Kolmogorov numbers of such operators can be found in [8]. Let N ≥ 2 be arbitrary, and set λ := min{1, 1/r}. Then for the standard summation operator S_{1}^{N} : ℓ_{p}^{N} → ℓ_{q}^{N} with

S_{1}^{N}((x_{i})_{i=1}^{N}) := (∑_{j<i} x_{j})_{i=1}^{N}

we have

Proposition 5.4. There are universal constants c_{0}, c_{1}, c_{2} > 0 such that the estimate

c_{1} \frac{N^{1/r}}{n} ≤ e_{n}(S_{1}^{N} : ℓ_{p}^{N} → ℓ_{q}^{N}) ≤ c_{2} \frac{N^{1/r}}{n}  (5.10)

holds for all n ≤ c_{0}N^{λ}.

Proof. We start with the upper estimate. Therefore choose m ∈ ℕ such that

2^{m} ≤ N < 2^{m+1}.

v_{k}’s are given now by

v_{k} = \begin{cases}1 & for k ≤ 0 \\ 2^{k} & if 1 ≤ k ≤ m \\ ∞ & when k > m. \end{cases}  (5.11)

Thus δ_{k}(a, b) = 2^{k/p′+k/q} = 2^{k/r} for k = 0, …, m − 1, and for k = m we have δ_{m}(a, b) = 2^{m/p′}(N − 2^{m})^{1/q} ≤ 2^{m/r}. By Theorem 2.2 we obtain

sup_{n∈ℕ}n e_{n}(S_{1}^{N}) ≤ c (∑_{k=0}^{m} δ_{k}^{r})^{1/r} ≤ c 2^{m/r} ≤ c′N^{1/r}.

For the lower estimate, set

I_{j} := [\frac{j − 1}{N}, \frac{j}{N} ], 1 ≤ j ≤ N,

and denote by P_{N} the conditional expectation operator mapping from L_{p}[0, 1] to L_{p}[0, 1] generated by the partition I_{1}, … I_{N} of [0, 1], i.e.,

P_{N}(f) := ∑_{j=1}^{N} 1_{I_{j}}|I_{j}|^{−1} ∫_{I_{j}} f(t) dt.

It is easily checked that ∥P_{N}∥ = 1. Further we define φ_{p} to be an embedding of ℓ_{p}^{N} into L_{p}[0, 1] with

φ_{p}((x_{j})) := ∑_{j=1}^{N} x_{j}1_{I_{j}}|I_{j}|^{−1/p}.

Recalling |I_{j}| = N^{−1}, we derive easily

(φ_{q} ∘ (N^{−1/r}S_{1}^{N}) ∘ φ_{p}^{−1} ∘ P_{N} )f = ∑_{k=1}^{N} 1_{I_{k}} (∑_{j<k} ∫_{I_{j}} f dt).  (5.12)

Next define R_{N} : L_{p}[0, 1] → L_{q}[0, 1] via

(R_{N}f)(s) := ∑_{k=1}^{N}( ∫_{\frac{k−1}{N}}^{s} f(t) dt) 1_{I_{k}}(s).

Using (5.12), we find

T_{1} = R_{N} + (φ_{q} ∘ (N^{−1/r}S_{1}^{N}) ∘ φ_{p}^{−1} ∘ P_{N} ) .  (5.13)

We derive an upper bound for ∥R_{N}∥ by estimating (we assume q < ∞; the case q = ∞ may be treated in the same way)

∥R_{N}f∥_{q} = (∑_{k=1}^{N} ∫_{I_{k}} | ∫_{\frac{k−1}{N}}^{s} f(t) dt|^{q} ds)^{1/q}

≤ (∑_{k=1}^{N} ∫_{I_{k}} (s − \frac{k − 1}{N})^{q/p′} ds (∫_{I_{k}} |f(t)|^{p} dt)^{q/p})^{1/q}

≤ N^{−1/r}(∑_{k=1}^{N}(∫_{I_{k}} |f(t)|^{p} dt)^{q/p})^{1/q}

The last sum can either be estimated by ∥f∥_{p} when p ≥ q, or by N^{1/q−1/p}∥f∥_{p} if p ≤ q. This shows that ∥R_{N}∥ ≤ N^{−λ}.

From now on, d_{1}, d_{2}, … shall denote fixed constants (that is, the value of d_{i} remains the same at each occurrence). Because of the additivity of the entropy numbers (see, e.g., [2]) and equality (5.13) we come to

e_{n}(T_{1}) ≤ ∥R∥ + e_{n}(φ_{q} ∘ (N^{−1/r}S_{1}^{N}) ∘ φ_{p}^{−1} ∘ P_{N}) ≤ d_{1}N^{−λ} + d_{2}N^{−1/r}e_{n}(S_{1}^{N}).

Since it is well-known that e_{n}(T_{1}) ≥ d_{3}/n (cf. [7]), this reveals

d_{2}N^{−1/r}e_{n}(S_{1}^{N}) ≥ \frac{d_{3}}{n} − \frac{d_{1}}{N^{λ}}  (5.14)

for all n, N ∈ ℕ. Let c_{0} := d_{3}(2d_{1})^{−1}, then (5.14) provides that for n ≤ c_{0}N^{λ} we have

N^{−1/r}e_{n}(S_{1}^{N}) ≥ \frac{d_{3}}{2d_{2}n}.

For p < q, Proposition 5.4 is somewhat unsatisfying, because we can estimate e_{n}(S_{1}^{N}) only for n ≤ c_{0}N^{1/r}. Recall that in this case, 1/r < 1. We leave this as an open problem.

Problem.^{1} Find optimal estimates for e_{n}(S_{1}^{N} : ℓ_{p}^{N} → ℓ_{q}^{N}) when p < q and c_{0}N^{1/r} ≤ n ≤ N.

6. Some Probabilistic Applications

Let T be an operator from a (separable) Hilbert space H into a Banach space E, and let (ξ_{k})_{k=1}^{∞} be an i.i.d. sequence of N (0, 1)-distributed random variables. Then the behaviour of e_{n}(T) as n → ∞ is known to be closely related with the properties of the E-valued Gaussian random variable

X := ∑_{k=1}^{∞} ξ_{k} · T f_{k} ,  (6.1)

where (f_{k})_{k=1}^{∞} denotes some orthonormal basis in H. For example, in [4] it was proved that ∑_{n=1}^{∞} n^{−1/2}e_{n}(T) < ∞ implies the a.s. convergence of (6.1) in E, while, conversely, the a.s. convergence of (6.1) yields sup_{n} n^{1/2}e_{n}(T) < ∞ (cf. [14]). A faster decay of e_{n}(T) as n → ∞ is reflected by a “better” small ball behaviour of X. More precisely the following holds (cf. [9], [11] and [3]):

Proposition 6.1. Let 0 < α < 2 and T : H → E generating X via (6.1). Then the following statements are equivalent:

(i) \overline{lim}_{n→∞} n^{1/α}e_{n}(T) := c_{α}(T) < ∞.

(ii) \underline{lim}_{ε→0} ε^{\frac{2α}{2−α}} log ℙ(∥X∥_{E} < ε) := −d_{α}(X) > −∞.

1Recently M. Lifshits kindly showed us a proof of the following fact: For any h > 0 and any pair (p, q) ≠ (1, ∞) there is c(p, q, h) > 0 such that e_{n}(S^{N}) ≥ c(p, q, h)n^{−1}N^{1/r}, n ≥ (1−h)N.

Since the upper estimate is valid for all n ∈ ℕ, this almost answers the problem.

Moreover,

d_{α}(X) ≤ c_{0} · c_{α}(T)^{\frac{2α}{2−α}}  (6.2)

for some universal c_{0} > 0. Conversely, if T satisfies

\underline{lim}_{n→∞} n^{1/α}e_{n}(T) := κ_{α}(T) > 0 ,

then

\overline{lim}_{ε→0} ε^{\frac{2α}{2−α}} log ℙ(∥X∥_{E} < ε) := −λ_{α}(X) < 0

and

λ_{α}(X) ≥ c′_{0} κ_{α}(T)^{\frac{2α}{2−α}} .

We want to apply these relations between the entropy and the small ball behaviour to the summation operator S_{a,b} : ℓ_{2}(ℤ) → ℓ_{q}(ℤ), 1 ≤ q ≤ ∞. To do so, let W = (W (t))_{t≥0} be a standard Wiener process over [0, ∞). Taking the natural unit vector basis (e_{k})_{k∈ℤ} in ℓ_{2}(ℤ) leads to

X = ∑_{k=−∞}^{∞} ξ_{k}S_{a,b}(e_{k}),  (6.3)

where the ℓ_{q}(ℤ)-valued Gaussian random variable X = (X_{k})_{k∈ℤ} is defined by

X_{k} := α_{k}W (t_{k})  (6.4)

with t_{k} := ∑_{j<k} β_{j}^{2} being an increasing sequence of nonnegative numbers such that lim_{k→−∞} t_{k} = 0. In our situation we have r = \frac{2q}{2+q} and v_{k}’s given in (2.6) can be calculated by

v_{k} = sup{m ∈ ℤ : t_{m} < 2^{k}}.  (6.5)

With this notation the following is valid:

Theorem 6.2. Let (α_{k})_{k∈ℤ} be a sequence of nonnegative numbers satisfying (2.1) and let (t_{k})_{k∈ℤ} be a nondecreasing sequence with lim_{k→−∞} t_{k} = 0. For X =

(X_{k})_{k∈ℤ} defined by (6.4) and v_{k}’s as in (6.5) we have the following. (i) If

∑_{k=−∞}^{∞} 2^{\frac{kq}{2+q}} · ( ∑_{l=v_{k}+1}^{v_{k+1}} α_{l}^{q})^{\frac{2}{2+q}} < ∞,  (6.6)

then

lim_{ε→0} ε^{2} log ℙ(∥X∥_{q} < ε) = 0.  (6.7)

(ii) Assume that [t_{k+1} − t_{k}]^{−1}α_{k}^{q} is monotone near ±∞. Then (6.7) holds provided that

∑_{k=−∞}^{∞} [t_{k} − t_{k−1}]^{\frac{q}{2+q}} · αk^{\frac{2q}{2+q}} < ∞ .  (6.8)

Proof. Combining Proposition 6.1 for α = 1 with Theorem 2.2, one easily sees that (6.6) implies

\underline{lim}_{ε→0} ε^{2} log ℙ(∥X∥_{q} < ε) = −d_{1}(X) > −∞.  (6.9)

Yet by the second part of Theorem 2.2 and estimate (6.2) we even have d_{1}(X)= 0 which completes the proof of (i). Assertion (ii) is a direct consequence of Proposition 2.4. Recall that β_{k}^{2} = t_{k+1} − t_{k}.

Remarks. (1) Theorem 2.3 shows that, in general, condition (6.6) cannot be replaced by the weaker assumption (6.8). More precisely, for each q ∈ [1, ∞] there are α_{k}’s and t_{k}’s satisfying (6.8), yet

\underline{lim}_{ε→0} ε^{2} log ℙ(∥X∥_{q} < ε) = −∞ .

(2) In the case q = ∞ Theorem 6.2 reads as follows. Suppose that

∑_{k=−∞}^{∞} 2^{k} sup_{v_{k}<l≤v_{k+1}} α_{l}^{2} < ∞.  (6.10)

Then

lim_{ε→0} ε^{2} log ℙ(sup_{k∈ℤ}α_{k}|W (t_{k})| < ε) = 0.  (6.11)

Again, (6.11) becomes wrong under a weaker condition

∑_{k=−∞}^{∞}[t_{k} − t_{k−1}] · α_{k}^{2} < ∞ .  (6.12)

Recall that (6.12) suffices for α_{k}’s monotone near ±∞ .

Observe that Theorem 6.2 provides us with sufficient conditions for

− log ℙ(∥X∥_{q} < ε) ⪯ ε^{−α}  (6.13)

in the case α = 2. Of course, similar conditions would also be of interest for α’s different of 2, yet we could not handle this problem by our methods. Recall that we have derived upper bounds for the entropy of summation operators by the corresponding results about integration operators. And there only the case α = 2 has been treated. But for t_{k}’s not increasing too fast we can prove necessary conditions for (6.13) and all α > 0. To be more precise, suppose that t_{k}’s satisfy

sup \{\frac{t_{k+1}}{t_{k}} : t_{k} > 0\} < ∞ .  (6.14)

Then the following holds.

Proposition 6.3. Let X = (X_{k})_{k∈ℤ} be a Gaussian sequence defined by (6.4) with t_{k}’s satisfying (6.14). If

\underline{lim}_{ε→0} ε^{α} log ℙ(∥X∥_{q} < ε) > −∞  (6.15)

for some α > 0, then necessarily

sup_{n≥1} n^{1/α+1/q} (δ_{n−1}^{∗} + δ_{−n}^{∗}) < ∞  (6.16)

with

δ_{n} = 2^{n/2} ( ∑_{l=v_{k}+1}^{v_{k+1}} α_{l}^{q})^{1/q} , n ∈ ℤ ,  (6.17)

and v_{k}’s defined by (6.5).

Proof. Using Proposition 6.1, we see that (6.15) implies

sup_{n≥1} n^{ρ}e_{n}(S_{a,b}) < ∞,  (6.18)

where, as before, β_{k}^{2} = t_{k+1} − t_{k} and ρ = 1/α + 1/2. Let us first treat the case t_{k} > 0 for all k ∈ ℤ and sup_{k} t_{k} = ∞. Then condition (6.14) implies

sup_{k∈ℤ}|{k′ : v_{k} = v_{k′}}| < ∞  (6.19)

(in fact, here (6.14) and (6.19) are even equivalent), i.e., condition (2.12) is satisfied. Thus Theorem 2.5 applies and by (6.16) we obtain

sup_{n≥1} n^{1/s} (δ_{n−1}^{∗} + δ_{−n}^{∗}) < ∞ ,

where

1/s = ρ − 1/2 + 1/q = 1/α + 1/q .

This completes the proof in this case.

Next assume t_{k} > 0 and sup_{k} t_{k} < ∞. Then by (6.14) there is k_{0} ∈ ℕ such that

sup_{k≤k_{0}} |{k′ : v_{k} = v_{k′}}| < ∞ .

Here Proposition 2.6 applies and completes the proof as before.

If we have sup_{k} t_{k} < ∞ and t_{k} = 0 for k ≤ k̃,

|{n ∈ ℤ : δ_{n} ≠ 0}| < ∞ ,

so there is nothing to prove.

Finally, if t_{k} = 0 for k ≤ k̃ and sup_{k} t_{k} = ∞, this time (6.14) yields

sup_{k≥k_{0}} |{k′ : v_{k} = v_{k′}}| < ∞

for a certain k_{0} ∈ ℕ, and we may proceed as in the preceding case.

Next we ask for conditions which ensure − log ℙ(∥X∥_{q} < ε) ⪰ ε^{−α} for some α > 0.

Proposition 6.4. Let (t_{k})_{k∈ℤ} and X be as in Proposition 6.3. If for some α > 0 and δ_{n}’s defined by (6.17)

\underline{lim}_{n→∞} n^{1/α+1/q} (δ_{n−1}^{∗} + δ_{−n}^{∗}) > 0 ,

then necessarily

\overline{lim}_{ε→0} ε^{α} log ℙ(∥X∥_{q} < ε) < 0 .

Proof. This can be proved exactly as Proposition 6.3 by using Theorem 2.5, Proposition 2.6 and Proposition 6.1.

Let us illustrate the preceding results by an example. We define random variables (Y_{k})_{k≥1} by

Y_{k} := (\frac{1}{2^{k}} ∑_{l=2^{k}}^{2^{k+1}−1} |\frac{W (l)}{l^{1/2}} |^{q})^{1/q} , q < ∞ ,

and

Y_{k} := sup_{2^{k}≤l<2^{k+1}} |\frac{W (l)}{l^{1/2}} | , q = ∞ .

Proposition 6.5. Let (γ_{k})_{k≥1} be a sequence of nonnegative real numbers, and for q ∈ [1, ∞], let 1/r = 1/2 + 1/q.

(i) If ∥(γ_{k})_{k∈ℕ}∥_{r} < ∞, then lim_{ε→0} ε^{2} log ℙ( ∑_{k=1}^{∞} γ_{k}^{q} Y_{k}^{q} < ε^{q}) = 0 .

(ii) For γ_{k} decreasing and − log ℙ( ∑_{k=1}^{∞} γ_{k}^{q} Y_{k}^{q} < ε^{q}) ⪯ ε^{−α} we necessarily

have sup_{k≥1} k^{1/α+1/q}γ_{k} < ∞ .

(iii) Suppose that γ_{k}’s are decreasing with γ_{k} ⪰ k^{−1/α−1/q} for a certain α > 0.

Then it follows − log ℙ( ∑_{k=1}^{∞} γ_{k}^{q} Y_{k}^{q} < ε^{q}) ⪰ ε^{−α} .

In all three cases, for q = ∞, one has to read ℙ(sup_{k≥1} γ_{k}Y_{k} < ε) everywhere.

Proof. We set

α_{l} = 2^{−k/q} γ_{k} l^{−1/2} , 2^{k} ≤ l < 2^{k+1}

for any k ≥ 0, and α_{l} = 0 if l < 0. Then it is easy to obtain

∥(γ_{k}Y_{k})_{k∈ℤ}∥_{q} = ∥X∥_{q}  (6.20)

with X = (X_{l})_{l∈ℤ} = (α_{l} W (l))_{l∈ℤ}. Let b be given by β_{k} = 1 if k ≥ 0, β_{k} = 0 otherwise. Then t_{l} = l for l ≥ 0, and t_{l} = 0 otherwise. For v_{k}’s defined in (2.6) we have v_{k} = 0 for k ≤ 0 and v_{k} = 2^{k} − 1 for k ≥ 1. Thus

δ_{k}(a, b) = 2^{k/2−k/q}γ_{k} ( ∑_{l=2^{k}}^{2^{k+1}−1} l^{−q/2})^{1/q}

if q < ∞ and

δ_{k}(a, b) = 2^{k/2}γ_{k}( sup_{2^{k}≤l<2^{k+1}} l^{−1/2}) = γ_{k}

for q = ∞. In any case we obtain

2^{−1/2}γ_{k} ≤ δ_{k}(a, b) ≤ γ_{k} ,  (6.21)

so that (γ_{k})_{k∈ℕ} ∈ ℓ_{r}(ℕ) iff (δ_{k})_{k∈ℕ} ∈ ℓ_{r}(ℕ), and (i) follows from Theorem 6.2. Conversely, if γ_{k}’s are decreasing, by (6.21) we have sup_{k≥1} k^{1/α+1/q}γ_{k} < ∞ iff the same holds for δ_{k}^{∗}’s. So (ii) is a consequence of Proposition 6.3 and (6.20). Similarly, assertion (iii) is proved.

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(Received 8.07.2000)

Authors’ address: Friedrich–Schiller–Universität Jena Ernst–Abbe–Platz 1–4 07743 Jena, Germany E-mail: jakob@creutzig.de, lindew@minet.uni-jena.de