Difference functions of periodic L_{p} functions

Tamás Mátrai

Thesis

Department of Analysis Eötvös Loránd University

Budapest

Supervisor: Tamás Keleti

2002

Contents

1 Introduction 3 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Notation and Results . . . . . . . . . . . . . . . . . . . . . . . 3

2 Preliminaries 5

3 Basic inclusions 7 3.1 Powers less than 1 . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Powers greater than 1 . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 A small power . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.4 The inclusion theorems . . . . . . . . . . . . . . . . . . . . . . 16

4 Upper bounds 17

5 Lower bounds 19 5.1 Non-ejective sets . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 General N-sets . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6 Ejectivity 23 6.1 A minimax theorem . . . . . . . . . . . . . . . . . . . . . . . . 23 6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1 Introduction

1.1 Background

About fifty years ago de Bruijn proved that if a periodic measurable real function f has the property that f(x+h)−f(x) is square integrable for every h ∈ ℝ, then f itself is also square integrable (see [8]). Thirteen years later Carroll proved the same statement for p integrable functions (1 ≤ p ≤ ∞) and a few years later this result was generalized to every compact Abelian group.

But what happens if we know that the difference functions f(x+h)−f(x) are p integrable only for some restricted values of h? This question has been extensively studied by Tamás Keleti in [1]. In this paper we continue his investigations.

Let us call a set H of the Abelian group G small if there is a function f with difference functions in a fixed L_{p} class for every h ∈ H but f itself does not belong to L_{p}. The previous results imply that the whole group is not small. So a small set is a proper subset of G and by continuity arguments one can show that a small set can be covered by a proper F_{σ} subgroup of G. Tamás Keleti showed that pseudo-Dirichlet sets are small. The reason of this is that pseudo-Dirichlet sets are “very periodic” (for the precise definition see below). He also realized that the property of being small is closely related to the non-ejectivity property of sets. It was known that among compact sets the non-ejective sets are exactly the N-sets and the class of N-sets is closed under taking generated subgroup. For these reasons Tamás Keleti conjectured that a set is small if and only if it is an N-set. We prove partially this conjecture.

1.2 Notation and Results

Let 𝕋 = ℝ/ℤ denote the circle group and let L_{0} be the class of measurable complex valued functions on 𝕋. Let G ⊂ F ⊂ L_{0} be two classes of functions on 𝕋. Following the notations in [1] we denote by ℌ(F, G) those subsets H of 𝕋 for which there exists a function f ∈ F \ G with difference functions Δ_{h}f(x) = f(x + h) − f(x) belonging to G for every h ∈ H. That is,

ℌ(F, G) = {H ⊂ 𝕋 : (∃f ∈ F \ G) (∀h ∈ H) Δ_{h}f ∈ G} .

For the general properties of ℌ(F, G) see [1].

Let L_{p} (0 < p < ∞) denote the class of those measurable functions on 𝕋 for which ||f||_{L_{p}} = (∫_{𝕋} |f|^{p})^{\frac{1}{p}} < ∞. It is clear that q ≤ p implies L_{p} ⊂ L_{q}. Let

L_{p^{<}} = ⋂_{0<q<p} L_{q},

a class containing L_{p}. For these function classes the classes ℌ(F, G) are related to some classes of thin sets in harmonic analysis. We collect here those classes which arise in our proofs and results.

Definition 1.1. A set H ⊂ 𝕋 is an N-set if there exists a trigonometric series on 𝕋 which is absolute convergent on H but not absolute convergent everywhere. The class of N-sets on 𝕋 will be denoted by 𝔑.

Definition 1.2. A set H ⊂ 𝕋 is a Dirichlet set if there is an increasing sequence of positive integers (q_{n}) such that sin(q_{n}πh) → 0 uniformly on H. The class of Dirichlet sets on 𝕋 will be denoted by 𝔇.

Definition 1.3. A set H ⊂ 𝕋 is a pseudo-Dirichlet set if there is an increasing sequence of positive integers (q_{n}) and a sequence ε_{n} converging to zero such that for every x ∈ H there exists a k(x) for which | sin(q_{n}πx)| < ε_{n} if n > k(x). The class of pseudo-Dirichlet sets on 𝕋 will be denoted by 𝔭𝔇.

Definition 1.4. A Borel set H ⊂ 𝕋 is a weak-Dirichlet set if for every probability measure μ supported by H,

lim sup_{|n|→∞} |μ̂(n)| = 1,

where

μ̂(n) = ∫_{𝕋} e^{−2πint}dμ(t)

stands for the Fourier-transform of μ. The class of weak-Dirichlet sets on 𝕋 will be denoted by 𝔴𝔇, while 𝔠𝔴𝔇 will stand for the class of compact weak-Dirichlet sets.

Remark 1.5. An easy computation shows that we get an equivalent definition of weak-Dirichlet sets by requiring

inf_{n≠0} ∫_{𝕋} |e^{−2πint} − 1|^{2} dμ(t) = 0

for every probability measure μ supported by H.

𝔉_{σ} stands for the class of subsets of 𝕋 which can be covered by a proper F_{σ} subgroup of 𝕋.

With these notations the results of T. Keleti can be summarized as follows ([1] Theorem 2.5, [2] Theorem 4.10).

Theorem 1.6. For every 0 < p < q < ∞,

𝔭𝔇 ⊂ ℌ(L_{p}, L_{q}) ⊂ 𝔉_{σ}.

He also proves the following inclusion ([1] Theorem 3.2).

Theorem 1.7. If H ⊂ 𝕋 is not an N-set, f : 𝕋 → ℂ is a measurable function with Δ_{h}f ∈ L_{∞} for every h ∈ H, then f ∈ L_{∞^{<}}.

These inclusions motivated the conjecture

ℌ(L_{p}, L_{q}) = 𝔑 (0 < p < q < ∞),

which we confirm partially in our main theorem:

Theorem 1.8. 1. For every 0 ≤ p < q ≤ 2,

ℌ(L_{p}, L_{q}) = 𝔑.

2. For every 0 ≤ p < q < ∞,

ℌ(L_{p}, L_{q^{<}}) = 𝔑.

In particular, if 0 ≤ p < ∞ and H ⊂ 𝕋 is not an N-set, f : 𝕋 → ℂ is a measurable function with Δ_{h}f ∈ L_{p} for every h ∈ H, then f ∈ L_{p^{<}}.

2 Preliminaries

For proving that the classes in question are contained in 𝔑 we use the fact that on the L_{2} class the Fourier transform is an isometric isomorphism. For 0 < p < ∞ let l_{p} denote the complex Banach-space of the two-way infinite sequences c = (c_{n})_{n=−∞}^{∞} of complex numbers for which

||c||_{l_{p}} = (∑ |c_{n}|^{p})^{\frac{1}{p}} < ∞.

For an f : 𝕋 → ℂ measurable function let

𝔉f(n) = ∫_{𝕋} f(t)e^{2πint}dt

denote the Fourier transform of f. The Theorem of Young and Hausdorff says that f ∈ L_{2} if and only if 𝔉f ∈ l_{2}, moreover every sequence of l_{2} can be obtained as the Fourier transform of a function belonging to the L_{2} class. According to the Parseval formula

||f||_{L_{2}} = ||𝔉f||_{l_{2}}.

The following two theorems shows the close relation of N-sets and weak-Dirichlet sets (see [7], page 190, Theorem 1.5. and Corollary 1.6. or [6], page 49).

Theorem 2.1. Any increasing union of compact weak-Dirichlet sets is an N-set.

Theorem 2.2. Any N-set can be obtained as an increasing countable union of compact weak-Dirichlet sets.

Finally we state a result giving an equivalent property of being a weak-Dirichlet set (see [5], page 162. Remark 4.3). We will use this theorem for proving that 𝔑 is contained in the classes we examine. For an h ∈ 𝕋 and A ⊂ 𝕋 we introduce the notation

Δ_{h}A = ((A + h) \ A) ∪ (A \ (A + h)) .

Definition 2.3. A compact set H ⊂ 𝕋 is non-ejective if for every 0 ≤ x ≤ 1,

inf_{\begin{matrix}A⊂𝕋 \\ λ(A)=x \end{matrix}} sup_{h∈H} λ(Δ_{h}A) = 0.

Theorem 2.4. Let H ⊂ 𝕋 be a compact set. Then H is weak-Dirichlet if and only if H is non-ejective

Unfortunately the proof of this theorem has not been published yet. In the sequel we will use only the implication weak-Dirichlet → non-ejective. The other (easy) direction will follow from our results.

In the proofs ℕ and ℤ will stand for the set of positive integers including zero and integers respectively.

The translation on 𝕋 by h ∈ 𝕋 will be denoted by γ_{h}.

For a function f : 𝕋 → ℂ and a, b ∈ ℝ ∪ {±∞} let

[a ≤ f < b] = {x ∈ 𝕋 : a ≤ f(x) < b}.

For a set K ⊂ 𝕋, χ_{K} denotes the characteristic function of K.

3 Basic inclusions

In this section we prove the following two inclusion theorems between the classes ℌ(L_{p}, L_{q}) and ℌ(L_{r}, L_{s^{<}}).

Theorem 3.1. For every 0 ≤ p < q < ∞, 0 ≤ r < s < ∞ with q ≤ s,

ℌ(L_{p}, L_{q}) ⊂ ℌ(L_{r}, L_{s}).

Theorem 3.2. For every 0 ≤ p < q < ∞ the classes

ℌ(L_{p}, L_{q^{<}})

coincide.

They will be necessary to transfer the results obtained for L_{1} or L_{2} to arbitrary L_{p} class.

The proof of these theorems uses some properties of powers of functions. These properties are described in Lemma 3.4, Lemma 3.6 and Lemma 3.8.

3.1 Powers less than 1

Definition 3.3. For a measurable function f : 𝕋 → ℂ,

L(f) = sup {0 ≤ p : f ∈ L_{p}}

denotes the class of f.

The following lemma describes the behavior of taking power f^{α} for α ≤ 1.

Lemma 3.4. Let 0 ≤ p ≤ r < ∞ and h ∈ 𝕋. If f ∈ L_{p}, 0 ≤ f and Δ_{h}f ∈ L_{r} then for every 0 < α ≤ 1 we have f^{α} ∈ L_{\frac{p}{α}} and Δ_{h}f^{α} ∈ L\frac{r}{α} with

||Δ_{h}f^{α}||_{L_{\frac{r}{α}}} ≤ ||Δ_{h}f||_{L_{r}}^{α}.

We also have

L(f^{α}) = \frac{L(f)}{α}.

Proof. The first and third statement being obvious we prove only the second one. It is easy to see that for 0 < α ≤ 1 and a, b ∈ ℂ,

||a|^{α} − |b|^{α}| ≤ |a − b|^{α} .

From this with

a = f(x + h), b = f(x)

we have

|f^{α}(x + h) − f^{α}(x)|^{\frac{r}{α}} ≤ |f(x + h) − f(x)|^{α \frac{r}{α}} = |f(x + h) − f(x)|^{r} ,

so

Δ_{h}f^{α} ∈ L_{r}

and

||Δ_{h}f^{α}||_{L_{\frac{r}{α}}} ≤ ||Δ_{h}f||_{L_{r}}^{α}.■

3.2 Powers greater than 1

Now we examine f^{α} for 1 < α. As we shall see, this power does not behave as well as it did for α ≤ 1, there are more technical difficulties for less result.

Lemma 3.5. Let 0 < δ < 1, 1 ≤ b ≤ b + b^{δ} ≤ a be reals. Then for every 1 ≤ γ we have

(a^{γ} − b^{γ}) ≤ 2^{γ} (a − b)^{γ+ \frac{(1−δ)(γ−1)}{δ}} .

Proof. We show that the value of the fraction

\frac{a^{γ} − b^{γ}}{(a − b)^{γ}}

is maximal when a = b + b^{δ}. This is true since

(\frac{x^{γ} − b^{γ}}{(x − b)^{γ}} )′ = (x − b)^{−1−γ}γb (b^{γ−1} − x^{γ−1}) ≤ 0

for b ≤ x and 1 ≤ γ, so the fraction

\frac{x^{γ} − b^{γ}}{(x − b)^{γ}}

is a decreasing function of x for b ≤ x.

For every 1 ≤ γ and 0 ≤ t ≤ 1 we have

(1 + t)^{γ} ≤ 1 + 2^{γ}t,  (1)

since the left-hand side of the expression is convex while the right-hand side is linear in t, and the inequality holds for t = 0 and t = 1.

Applying (1) with t = \frac{1}{y} we have

(1 + y)^{γ} − y^{γ} ≤ 2^{γ}y^{γ−1},

and with y = b^{1−δ} we get

\frac{a^{γ} − b^{γ}}{(a − b)^{γ}} ≤ \frac{(b + b^{δ})^{γ} − b^{γ}}{(b + b^{δ} − b)^{γ}} = (1 + b^{1−δ})^{γ} − b^{(1−δ)γ} ≤

≤ 2^{γ}b^{(1−δ)(γ−1)} ≤ 2^{γ}(a − b)^{\frac{(1−δ)(γ−1)}{δ}} ,

as stated.■

Lemma 3.6. Let 0 < q < r < ∞, H ⊂ 𝕋 and f : 𝕋 → ℂ such that L(f) = q and Δ_{h}f ∈ L_{r} for every h ∈ H. Then there is a continuous function ρ : (0, q) → ℝ such that p < ρ(p) and for every 0 < p < q there exists a function f_{p} such that L(f_{p}) = p and ρ(p) ≤ L(Δ_{h}f_{p}) for every h ∈ H.

Proof. Let f_{p} = |f|^{\frac{q}{p}}. Again, it is obvious that L(f_{p}) = p.

Fix an h ∈ H and let 0 < δ < 1, p < s be arbitrary. The value of δ will be chosen later. Let

a(x) = max(|f(x + h)|, |f(x)|),
b(x) = min(|f(x + h)|, |f(x)|),
A = {x ∈ 𝕋 : b(x) + b^{δ}(x) ≤ a(x)} ,
A′ = {x ∈ 𝕋 : 1 ≤ b(x)} .

Then

||Δ_{h}f_{p}||_{L_{s}}^{s} = ∫_{𝕋} ||f(x + h)|^{\frac{q}{p}} − |f(x)|^{\frac{q}{p}} |^{s} dx =
= ∫_{𝕋} (a^{\frac{q}{p}} (x) − b^{\frac{q}{p}} (x))^{s} dx =
= ∫_{A∩A′} (a^{\frac{q}{p}} (x) − b^{\frac{q}{p}} (x))^{s} dx+

+ ∫_{(𝕋\A)∩A′} (a^{\frac{q}{p}} (x) − b^{\frac{q}{p}} (x))^{s} dx + ∫_{𝕋\A′} (a^{\frac{q}{p}} (x) − b^{\frac{q}{p}} (x))^{s} dx.

We estimate the integrals separately. For the first term, by applying Lemma 3.5 with γ = \frac{q}{p}, we get

∫_{A} (a^{\frac{q}{p}} (x) − b^{\frac{q}{p}} (x))^{s} dx ≤

≤ 2^{\frac{q}{p}} ∫_{𝕋} (a(x) − b(x))^{s \frac{q}{p} +s^{\frac{(1−δ)( \frac{q}{p} −1)}{δ}}} .  (2)

Since Δ_{h}f ∈ L_{r} means

∫_{𝕋} |f(x + h) − f(x)|^{r} dx = ∫_{𝕋} (a(x) − b(x))^{r} dx < ∞,

(2) is finite if

s\frac{q}{p} + s\frac{(1 − δ)(\frac{q}{p} − 1)}{δ} ≤ r,

that is if

s ≤ \frac{rp}{q + \frac{(1−δ)(q−p)}{δ}} .  (3)

For the second term, again from Lemma 3.5 with γ = \frac{q}{p}, we get

∫_{(𝕋\A)∩A′} (a^{\frac{q}{p}} (x) − b^{\frac{q}{p}} (x))^{s} dx ≤ ∫_{𝕋} ((b + b^{δ})^{\frac{q}{p}} − b^{\frac{q}{p}} )^{s} ≤

≤ 2^{s \frac{q}{p}} ∫_{𝕋} b^{sδ( \frac{q}{p} +^{\frac{(1−δ)( \frac{q}{p} −1)}{δ}} )} ≤ 2^{s \frac{q}{p}} ∫_{𝕋} b^{s(δ+ \frac{q}{p} −1)} ≤ 2^{s \frac{q}{p}} ∫_{𝕋} |f|^{s(δ+ \frac{q}{p} −1)}, which is finite if

s (δ + \frac{q}{p} − 1) < q,

that is if

s < \frac{p}{1 − (1 − δ)\frac{p}{q}} .  (4)

To estimate the third term we use that if 0 ≤ b ≤ 1, b ≤ a then

a^{α} ≤ 2^{α} (a − b)^{α} + 2^{α},

for 0 ≤ a ≤ 2 by the second term, for 2 ≤ a by the first term. Applying this with α = s\frac{q}{p}, for the third term we have

∫_{𝕋\A′} (a^{\frac{q}{p}} (x) − b^{\frac{q}{p}} (x))^{s} dx ≤ ∫_{𝕋\A′} a^{s \frac{q}{p}} ≤ ∫_{𝕋\A′} 2^{s \frac{q}{p}} ((a − b)^{s \frac{q}{p}} + 1) ,

which is finite whenever

s\frac{q}{p} ≤ r,

that is if

s ≤ \frac{pr}{q}.  (5)

With the unique 0 < δ < 1 fulfilling

\frac{1 − δ}{δ} (q − p) = \frac{r − q}{2} ,

the right-hand side of (3) and (4) is bigger than p. Fixing this δ let

ρ = min \{\frac{rp}{q + \frac{(1−δ)(q−p)}{δ}} , \frac{p}{1 − (1 − δ)\frac{q}{p}} , \frac{pr}{q}\} .

Then p < ρ, and (3), (4) and (5) are satisfied for s < ρ. This proves the statement.■

3.3 A small power

To put a function f with L(f) = 0 in a L_{p} class for 0 < p it must be composed by a function which increases more slowly than x^{α} for any 0 < α. This is done in the following lemmas. They will help to deal with the p = 0 case.

Lemma 3.7. If Φ : [0, ∞) → [0, ∞) is a concave function with

Φ(1) ≤ 1,

f : 𝕋 → ℝ is a function and Δ_{h}f ∈ L_{p} for some 0 ≤ p < ∞ and h ∈ 𝕋, then

Δ_{h}Φ(|f|) ∈ L_{p}

with ||Δ_{h}Φ(|f|)||_{L_{p}} bounded by a function of ||Δ_{h}f||_{L_{p}}.

Proof. Since 0 ≤ Φ, the concavity of Φ and Φ(1) ≤ 1 imply Φ(x) ≤ x+1. Using the concavity and Φ(x) ≤ x + 1 we get

|Φ(|f(x + h)|) − Φ(|f(x)|)| ≤ Φ(||f(x + h)| − |f(x)||) − Φ(0) ≤
≤ ||f(x + h)| − |f(x)|| + 1 ≤ |f(x + h) − f(x)| + 1,

which clearly implies the statement.

Lemma 3.8. Let (λ_{i})_{i=0}^{∞} be a given sequence of nonnegative reals with

∑_{i=0}^{∞} λ_{i} = 1,  (6)

∑_{i=1}^{∞} iλ_{i} = ∞.  (7)

Then there is a Φ : [0, ∞) → [0, ∞) concave function with Φ(1) ≤ 1 such that

∑_{i=0}^{∞} Φ(i)λ_{i} < ∞,  (8)

∑_{i=0}^{∞} Φ^{3}(i)λ_{i} = ∞.  (9)

Proof. From (6) we have an increasing sequence (i_{k})_{k=0}^{∞} of integers with

∑_{i_{k}≤i} λ_{i} ≤ \frac{1}{2^{k}} .  (10)

We define recursively a strictly increasing sequence (a_{k})_{k=0}^{∞} of integers fulfilling

max{k, i_{k}} ≤ a_{k},  (11)

a_{k} − a_{k−1} ≤ a_{k+1} − a_{k}  (12)

and

\frac{1}{k^{2}} ≤ ∑_{a_{k}≤i<a_{k+1}} (k + \frac{i − a_{k}}{a_{k} − a_{k−1}} ) λ_{i}  (13)

for every k = 1, 2, …

Let a_{0} = 0, a_{1} = i_{1}. If a_{k} is already defined we define a_{k+1} simply by taking its value large enough. Conditions (11) and (12) can easily be satisfied, while (7) shows that (13) also holds for a large a_{k+1}, since the coefficients in (13) grow linearly in i.

If there is a given strictly increasing sequence (a_{k})_{k=0}^{∞} of integers with a_{0} = 0 and a sequence (b_{k})_{k=0}^{∞} of reals, we denote by Φ_{(a_{k}),(b_{k})} the function for which Φ(a_{k}) = b_{k} and linear on the intervals [a_{k}, a_{k+1}]. With this notation let Φ = Φ_{(a_{k}),(k)}, a concave function by (12) and Φ(1) ≤ 1 by (11).

Since

∑_{i=0}^{∞} Φ(i)λ_{i} = ∑_{k=0}^{∞} ∑_{a_{k}≤i<a_{k+1}} Φ(i)λ_{i} ≤

≤ ∑_{k=0}^{∞} ∑_{a_{k}≤i<a_{k+1}}(k + 1)λ_{i} ≤ ∑_{k=0}^{∞} ∑_{a_{k}≤i}(k + 1)λ_{i} ≤

≤ ∑_{k=0}^{∞} ∑_{i_{k}≤i}(k + 1)λ_{i} ≤ ∑_{k=0}^{∞} \frac{(k + 1)}{2^{k}} < ∞,

we have property (8). If we knew

\frac{1}{k^{2}} ≤ ∑_{a_{k}≤i} Φ(i)λ_{i}  (14)

for infinitely many k’s, then, since k ≤ Φ(i) for a_{k} ≤ i, we would have

1 ≤ ∑_{a_{k}≤i} Φ^{3}(i)λ_{i}

for infinitely many k’s, which would show that the series of (9) cannot converge. So if (14) holds only for a finite set of k’s then we modify Φ to have it for infinitely many k’s.

If

∑_{a_{k}≤i} Φ(i)λ_{i} < \frac{1}{k^{2}} ,

then there is a 0 < d such that with the sequence b_{l} = l, if l ≤ k and b_{l} = l+d, if k < l, we have

∑_{a_{k}≤i} Φ_{(a_{k}),(b_{k})}(i)λ_{i} = \frac{1}{k^{2}}  (15)

and the function Φ_{(a_{k}),(b_{k})} is still concave. This follows from (13), which says that if Φ would increase on [a_{k}, a_{k+1}] as is does on [a_{k−1}, a_{k}] then

\frac{1}{k^{2}} ≤ ∑_{a_{k}≤i<a_{k+1}} Φ(i)λ_{i}.

Moreover, if this modification is done for a k bigger than 2, then also Φ(1) ≤ 1 remains true.

We repeat this modification step finitely or infinitely many times to get, as a monotone increasing point-wise limit of concave functions, a concave function Φ for which (14) holds for infinitely many k’s, so we certainly have property (9). We show that we did not loose property (8).

Either there were only finitely or infinitely many modification steps, in the jth step, say at a_{k_{j}}, the value of the sum

∑_{i=1}^{∞} Φ(i)λ_{i}

could increase by at most \frac{1}{k_{j}^{2}} . Since

∑_{j=1}^{∞} \frac{1}{k_{j}^{2}} ≤ ∑_{j=1}^{∞} \frac{1}{j^{2}} < ∞,

∑_{i=0}^{∞} Φ(i)λ_{i}

must converge, so the proof is complete.■

As an application, we can prove the last lemma of this section.

Lemma 3.9. Let 0 < s < ∞, H ⊂ 𝕋 and f : 𝕋 → ℝ be such that L(f) = 0 and Δ_{h}f ∈ L_{s} for every h ∈ H. Then there is a function 0 ≤ f_{1} such that f_{1} ∈ L_{1} \L_{3} and Δ_{h}f_{1} ∈ L_{4} for every h ∈ H with ||Δ_{h}f_{1}||_{L_{4}} bounded by some function of ||Δ_{h}f||_{L_{s}}.

Proof. If s < 4, then let

f̃ = [|f|^{\frac{s}{4}} ] ,

else we set

f̃ = [|f|] ,

where [.] denotes the integer part. In the first case, by Lemma 3.4, we have Δ_{h}f̃ ∈ L_{4} with

||Δ_{h}f̃||_{L_{4}} ≤ ||Δ_{h}f||_{L_{s}} + 2  (16)

for every h ∈ H. In the second case (16) holds obviously for every h ∈ H. Since L(f) = 0, we have L(f̃) = 0 in both cases.

By applying Lemma 3.8 with

λ_{i} = λ ([f̃ = i]) ,

we get the concave function Φ for which f_{1} = Φ ∘ f̃ ∈ L_{1} \ L_{3}. According to Lemma 3.7, Δ_{h}f_{1} ∈ L_{4} with ||Δ_{h}f_{1}||_{L_{4}} bounded by some function of ||Δ_{h}f||_{L_{s}} for every h ∈ H. This proves the statement.■

Corollary 3.10. Let 0 < s < ∞, H ⊂ 𝕋 and f : 𝕋 → ℝ be such that L(f) = 0 and Δ_{h}f ∈ L_{s} for every h ∈ H. Then

H ∈ ℌ(L_{u}, L_{v}) ∩ ℌ(L_{u}, L_{v^{<}})

for every 0 ≤ u < v < ∞.

Proof. Let f_{1} be as in the previous Lemma.

If L(f_{1}) ≤ \frac{u+v}{2} or 4 ≤ v, we apply Lemma 3.4 for f_{1} with r = 4 and

α = min \{\frac{4}{v}, \frac{2L(f_{1})}{u + v}\} .

We get a function f^{⋆} with u < L(f^{⋆}) < v and Δ_{h}f^{⋆} ∈ L_{v} for every h ∈ H.

If \frac{u+v}{2} < L(f_{1}) and v < 4, we apply Lemma 3.6 for f_{1} with q = L(f_{1}), r = 4 to have a u < p < v and again a function f^{⋆} with L(f^{⋆}) = p and Δ_{h}f^{⋆} ∈ L_{v} for every h ∈ H. This can be done by the continuous dependence of ρ from p. Thus in both cases we get

H ∈ ℌ(L_{u}, L_{v}) ∩ ℌ(L_{u}, L_{v^{<}}).■

3.4 The inclusion theorems

Now we can prove the two inclusion theorems.

Proof of Theorem 3.1. Suppose that for a set H ⊂ 𝕋 we have H ∈ ℌ(L_{p}, L_{q}), that is there is a function f ∈ L_{p} \ L_{q} with Δ_{h}f ∈ L_{q} for every h ∈ H. Then by Lemma 3.4 with α = \frac{q}{s} we have f̃ = |f|^{\frac{q}{s}} ∈ L_{\frac{ps}{q}} \ L_{s} and Δ_{h}f̃ ∈ L_{s} for every h ∈ H. So

H ∈ ℌ(L_{r}, L_{s})

if r < L(f̃).

Suppose now that L(f̃) ≤ r < s. If 0 < L(f̃), then by applying Lemma 3.4 with α = \frac{2L(f̃)}{r+s} we have a function \overline{f} ∈ L_{r} \ L_{s} with Δ_{h}\overline{f} ∈ L_{s \frac{r+s}{2L(f̃)}} ⊂ L_{s} for every h ∈ H. Thus H ∈ ℌ(L_{r}, L_{s}).

If L(f̃) = 0 then the statement follows from Corollary 3.10.■

Proof of Theorem 3.2. It is enough to prove that

ℌ(L_{p}, L_{q^{<}}) ⊂ ℌ(L_{r}, L_{s^{<}})

for every 0 ≤ p < q < ∞, 0 ≤ r < s < ∞. Let H ∈ ℌ(L_{p}, L_{q^{<}}). We distinguish two cases.

Case 1. (q ≤ s) We proceed as in the proof of Theorem 3.1. By Lemma 3.4 with α = \frac{q}{s} we have f̃ = |f|^{\frac{q}{s}} ∈ L\frac{s}{q} \ L_{s^{<}} and Δ_{h}f̃ ∈ L_{s^{<}} for every h ∈ H, and we are done if r < L(f̃).

Now if 0 < L(f̃) ≤ r < s then again by Lemma 3.4 with α = \frac{2L(f̃)}{r+s} we have the function \overline{f} ∈ L_{r} \ L_{s^{<}} with Δ_{h}\overline{f} ∈ L_{(s \frac{r+s}{2L(f̃)} )}^{<} ⊂ L_{s} for every h ∈ H

which shows H ∈ ℌ(L_{r}, L_{s^{<}}).

If L(f̃) = 0 then the statement follows from Corollary 3.10.

Case 2. (s < q) Let us consider first 0 < L(f). By Lemma 3.6 we have a pair u, v satisfying 0 < u < v < s and a function f̃ with L(f̃) = u and Δ_{h}f̃ ∈ L_{v^{<}} for every h ∈ H. This f̃ shows that H ∈ ℌ(L_{u}, L_{v^{<}}). Now we are in the situation of Case 1., so we have H ∈ ℌ(L_{r}, L_{s^{<}}).

Now if L(f) = 0 then again the statement follows from Corollary 3.10.■

4 Upper bounds

The proof of the upper bounds uses an idea of de Bruijn proving the weak difference property for the L_{2} class (see [8]). The trick is to average the Fourier coefficients of the functions Δ_{h}f for different h’s.

Theorem 4.1. For every 1 ≤ p < 2,

ℌ(L_{p}, L_{2}) ⊂ 𝔑.

Proof. Let H ⊂ 𝕋 be not an N-set and suppose that for a function f ∈ L_{p} we have Δ_{h}f ∈ L_{2} for every h ∈ H. We show that f ∈ L_{2}.

For 0 < M let

A_{M} = {h ∈ H : ||Δ_{h}f||_{L_{2}}^{2} ≤ M} .

It is easy to see that A_{M} is compact and for M < N we have A_{M} ⊂ A_{N}. Since according to our assumption the increasing union

H = ⋃_{M∈ℕ}A_{M}

is not an N-set, we have from Theorem 2.1 that for an M ∈ ℕ the set A_{M} is not weak-Dirichlet.

Let A_{M} ∉ 𝔴𝔇. This means that there exists a probability measure μ supported by A_{M} for which

lim sup_{|n|→∞} |μ̂(n)| < 1.

Thus for an η > 0 sufficiently small and an n_{0} we have

η < |μ̂(n) − 1|  (17)

for every |n| > n_{0}.

For every h ∈ A_{M} we have

N(h) = ||Δ_{h}f||_{L_{2}}^{2} = ||𝔉Δ_{h}f||_{l_{2}}^{2} = ∑ |𝔉(f ∘ γ_{h})(n) − 𝔉f(n)|^{2} =

= ∑ |(𝔉f)(n) (e^{−2πinh} − 1)|^{2} .

N(h), as a function of h, is measurable and bounded by M on the support of μ. So by the Hölder inequality and the Fubini Theorem,

||(𝔉f) (μ̂ − 1)||_{l_{2}}^{2} = ∑ |𝔉f|^{2} (n) |μ̂ − 1|^{2} (n) =
= ∑ |𝔉f|^{2} (n) |∫_{𝕋} (e^{−2πinh} − 1) dμ(h)|^{2} ≤
≤ ∑ |𝔉f|^{2} (n) ∫_{𝕋} |e^{−2πinh} − 1|^{2} dμ(h) =

= ∫_{𝕋} ∑ |(𝔉f) (n) (e^{−2πinh} − 1)|^{2} dμ(h) = ∫_{𝕋} N(h)dμ(h) ≤ M.

From this and (17) we have

||𝔉f||_{l_{2}}^{2} ≤ ∑_{j=−n_{0}}^{n_{0}} |𝔉f|^{2} (j) + \frac{M}{η} .

Thus 𝔉f ∈ l_{2}, which implies f ∈ L_{2} by the Theorem of Hausdorff and Young. This proves the statement.■

Corollary 4.2. 1. For every 0 ≤ p < q ≤ 2,

ℌ(L_{p}, L_{q}) ⊂ 𝔑.

2. For every 0 ≤ p < q < ∞,

ℌ(L_{p}, L_{q^{<}}) ⊂ 𝔑.

1. Let 0 ≤ p < q ≤ 2. By Theorem 3.1 and Theorem 4.1 we have

ℌ(L_{p}, L_{q}) ⊂ ℌ(L_{1}, L_{2}) ⊂ 𝔑,

as stated.

2. For 0 ≤ p < q < ∞ we have by Theorem 3.2

ℌ(L_{p}, L_{q^{<}}) ⊂ ℌ(L_{1}, L_{2^{<}}).

Since

ℌ(L_{1}, L_{2^{<}}) ⊂ ⋃_{1<r<2} ℌ(L_{1}, L_{r}) ⊂ 𝔑,

our statement follows.■

5 Lower bounds

The lower bounds are based on the non-ejectivity property of weak-Dirichlet sets.

5.1 Non-ejective sets

Lemma 5.1. For every compact non-ejective set H ⊂ 𝕋, there is a function f : 𝕋 → ℝ such that L(f) = 0 and Δ_{h}f ∈ L_{1} with

||Δ_{h}f||_{L_{1}} ≤ 2

for every h ∈ H.

Proof. By the non-ejectivity of H, for every fixed n ∈ ℕ and for every 0 < ε there exists a set K_{n}(ε) such that

λ (K_{n}(ε)) = \frac{1}{n^{2}}  (18)

and

λ (Δ_{h}K_{n}(ε)) ≤ ε, ∀h ∈ H.  (19)

Let

f_{n} = 2^{n}χ_{K_{n}( \frac{1}{4^{n}})}

and

f = ∑_{n=1}^{∞} f_{n},

as a point-wise limit of measurable functions. We show that f < ∞ almost everywhere, L(f) = 0 and Δ_{h}f ∈ L_{1} with ||Δ_{h}f||_{L_{1}} ≤ 2 for every h ∈ H.

By (18) we have

∑ λ (K_{n} (\frac{1}{4^{n}} )) = ∑ \frac{1}{n^{2}} < ∞,

so the first statement follows from the Borel-Cantelli Lemma.

Since f_{n} ≤ f for every n ∈ ℕ and for every 0 < α we have

||f_{n}||_{L_{α}} = \frac{1}{n^{\frac{2}{α}}} 2^{n} → ∞

if n → ∞, we can conclude L(f) = 0.

From f < ∞ almost everywhere we have that the series of the difference functions

∑_{n=1}^{∞} Δ_{h}f_{n}

converges almost everywhere to Δ_{h}f. Since for every N, M ∈ ℕ with N < M we have

||∑_{n=N}^{M} Δ_{h}f_{n}||_{L_{1}} ≤ ∑_{n=N}^{M} ||Δ_{h}f_{n}||_{L_{1}} ≤

≤ ∑_{n=N}^{M} \frac{2}{2^{n}} ≤ \frac{1}{2^{N−2}} ,

the series

∑_{n=1}^{∞} Δ_{h}f_{n}

is Cauchy in L_{1}, so it is convergent point-wise and in L_{1} too, so the point-wise limit Δ_{h}f belongs to L_{1} with

||Δ_{h}f||_{L_{1}} = ||∑_{n=1}^{∞} Δ_{h}f_{n}||_{L_{1}} ≤ ∑_{n=1}^{∞} ||Δ_{h}f_{n}||_{L_{1}} ≤ ∑_{n=1}^{∞} \frac{2}{2^{n}} ≤ 2,

which completes the proof.■

Corollary 5.2. For every H ∈ 𝔠𝔴𝔇 there is a function f : 𝕋 → ℝ such that L(f) = 0 and Δ_{h}f ∈ L_{1} with

||Δ_{h}f||_{L_{1}} ≤ 2

for every h ∈ H.

Proof. Since by Theorem 2.4 every compact weak-Dirichlet set is nonejective, the statement follows from the previous lemma.■

5.2 General N-sets

On the account of Theorem 2.2, the remaining task is to examine the behavior of ℌ(L_{p}, L_{q}) for compact increasing union.

Lemma 5.3. Let 1 ≤ p < q ≤ r < ∞ and suppose we have an increasing sequence of sets (H_{i}) such that for every i ∈ ℕ there is a function f_{i} ∈ L_{p} \L_{q} such that for every h ∈ H_{i} we have ||Δ_{h}f_{i}||_{L_{r}} ≤ K_{i}. Then for

H = ⋃_{i=1}^{∞} H_{i}

there is a function f ∈ L_{p} \ L_{q} such that ||Δ_{h}f||_{L_{r}} < ∞ for every h ∈ H.

Proof. By taking absolute value we can suppose that 0 ≤ f_{i} for every i ∈ ℕ. For every M ∈ ℝ and i ∈ ℕ, with the notation f_{i} ∧ M = min(f_{i}, M), we have

||f_{i} ∧ M||_{L_{p}} ≤ ||f_{i}||_{L_{p}}

and

||Δ_{h}(f_{i} ∧ M)||_{L_{r}} ≤ ||Δ_{h}f_{i}||_{L_{r}} ≤ K_{i}

for any h ∈ H_{i}, while ||f_{i} ∧ M||_{L_{q}} → ∞ for M → ∞. So we can find an M_{i} ∈ ℝ such that

2^{i} max(K_{i}, ||f_{i}||_{L_{p}}) ≤ ||f_{i} ∧ M_{i}||_{L_{q}} .

Let

f = ∑_{i=1}^{∞} \frac{f_{i} ∧ M_{i}}{2^{i} max(K_{i}, ||f_{i}||_{L_{p}})},  (20)

as a point-wise limit of measurable functions.

It is obvious that f ∈ L_{p} \ L_{q}, specially f < ∞ almost everywhere. Thus the series (20) is absolute convergent almost everywhere, so for every h ∈ H the series

∑_{i=1}^{∞} Δ_{h} \frac{f_{i} ∧ M_{i}}{2^{i} max(K_{i}, ||f_{i}||_{L_{p}})}

converges point-wise to Δ_{h}f almost everywhere. If h ∈ H_{j} for a j ∈ ℕ, then for every N, M ∈ ℕ with j < N < M we have

||∑_{i=N}^{M} Δ_{h} (\frac{f_{i} ∧ M_{i}}{2^{i} max(K_{i}, ||f_{i}||_{L_{p}})})||_{L_{r}} ≤ ∑_{i=N}^{M} ||Δ_{h} (\frac{f_{i} ∧ M_{i}}{2^{i} max(K_{i}, ||f_{i}||_{L_{p}})})||_{L_{r}} =

= ∑_{i=N}^{M} \frac{||Δ_{h}(f_{i} ∧ M_{i})||_{L_{r}}}{2^{i} max(K_{i}, ||f_{i}||_{L_{p}})} ≤ ∑_{i=N}^{M} \frac{1}{2^{i}} ≤ \frac{1}{2^{N−1}} ,

so the series

∑_{i=1}^{∞} Δ_{h} \frac{f_{i} ∧ M_{i}}{2^{i} max(K_{i}, ||f_{i}||_{L_{p}})}

is Cauchy, hence convergent in L_{r}. So we have that Δ_{h}f itself is in L_{r} for every h ∈ H, which completes the proof.■

Finally we can prove the remaining statements of Theorem 1.8.

Theorem 5.4. For every 0 ≤ p < q < ∞,

1.

𝔑 ⊂ ℌ(L_{p}, L_{q^{<}}).

2.

𝔑 ⊂ ℌ(L_{p}, L_{q}).

Proof.

1. Let H ∈ 𝔑. By Theorem 2.2 there is an increasing sequence of sets (H_{i}) ⊂ 𝔠𝔴𝔇 with

H = ⋃_{i=1}^{∞} H_{i}.

By Corollary 5.2, for any i = 1, 2, … we have a function f_{i} such that L(f_{i}) = 0 and Δ_{h}f_{i} ∈ L_{1} with

||Δ_{h}f_{i}||_{L_{1}} ≤ 2

for every h ∈ H_{i}. We can apply Lemma 3.9 to obtain a sequence of functions f̂_{i} fulfilling f̂_{i} ∈ L_{1} \ L_{3} and Δ_{h}f̂_{i} ∈ L_{4} with

||Δ_{h}f̂_{i}||_{L_{1}} ≤ K

for some K ∈ ℝ, for every h ∈ H_{i}. Thus by Lemma 5.3 there is a function f ∈ L_{1} \ L_{3} for which Δ_{h}f ∈ L_{4} for every h ∈ H. This shows H ∈ ℌ(L_{1}, L_{4^{<}}), and by Theorem 3.2 the proof is complete.

2. Let H ∈ 𝔑. We know from the preceding that H ∈ ℌ(L_{p}, L_{q^{<}}) for every 0 ≤ p < q < ∞, so from

ℌ(L_{p}, L_{q^{<}}) ⊂ ⋃_{r<q} ℌ(L_{p}, L_{r})

we have H ∈ ℌ(L_{p}, L_{r}) for some r < q. By Theorem 3.1 we have

ℌ(L_{p}, L_{r}) ⊂ ℌ(L_{p}, L_{q}),

thus H ∈ ℌ(L_{p}, L_{q}), which completes the proof.■

Corollary 4.2 and Theorem 5.4 together prove our main theorem.

6 Ejectivity

As a corollary of Lemma 5.1 and Theorem 4.2 we get that every non-ejective set is an N-set, in particular every compact non-ejective set is a compact N-set, hence a compact weak-Dirichlet set. This proves Theorem 2.4 in one direction. However, as Imre Ruzsa showed, the averaging technique of Theorem 4.1 can be applied directly to the ejective set, which gives a direct proof [9].

6.1 A minimax theorem

In the main theorem of the preceding we have proved in particular that for a weak-Dirichlet set H there is a function f ∈ L_{2} \ L_{3} such that every h ∈ H the difference functions Δ_{h}f are so small that they may belong to any fixed higher class. The following theorem shows a quantitative analogue of this result in the class L_{2}. In the proof of Ruzsa Imre this result is the first step for proving that every weak-Dirichlet set is non-ejective. Here we give a different proof.

For a set H ⊂ 𝕋, M(H) denotes the set of probability measures supported by H. Let

F = {f : 𝕋 → ℂ : f ∈ L_{2}, ||f||_{L_{2}} = 1, ∫_{𝕋} f = 0} ,

while _{ℝ}l_{p} ⊂ l_{p} stands for the real Banach-space of two-way infinite p-sum-mable sequences.

Theorem 6.1. Let H ⊂ 𝕋 be a compact set. Then

inf_{f∈F} sup_{h∈H} ||Δ_{h}f||_{L_{2}}^{2} = sup_{μ∈M(H)} inf_{n≠0} ∫_{𝕋} |e^{−2πinh} − 1|^{2}dμ(h).

Proof. If for an f ∈ F we have

sup_{h∈H} ||Δ_{h}f||_{L_{2}}^{2} ≤ ε^{2},

then by the Parseval formula

∑ |𝔉f(n)|^{2} = ||f||_{L_{2}}^{2} = 1

and

∑ |𝔉f(n)|^{2} |e^{−2πinh} − 1|^{2} = ||𝔉Δ_{h}f||_{l_{2}}^{2} = ||Δ_{h}f||_{L_{2}} ≤ ε^{2},

thus for every μ ∈ M(H) we have by the Fubini Theorem

∑ |𝔉f(n)|^{2} ∫_{𝕋} |e^{−2πinh} − 1|^{2} dμ(h) =

∫_{𝕋} ∑ |𝔉f(n)|^{2} |e^{−2πinh} − 1|^{2} dμ(h) ≤ ε^{2}.

Since 𝔉f(0) = 0 for our f ∈ F, we get

inf_{n≠0} ∫_{𝕋} |e^{−2πinh} − 1|^{2}dμ(h) ≤ ε^{2},

which proves

sup_{μ∈M(H)} inf_{n≠0} ∫_{𝕋} |e^{−2πinh} − 1|^{2}dμ(h) ≤ inf_{f∈F} sup_{h∈H} ||Δ_{h}f||_{L_{2}}^{2}.

For the other direction let

ε^{2} = inf_{f∈F} sup_{h∈H} ||Δ_{h}f||_{L_{2}}^{2}.

We denote by (c_{0}, ||.||_{∞}) the real Banach space of two-way infinite nullconvergent sequences, that is (c_{n})_{n=−∞}^{∞} ∈ c_{0} if and only if

lim_{n→∞} |c_{n}| + |c_{−n}| = 0.

For every positive n ∈ ℕ let

e_{n} = (…, 0, \underbrace{1, …, 1, 1}_{n}, 0, \underbrace{1, 1, …, 1}_{n}, 0, …) ∈ l_{∞}.

For an h ∈ 𝕋, let

a(h) = (a_{n}(h))_{n=−∞}^{∞} = (e^{−2πinh} − 1)_{n=−∞}^{∞} ,

and for every Q ∈ ℕ consider the following subsets of c_{0}:

W_{Q} = {P_{m} ((|a_{n}(h)|^{2})) |m ∈ ℕ, Q ≤ m, h ∈ H} ,

where P_{m} : l_{∞} → c_{0},

P_{m} ((c_{n})_{n=−∞}^{∞}) = (…, 0, c_{−m}, c_{−m+1}, …, c_{0}, …, c_{m−1}, c_{m}, 0, …)

is the projection to the central 2m+1 coordinates. Let K_{Q} denote the closure of the convex hull of W_{Q} in the ||.||_{∞} norm. We show that for every 0 ≤ δ and n, Q ∈ ℕ,

(ε^{2} − δ)e_{n} ∈ K_{Q}.

Suppose that this is not true for a Q and n and consider the Minkowsky sum K of K_{Q} and the segment [0, \frac{δ}{2}]e_{n} and let

γ = sup{0 < τ < 1 : τe_{n} ∈ K}.

We have \frac{δ}{2} ≤ γ ≤ ε^{2} − \frac{δ}{2}.

Since K is a convex subset of the locally convex Banach space l_{∞} and γe_{n} is one point on its boundary, by the Theorem of Hahn and Banach there is a functional ϕ ∈ c_{0}^{⋆} with norm 1 such that

ϕ(c) ≤ γ, ∀c ∈ K  (21)

and ϕ(e_{n}) = 1. Since c_{0}^{⋆} = _{ℝ}l_{1}, ϕ can be represented as a (z_{n})_{n=−∞}^{∞} ∈ _{ℝ}l_{1} sequence, for which we have

∑_{j=−∞}^{∞} |z_{j}| = ||(z_{n})||_{L_{1}} = 1 = ϕ(e_{n}) = ∑_{j=−n}^{n} z_{j},

that is 0 ≤ z_{j} and z_{m} = 0 for n < |m| and for n = 0. Let (d_{n})_{n=−∞}^{∞} ∈ _{ℝ}l_{2} be the sequence for which 0 ≤ d_{n} = \sqrt{z_{n}}. Then, from (21), || (a_{n}(h)d_{n}) ||_{l_{2}}^{2} ≤ γ for every h ∈ H.

Consider now f = 𝔉 ((d_{n})_{n=−∞}^{∞}). By the Parseval formula ||f||_{L_{2}}^{2} = 1, ∫_{𝕋} f = 0, so f ∈ F, but for every h ∈ H,

||Δ_{h}f||_{L_{2}}^{2} = ||𝔉 (Δ_{h}f) ||_{l_{2}}^{2} = || (a_{n}(h)d_{n}) ||_{l_{2}}^{2} ≤ γ ≤ ε^{2} − \frac{δ}{2},

which is a contradiction.

This means that for every 0 < δ and Q ∈ ℕ there is a convex combination (w_{n}^{Q})_{n=−∞}^{∞} of certain elements of W_{Q} such that

(ε^{2} − δ) ≤ w_{j}^{Q}

for every 0 ≠ |j| ≤ Q. By reformulation, for every Q ∈ ℕ there is a μ_{Q} convex combination of Dirac measures on 𝕋 such that for every |j| ≤ Q, j ≠ 0,

ε^{2} − δ ≤ (∫_{𝕋} P_{Q}(|a(h)|^{2})dμ_{Q}(h)) (j) = P_{Q} (∫_{𝕋} |a(h)|^{2}dμ_{Q}(h)) (j),

that is

ε^{2} − δ ≤ ∫_{𝕋} |a_{j}|^{2}(h)dμ_{Q}(h) = ∫_{𝕋} |e^{−2πijh} − 1|^{2} dμ_{Q}(h)

for every |j| ≤ Q, j ≠ 0. Moreover μ_{Q} is a probability measure.

The Banach space of measures on 𝕋 being the dual space of the continuous functions on 𝕋, we can take a subsequence μ_{Q_{j}} weakly converging to a μ probability measure, and since suppμ_{Q} ⊂ H for every Q ∈ ℕ and H is compact we have suppμ ⊂ H also.

By the of definition of weak limit we get

ε^{2} − δ ≤ lim_{j→∞} ∫_{𝕋} |e^{−2πinh} − 1|^{2} dμ_{Q_{j}}(h) = ∫_{𝕋} |e^{−2πint} − 1|^{2} dμ(h)

for every n ≠ 0. Taking δ → 0 we obtain

inf_{f∈F} sup_{h∈H} ||Δ_{h}f||_{L_{2}}^{2} ≤ sup_{μ∈M(H)} inf_{n≠0} ∫_{𝕋} |e^{−2πinh} − 1|^{2}dμ(h),

so the proof is complete.■

Remark 6.2. The argument of Imre Ruzsa for proving that every compact weak-Dirichlet set is non-ejective goes on the following way. If the set H ⊂ 𝕋 is a compact weak-Dirichlet set, then by Remark 1.5,

sup_{μ∈M} inf_{n≠0} ∫_{𝕋} |e^{−2πinh} − 1|^{2}dμ(h) = 0.

So by Theorem 6.1 for every 0 < ε there is a function f^{(ε)} ∈ L_{2} with ||f^{(ε)}||_{L_{2}} = 1 and ∫_{𝕋} f^{(ε)} = 0 such that ||Δ_{h}f^{(ε)}||_{L_{2}} ≤ ε for every h ∈ H.

For a sequence (ε_{n}) ∈ {−1, +1}^{ℤ} let

f_{(ε_{n})} = 𝔉^{−1}((ε_{n}(𝔉f)(n))),

that is the function obtained from f by signing its Fourier coefficients with (ε_{n}). By averaging on the signing, Imre Ruzsa showed that there is a sequence (ε_{n}) ∈ {−1, +1}^{ℤ} and a u ∈ ℝ such that for the sub-level set A = [f_{(ε_{n})}^{(ε)} < u] we get

sup_{h∈H} λ(Δ_{h}A) ≤ ελ(A),

from which the non-ejectivity of H follows easily.

6.2 Conclusions

There is no reason to think that the conjecture of Tamás Keleti fails for the class ℌ(L_{p}, L_{q}) if 2 ≤ q. Our proof brakes down at q = 2 only because of that caprice of the Fourier transform that it happens to be isometric isomorphism just on the L_{2} class. By using an analogue of the argument of Ruzsa Imre we could avoid the Fourier transform and prove the conjecture for the remaining classes. On the other hand, for 2 < q we have no Theorem 6.1 like statement for the L_{q} class. But from a function f ∉ L_{q} with Δ_{h} ∈ L_{q} for every h ∈ H one may can construct a function for which a sub-level set works on the same way. A difficulty can be that not every function has appropriate sub-level sets even in the L_{2} case, and for 2 < q we cannot average on the Fourier coefficients.

References

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