Periodic Lip^{α} functions with Lip^{β} difference functions ^{∗}

Tamás Keleti ^{†}

December 21, 2004

1 Introduction

In [6] the following notion was introduced: Let G be either the additive subgroup of reals R or the circle group T = R/Z. Let F and G be classes of functions on G with F ⊃ G. We denote by ℌ(F, G) the class of those subsets H of G, for which a function f ∈ F can have difference functions Δ_{h}f(x) = f(x+h)−f(x) in G for every h ∈ H without f belonging to G. That is,

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

We denote by Lip^{α} the class of functions f on T for which there exists an L > 0 such that |f(x) − f(y)| ≤ L|x − y|^{α} for every x, y ∈ T. (Sometimes we identify T with [0, 1). If a ∈ T then by |a| we mean min(a, 1 − a).)

It was proved in [6] (Theorem 4.10) that for 0 < α < β ≤ 1 we have ℌ(Lip^{α}, Lip^{β}) ⊂ 𝔉_{σ} where 𝔉_{σ} denotes the family of those subsets of T that can be covered by a proper F_{σ} subgroup of T. Generalizing a result of M. Balcerzak, Z. Buczolich and M. Laczkovich [1], it was also proved in [6] (Theorem 5.3) that equality holds if β = 1. In this paper we investigate the case when 0 < α < β < 1.

A set H ⊂ T is called a pseudo-Dirichlet set if there exists an increasing sequence of integers (q_{n}) and a sequence (ε_{n}) converging to zero such that for any x ∈ H there exists an n_{0}(x) such that | sin q_{n}πx| < ε_{n} if n ≥ n_{0}(x). We denote the family of pseudo-Dirichlet sets by p𝔇.

The pseudo-Dirichlet sets were considered by N. Bary [2] who proved that they are contained in the class of N_{0}-sets. Clearly p𝔇 contains all the Dirichlet sets; in fact, pseudo-Dirichlet sets are the countable increasing unions of Dirichlet sets (see [?]). These notions together with several other notions of thinnes in harmonic analysis is discussed among others in [3] and [4].

In this paper we prove that

p𝔇 ⊂ ℌ(Lip^{α}, Lip^{β}),

and that all the ℌ(Lip^{α}, Lip^{β}) classes are the same for any 0 < α < β < 1.

1991 Mathematical Reviews Classification. Primary 26A16; Secondary 26A33, 39A70, 42A16.

∗The author wishes to express his gratitude to Professor Miklós Laczkovich for his advice and encouragement to the completion of a PhD dissertation [5] which this paper is based on.

†Supported by OTKA grant F 019468.

2 p𝔇 ⊂ ℌ(Lip^{α}, Lip^{β})

We will use the following well known lemma (see e.g. [2] Chapter XI, pp. 691, Theorem 2):

Lemma 2.1 If 0 < γ < 1 and (q_{n}) is an increasing sequence of integers such that q_{n+1}/q_{n} > λ for a suitable λ > 1, then for any sequence (a_{n}) of complex numbers,

|a_{n}| = O(1/q_{n}^{γ})  ⟺  g(x) = ∑_{n=1}^{∞} a_{n}e^{2πiq_{n}x} ∈ Lip^{γ}.

Theorem 2.2 For any 0 < α < β < 1,

p𝔇 ⊂ ℌ(Lip^{α}, Lip^{β}).

That is, for any 0 < α < β < 1 and for any pseudo-Dirichlet set H, there exists a Lip^{α} function f such that Δ_{h}f is Lip^{β} for any h ∈ H but f is not Lip^{β}.

Proof. Let H be a pseudo-Dirichlet set. Take a sequence q_{1} < q_{2} < … and a sequence ε_{n} → 0 witnessing the pseudo-Dirichlet property of H. Selecting a suitable subsequence, we may assume that q_{n+1} > 2q_{n} for every n ∈ N. Let

f(x) = ∑_{n=1}^{∞} \frac{1}{q_{n}^{β}δ_{n}} e^{2πiq_{n}x},  where  δ_{n} = max(ε_{n}, 1/q_{n}^{β−α}).

Since \frac{1}{q_{n}^{β}δ_{n}} ≤ 1/q_{n}^{α} and q_{n+1}/q_{n} > 2 (n ∈ N), we can apply Lemma 2.1

to obtain f ∈ Lip^{α}. On the other hand, since δ_{n} → 0, \frac{1}{q_{n}^{β}δ_{n}} ≠ O(1/q_{n}^{β}), so

Lemma 2.1 implies that f ∉ Lip^{β}.

For an h ∈ H,

Δ_{h}f(x) = ∑_{n=1}^{∞} \frac{1}{q_{n}^{β}δ_{n}} (e^{2πiq_{n}h} − 1) e^{2πiq_{n}x}

and

|\frac{1}{q_{n}^{β}δ_{n}} (e^{2πiq_{n}h} − 1)| = \frac{1}{q_{n}^{β}δ_{n}} 2| sin πq_{n}h| ≤ \frac{1}{q_{n}^{β}δ_{n}} 2ε_{n} ≤ \frac{2}{q_{n}^{β}} ,

therefore, by Lemma 2.1, Δ_{h}f ∈ Lip^{β}.

Combining the previous theorem with the result of [6] mentioned in the Introduction, we get the following:

Corollary 2.3 For any 0 < α < β < 1,

p𝔇 ⊂ ℌ(Lip^{α}, Lip^{β}) ⊂ 𝔉_{σ}.

3 The classes ℌ(Lip^{α}, Lip^{β}) are the same for 0 < α < β < 1

The following easy lemmas were obtained in [6]:

Lemma 3.1 (Monotonicity Lemma) If F_{1} ⊃ F_{2} ⊃ G then ℌ(F_{1}, G) ⊃ ℌ(F_{2}, G).

Lemma 3.2 If F_{1} ⊃ F_{2} ⊃ F_{3} and ℌ(F_{1}, F_{2}) ⊂ ℌ(F_{2}, F_{3}) then ℌ(F_{1}, F_{3}) = ℌ(F_{2}, F_{3}).

We will need the notion of fractional integration. There are several different notions of fractional integrals (see e.g. the monograph [7]); here we use the so called Weyl fractional integral which is defined in the following way (see [7] p. 263 or [8] Vol. II p. 133):

Let f be an integrable function on T and suppose that ∫_{T} f = 0. Then, for any γ > 0 let

(1)  I_{γ}[f](x) = ∫_{T} f(t)Ψ_{γ}(x − t)dt,

where

(2)  Ψ_{γ}(t) = ∑_{n∈Z\\{0\}} \frac{e^{2πint}}{(2πin)^{γ}} .

It is known (see e.g. [8]) that the series in (2) converges everywhere on T \ {0} and the integral in (1) exists almost everywhere. (If f ∼ ∑ c_{n}e^{2πinx} then I_{γ}[f] ∼ ∑ c_{n} \frac{e^{2πinx}}{(2πin)^{γ}}.)

Since the operator I_{γ} is defined by a convolution it commutes with the translation operator and it is linear; that is,

(3)  I_{γ}[f(y + h)](x) = I_{γ}[f(y)](x + h),

and

(4)  I_{γ}[cf + dg] = cI_{γ}[f] + dI_{γ}[g].

Denote by Lip_{0}^{λ} the class of the functions of Lip^{λ} with integral 0 (over T). It is also well known (see e.g. [7] p. 275) that if γ, λ > 0 and γ + λ < 1, then I_{γ} is a bijection (actually it is an isomorphism) between the classes Lip_{0}^{λ} and Lip_{0}^{λ+γ}; that is,

(5)  I_{γ} : Lip_{0}^{λ} ↔ Lip_{0}^{λ+γ}  (λ + γ < 1).

Theorem 3.3 For any 0 < α_{1} < β_{1} < 1 and 0 < α_{2} < β_{2} < 1,

ℌ(Lip^{α_{1}}, Lip^{β_{1}}) = ℌ(Lip^{α_{2}}, Lip^{β_{2}}).

Proof. First we prove that if 0 < α < β and β + γ < 1 then

(6)  ℌ(Lip^{α+γ}, Lip^{β+γ}) = ℌ(Lip^{α}, Lip^{β}).

Indeed, (3) and (4) implies that the operator I_{γ} also commutes with the difference operator Δ_{h}; that is

(7)  Δ_{h}I_{γ}[f] = I_{γ}[Δ_{h}f].

It follows from (5) and (7) that if f_{0} ∈ Lip_{0}^{α} \ Lip_{0}^{β} and Δ_{h}f_{0} ∈ Lip_{0}^{β} for every h ∈ H then I_{γ}[f_{0}] ∈ Lip_{0}^{α+γ} \ Lip_{0}^{β+γ} and Δ_{h}I_{γ}[f_{0}] ∈ Lip_{0}^{β+γ} for every h ∈ H. Furthermore, if g_{0} ∈ Lip_{0}^{α+γ} \ Lip_{0}^{β+γ} and Δ_{h}g_{0} ∈ Lip_{0}^{β+γ} for every h ∈ H then I_{γ}^{−1}[g_{0}] ∈ Lip_{0}^{α} \ Lip_{0}^{β} and Δ_{h}I_{γ}^{−1}[g_{0}] ∈ Lip_{0}^{β} for every h ∈ H. Therefore if the function f : T → R witnesses that H ∈ ℌ(Lip^{α}, Lip^{β}) then I_{γ}[f_{0}] - where f_{0} = f − ∫_{T} f - witnesses that H ∈ ℌ(Lip^{α+γ}, Lip^{β+γ}); furthermore if the function g : T → R witnesses that H ∈ ℌ(Lip^{α+γ}, Lip^{β+γ}) then I_{γ}^{−1}[g_{0}] - where g_{0}(x) = g − ∫_{T} g - witnesses that H ∈ ℌ(Lip^{α}, Lip^{β}).

Now we prove that for any 0 < η < δ < β < 1,

(8)  ℌ(Lip^{β−δ}, Lip^{β}) = ℌ(Lip^{β−η}, Lip^{β}).

Indeed, by (7), ℌ(Lip^{β−δ}, Lip^{β−δ/2}) = ℌ(Lip^{β−δ/2}, Lip^{β}), which implies, by Lemma 3.2, that ℌ(Lip^{β−δ}, Lip^{β}) = ℌ(Lip^{β−δ/2}, Lip^{β}). Thus we have also ℌ(Lip^{β−δ}, Lip^{β}) = ℌ(Lip^{β−δ/2^{k}}, Lip^{β}) for any k ∈ N. Then, by the Monotonicity Lemma, ℌ(Lip^{β−δ}, Lip^{β}) = ℌ(Lip^{β−η}, Lip^{β}).

Finally, supposing that β_{1} ≤ β_{2} and applying (7) and (8), we get

ℌ(Lip^{α_{1}}, Lip^{β_{1}}) = ℌ(Lip^{α_{1}+β_{2}−β_{1}}, Lip^{β_{2}}) = ℌ(Lip^{α_{2}}, Lip^{β_{2}}).

Remark 3.4 Unfortunately this proof does not work if β_{1} or β_{2} equals 1. Namely, (5) is not true for λ + γ = 1. In this case I_{γ} is a bijection between Lip_{0}^{1−γ} and Λ_{∗}_{0}, the class of Zygmund functions on T with 0 integral.

(A function f is Zygmund, if for any x and h, |f(x+h)−2f(x)+f(x−h)| ≤ Ch. The class of Zygmund functions is denoted by Λ_{∗}. It is known (see e.g. [8] Vol. I p. 43-44 and Vol. II p. 138) that

Lip^{1}(R) ⊂ Λ_{∗} ⊂ Lip^{α}(R) ∀0 < α < 1,

and Λ_{∗} ≠ Lip^{1}(R).)

Therefore with this method we can only prove that

ℌ(Lip^{α}, Λ_{∗}(T)) = ℌ(Lip^{α_{1}}, Lip^{β_{1}})

for any 0 < α < 1 and 0 < α_{1} < β_{1} < 1.

However, if, for a fixed 0 < α < 1, one can find a linear operator I that commutes with the translation operator and I is a bijection between Lip^{1} and Lip^{α} (or between Lip^{1} and Λ_{∗}) then Theorem 3.3 would remain true for β_{1} = 1. This would give a complete answer to our question since, as we mentioned in the Introduction, it was proved in [6] that ℌ(Lip^{α}, Lip^{1}) = 𝔉_{σ}, so the existence of such an operator would imply that ℌ(Lip^{α}, Lip^{β}) = 𝔉_{σ} for any 0 < α < β ≤ 1.

References

[1] M. Balcerzak, Z. Buczolich and M. Laczkovich, Lipschitz differences and Lipschitz functions, Colloq. Math., 72 (1997), 319-324.

[2] N. K. Bary, Trigonometric series (in Russian), Moscow, 1961; English translation: A Treatise on Trigonometric Series, Macmillan, New York, 1964.

[3] L. Bukovský, N. N. Kholshchevnikova and M. Repický, Thin sets of harmonic analysis and infinite combinatorics, Real Analysis Exchange, 20 No. 2 (1994-1995), 454-509.

[4] S. Kahane, Antistable classes of thin sets in harmonic analysis, Illinois J. Math. 37 (1993), 186-223.

[5] T. Keleti, Difference functions of periodic measurable functions, PhD thesis, Eötvös Loránd University, Budapest, 1996 (http://www.cs.elte.hu/phd th/).

[6] T. Keleti, Difference functions of periodic measurable functions, submitted.

[7] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and differentiations and some applications (in Russian), Science and Technology, Minsk, 1987.

[8] A. Zygmund, Trigonometric series Vol. I-II, Cambridge University Press, 1959.

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