(1)http://jipam.vu.edu.au/ Volume 5, Issue 2, Article 22, 2004 ON BELONGING OF TRIGONOMETRIC SERIES TO ORLICZ SPACE S

http://jipam.vu.edu.au/

Volume 5, Issue 2, Article 22, 2004

ON BELONGING OF TRIGONOMETRIC SERIES TO ORLICZ SPACE

S. TIKHONOV

DEPARTMENT OFMECHANICS ANDMATHEMATICS

MOSCOWSTATEUNIVERSITY

VOROB’YOVY GORY, MOSCOW119899 RUSSIA.

tikhonov@mccme.ru

Received 18 July, 2003; accepted 27 December, 2003 Communicated by A. Babenko

ABSTRACT. In this paper we consider trigonometric series with the coefficients fromR⁺₀BV S class. We prove the theorems on belonging to these series to Orlicz space.

Key words and phrases: Trigonometric series, Integrability, Orlicz space.

2000 Mathematics Subject Classification. 42A32, 46E30.

1. INTRODUCTION

We will study the problems of integrability of formal sine and cosine series

(1.1) g(x) =

∞

X

n=1

λnsinnx,

(1.2) f(x) =

∞

X

n=1

λ_ncosnx.

First, we will rewrite the classical result of Young, Boas and Heywood for series (1.1) and (1.2) with monotone coefficients.

Theorem 1.1 ([1], [2], [11]). Letλ_n↓0.

If0≤α <2, then

g(x)

x^α ∈L(0, π)⇐⇒

∞

X

n=1

n^α−1λ_n<∞.

ISSN (electronic): 1443-5756 c

2004 Victoria University. All rights reserved.

This work was supported by the Russian Foundation for Fundamental Research (grant no. 03-01-00080) and the Leading Scientific Schools (grant no. NSH-1657.2003.1).

002-04

If0< α <1, then

f(x)

x^α ∈L(0, π)⇐⇒

∞

X

n=1

n^α−1λn<∞.

Several generalizations of this theorem have been obtained in the following directions: more general weighted functions γ(x) have been considered; also, integrability of g(x)γ(x) and f(x)γ(x)of orderphave been examined for different values ofp; finally, more general condi- tions on coefficients{λ_n}have been considered.

Igari ([3]) obtained the generalization of Boas-Heywood’s results. The author used the nota- tion of a slowly oscillating function.

A positive measurable functionS(t)defined on[D; +∞), D > 0is said to be slowly oscil- lating if lim

t→∞

S(κt)

S(t) = 1holds for allx >0.

Theorem 1.2 ([3]). Letλ_n ↓0,p≥1,and letS(t)be a slowly oscillating function.

If−1< θ <1, then

g^p(x)S ¹_x

x^pθ+1 ∈L(0, π)⇐⇒

∞

X

n=1

n^pθ+p−1S(n)λ^p_n<∞.

If−1< θ <0, then

f^p(x)S ¹_x

x^pθ+1 ∈L(0, π)⇐⇒

∞

X

n=1

n^pθ+p−1S(n)λ^p_n<∞.

Vukolova and Dyachenko in [10], considering the Hardy-Littlewood type theorem found the sufficient conditions of belonging of series (1.1) and (1.2) to the classesL_p forp > 0.

Theorem 1.3 ([10]). Letλ_n ↓0, andp >0.Then

∞

X

n=1

n^p−2λ^p_n <∞=⇒ψ(x)∈L^p(0, π),

where a functionψ(x)is either af(x)or ag(x).

In the same work it is shown that the converse result does not hold for cosine series.

Leindler ([5]) introduced the following definition. A sequencec:={cn}of positive numbers tending to zero is of rest bounded variation, or brieflyR⁺₀BV S, if it possesses the property

∞

X

n=m

|c_n−c_n+1| ≤K(c)c_m

for all natural numbersm, whereK(c)is a constant depending only onc.In [5] it was shown that the classR₀⁺BV S was not comparable to the class of quasi-monotone sequences, that is, to the class of sequencesc={c_n}such thatn^−αc_n ↓0for someα≥0. Also, in [5] it was proved that the series (1.1) and (1.2) are uniformly convergent overδ≤x≤π−δfor any0< δ < π.

In the same paper the following was proved.

Theorem 1.4 ([5]). Let{λ_n} ∈R⁺₀BV S,p≥1, and ¹_p −1< θ < ¹_p. Then

ψ^p(x)

x^pθ ∈L(0, π)⇐⇒

∞

X

n=1

n^pθ+p−2λ^p_n<∞, where a functionψ(x)is either af(x)or ag(x).

Very recently Nemeth [8] has found the sufficient condition of integrability of series (1.1) with the sequence of coefficients {λ_n} ∈ R⁺₀BV S and with quite general conditions on a weight function. The author has used the notation of almost monotonic sequences.

A sequenceγ :={γ_n}of positive terms will be called almost increasing (decreasing) if there exists constantC :=C(γ)≥1such that

Cγ_n≥γ_m (γ_n ≤Cγ_m) holds for anyn≥m.

Here and further, C, C_i denote positive constants that are not necessarily the same at each occurrence.

Theorem 1.5 ([8]). If{λ_n} ∈ R⁺₀BV S, and the sequence γ := {γ_n}such that{γ_nn^−2+ε}is almost decreasing for someε >0, then

∞

X

n=1

γ_n

n λ_n<∞=⇒γ(x)g(x)∈L(0, π).

Here and in the sequel, a functionγ(x) is defined by the sequenceγ in the following way:

γ ^π_n

:=γ_n, n∈Nand there exist positive constantsAandBsuch thatAγ_n+1 ≤γ(x)≤Bγ_n forx∈(_n+1^π ,^π_n).

We will solve the problem of finding of sufficient conditions, for which series (1.1) and (1.2) belong to the weighted Orlicz spaceL(Φ, γ). In particular, we will obtain sufficient conditions for series (1.1) and (1.2) to belong to weighted spaceL^p_γ.

Definition 1.1. A locally integrable almost everywhere positive functionγ(x) : [0, π]→[0,∞) is said to be a weight function. Let Φ(t) be a nondecreasing continuous function defined on [0,∞)such thatΦ(0) = 0and lim

t→∞Φ(t) = +∞. For a weightγ(x)the weighted Orlicz space L(Φ, γ)is defined by (see [9], [12])

(1.3) L(Φ, γ) =

h: Z π

0

γ(x)Φ(ε|h(x)|)dx <∞ for some ε >0

.

IfΦ(x) =x^pfor1≤p < ∞, when the weighted Orlicz spaceL(Φ, γ)defined by (1.3) is the usual weighted spaceL^p_γ(0, π).

We will denote (see [6]) by4(p, q) (0 ≤ q ≤ p) the set of all nonnegative functionsΦ(x) defined on [0,∞) such that Φ(0) = 0 and Φ(x)/x^p is nonincreasing andΦ(x)/x^q is nonde- creasing. It is clear that4(p, q)⊂ 4(p,0) (0< q ≤ p). As an example,4(p,0)contains the functionΦ(x) = log(1 +x).

2. RESULTS

The following theorems provide the sufficient conditions of belonging of f(x)and g(x)to Orlicz spaces.

Theorem 2.1. LetΦ(x) ∈ 4(p,0) (0 ≤ p). If{λ_n} ∈ R⁺₀BV S, and sequence{γ_n}is such that{γ_nn^−1+ε}is almost decreasing for someε >0, then

∞

X

n=1

γ_n

n²Φ(nλ_n)<∞=⇒ψ(x)∈L(Φ, γ),

where a functionψ(x)is either a sine or cosine series.

For the sine series it is possible to obtain the sufficient condition of its belonging to Orlicz space with more general conditions on the sequence{γ_n}but with stronger restrictions on the functionΦ(x).

Theorem 2.2. LetΦ(x) ∈ 4(p, q) (0 ≤ q ≤ p). If{λ_n} ∈ R⁺₀BV S, and sequence {γ_n}is such that{γ_nn^−(1+q)+ε}is almost decreasing for someε >0, then

∞

X

n=1

γ_n

n^2+qΦ(n²λ_n)<∞=⇒g(x)∈L(Φ, γ).

Remark 2.3. IfΦ(t) =t, then Theorem 2.2 implies Theorem 1.5, and ifΦ(t) =t^p with0< p and{γn= 1, n∈N}, then Theorem 2.1 is a generalization of Theorem 1.3. Also, ifΦ(t) =t^p with 1 ≤ p and{γn = n^αS(n), n ∈ N} with corresponding conditions on α andS(n), then Theorems 2.1 and 2.2 imply the sufficiency parts (⇐=) of Theorems 1.2 and 1.4.

3. AUXILIARY RESULTS

Lemma 3.1 ([4]). Ifa_n≥0, λ_n>0, and if p≥1, then

∞

X

n=1

λ_n

n

X

ν=1

a_ν

!p

≤C

∞

X

n=1

λ^1−p_n a^p_n

∞

X

ν=n

λ_ν

!p

.

Lemma 3.2 ([6]). LetΦ∈ 4(p, q) (0≤q ≤p)andtj ≥0,j = 1,2, . . . , n, n ∈N. Then (1): θ^pΦ (t)≤Φ (θt)≤θ^qΦ (t), 0≤θ ≤1, t≥0,

(2): Φ

n

P

j=1

tj

!

≤

n

P

j=1

Φ^p1∗ (tj)

!p^∗

, p^∗ = max(1, p).

Lemma 3.3. LetΦ∈ 4(p, q) (0≤q≤p). Ifλ_n>0, a_n≥0, and if there exists a constantK such that a_ν+j ≤Ka_ν holds for all j, ν ∈N, j ≤ν,then

∞

X

k=1

λ_kΦ

k

X

ν=1

a_ν

!

≤C

∞

X

k=1

Φ (ka_k)λ_k P∞

ν=kλ_ν kλ_k

p^∗

,

wherep^∗ = max(1, p).

Proof. Letξbe an integer such that2^ξ ≤k < 2^ξ+1. Then

k

X

ν=1

a_ν ≤

ξ−1

X

m=0 2^m+1−1

X

ν=2^m

a_ν +

k

X

ν=2^ξ

a_ν ≤C₁

ξ−1

X

m=0

2^ma₂^m+ 2^ξa₂^ξ

!

≤C₁

ξ

X

m=0

2^ma₂^m.

Lemma 3.2 implies

Φ

k

X

ν=1

aν

!

≤Φ C1 ξ

X

m=0

2^ma2^m

!

≤C₁^pΦ

ξ

X

m=0

2^ma₂^m

!

≤C

ξ

X

m=0

Φ^p1∗ (2^ma₂^m)

!^p∗

≤C

k

X

m=1

Φ^p1∗ (ma_m) m

!p^∗

.

By Lemma 3.1, we have

∞

X

k=1

λ_kΦ

k

X

ν=1

a_ν

!

≤C

∞

X

k=1

λ_k

k

X

m=1

Φ^p1∗ (mam) m

!^p∗

≤C

∞

X

k=1

Φ (ka_k)λ_k

P∞ ν=kλν

kλ_k p^∗

.

Note that this Lemma was proved in [7] for the case0< p≤1.

4. PROOFS OF THEOREMS

Proof of Theorem 2.1. Letx∈(_n+1^π ,^π_n]. Applying Abel’s transformation we obtain

|f(x)| ≤

n

X

k=1

λ_k+

∞

X

k=n+1

λ_kcoskx

≤

n

X

k=1

λ_k+

∞

X

k=n

|(λ_k−λ_k+1)D_k(x)|,

whereDk(x)are the Dirichlet kernels, i.e.

D_k(x) = 1 2 +

k

X

n=1

cosnx, k ∈N.

Since|D_k(x)|=O ¹_x

andλ_n∈R⁺₀BV S, we see that

|f(x)| ≤C

n

X

k=1

λ_k+n

∞

X

k=n

|λ_k−λ_k+1|

!

≤C

n

X

k=1

λ_k+nλ_n

! .

The following estimates for series (1.2) can be obtained in the same way:

|g(x)| ≤

n

X

k=1

λk+

∞

X

k=n+1

λksinkx

≤

n

X

k=1

λ_k+

∞

X

k=n

(λ_k−λ_k+1)De_k(x)

≤C

n

X

k=1

λ_k+n

∞

X

k=n

|λ_k−λ_k+1|

!

≤C

n

X

k=1

λ_k+nλ_n

! ,

whereDe_k(x)are the conjugate Dirichlet kernels, i.e.De_k(x) :=Pk

n=1sinnx, k ∈N.

Therefore,

|ψ(x)| ≤C

n

X

k=1

λ_k+nλ_n

! , where a functionψ(x)is either af(x)or ag(x).

One can see that if {λ_n} ∈ R⁺₀BV S, then{λ_n} is almost decreasing sequence, i.e. there exists a constantK ≥1such thatλ_n ≤Kλ_kholds for anyk ≤n. Then

(4.1) |ψ(x)| ≤C

n

X

k=1

λ_k+λ_n

n

X

k=1

1

!

≤C

n

X

k=1

λ_k.

We will use (4.1) and the fact that{λ_k}is almost decreasing sequence; also, we will use Lem- mas 3.2 and 3.3:

Z π

0

γ(x)Φ (|ψ(x)|)dx≤

∞

X

n=1

Φ C1 n

X

k=1

λk

!Z π/n

π/(n+1)

γ(x)dx

≤C₁^pπB

∞

X

n=1

γ_n n²Φ

n

X

k=1

λk

!

≤C

∞

X

k=1

Φ (kλ_k)γ_k k²

k γk

∞

X

ν=k

γ_ν ν²

!p^∗

,

where p^∗ = max(1, p). Since there exists a constant ε > 0 such that {γ_nn^−1+ε} is almost decreasing, then

∞

X

ν=k

γ_ν

ν² ≤C γ_k k^1−ε

∞

X

ν=k

ν^−ε−1 ≤Cγ_k k . Then

Z π

0

γ(x)Φ (|ψ(x)|)dx≤C

∞

X

k=1

γ_k

k²Φ (kλ_k).

The proof of Theorem 2.1 is complete.

Proof of Theorem 2.2. While proving Theorem 2.2 we will follow the idea of the proof of The- orem 2.1.

Letx∈(_n+1^π ,^π_n]. Then

|g(x)| ≤

n

X

k=1

kxλk+

∞

X

k=n+1

λksinkx (4.2)

≤

n

X

k=1

kxλ_k+

∞

X

k=n

(λ_k−λ_k+1)De_k(x)

≤C 1 n

n

X

k=1

kλ_k+nλ_n

!

≤C 1 n

n

X

k=1

kλ_k+ 1 nλ_n

n

X

k=1

k

!

≤C₁1 n

n

X

k=1

kλ_k.

Using Lemma 3.2, Lemma 3.3 and the estimate (4.2), we can write Z π

0

γ(x)Φ (|g(x)|)dx≤

∞

X

n=1

Φ C₁1 n

n

X

k=1

kλ_k

!Z π/n

π/(n+1)

γ(x)dx

≤C₁^pπB

∞

X

n=1

γ_n n^2+qΦ

n

X

k=1

kλ_k

!

≤C2

∞

X

k=1

Φ k²λk

γ_k k^2+q

k^1+q γ_k

∞

X

ν=k

γ_ν ν^2+q

!p^∗

, wherep^∗ = max(1, p).

By the assumption on{γ_n}, Z π

0

γ(x)Φ (|g(x)|)dx≤C

∞

X

k=1

γ_k

k^2+qΦ k²λ_k ,

and the proof of Theorem 2.2 is complete.

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