JJ II

(1)

volume 5, issue 2, article 22, 2004.

Received 18 July, 2003;

accepted 27 December, 2003.

Communicated by:A. Babenko

Abstract Contents

JJ II

J I

Home Page Go Back

Close Quit

Journal of Inequalities in Pure and Applied Mathematics

ON BELONGING OF TRIGONOMETRIC SERIES TO ORLICZ SPACE

S. TIKHONOV

Department of Mechanics and Mathematics Moscow State University

Vorob’yovy gory, Moscow 119899 RUSSIA.

EMail:tikhonov@mccme.ru

c

2000Victoria University ISSN (electronic): 1443-5756 002-04

(2)

On Belonging Of Trigonometric Series To Orlicz Space

S. Tikhonov

Title Page Contents

JJ II

J I

Go Back Close

Quit Page2of17

J. Ineq. Pure and Appl. Math. 5(2) Art. 22, 2004

http://jipam.vu.edu.au

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.

2000 Mathematics Subject Classification:42A32, 46E30.

Key words: Trigonometric series, Integrability, Orlicz space.

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]). Letλn ↓0.

If0≤α <2, then

g(x)

x^α ∈L(0, π)⇐⇒

∞

X

n=1

n^α−1λ_n<∞.

If0< α <1, then

f(x)

x^α ∈L(0, π)⇐⇒

∞

X

n=1

n^α−1λ_n<∞.

Several generalizations of this theorem have been obtained in the following directions: more general weighted functions γ(x)have been considered; also,

(4)

S. Tikhonov

Title Page Contents

JJ II

J I

Go Back Close

Quit Page4of17

integrability of g(x)γ(x)andf(x)γ(x)of orderphave been examined for dif- ferent values of p; finally, more general conditions on coefficients {λn} have been considered.

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

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

t→∞

S(κt)

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

Theorem 1.2 ([3]). Let λ_n ↓ 0, p ≥ 1, and let S(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 _x¹

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),

(5)

S. Tikhonov

Title Page Contents

JJ II

J I

Go Back Close

Quit Page5of17

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 sequence c := {c_n} of positive numbers tending to zero is of rest bounded variation, or briefly R⁺₀BV S, if it possesses the property

∞

X

n=m

|c_n−c_n+1| ≤K(c)c_m

for all natural numbers m, where K(c) is a constant depending only on c.In [5] it was shown that the class R₀⁺BV S was not comparable to the class of quasi-monotone sequences, that is, to the class of sequencesc={c_n}such that n^−αcn ↓ 0for someα ≥0. Also, in [5] it was proved that the series (1.1) and (1.2) are uniformly convergent overδ ≤ x ≤ π−δfor any 0< δ < π. In the same paper the following was proved.

Theorem 1.4 ([5]). Let{λn} ∈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 Sand with quite

(6)

S. Tikhonov

Title Page Contents

JJ II

J I

Go Back Close

Quit Page6of17

general conditions on a weight function. The author has used the notation of almost monotonic sequences.

A sequence γ := {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_idenote positive constants that are not necessarily the same at each occurrence.

Theorem 1.5 ([8]). If{λ_n} ∈R₀⁺BV S, and the sequenceγ :={γ_n}such that {γ_nn^−2+ε}is almost decreasing for someε >0, then

∞

X

n=1

γ_n

nλ_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 constantsAandB such thatAγ_n+1 ≤γ(x)≤Bγ_nforx∈(_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

(7)

S. Tikhonov

Title Page Contents

JJ II

J I

Go Back Close

Quit Page7of17

continuous function defined on[0,∞)such thatΦ(0) = 0and lim

t→∞Φ(t) = +∞.

For a weightγ(x)the weighted Orlicz spaceL(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

.

If Φ(x) = x^p for 1 ≤ p < ∞, when the weighted Orlicz space L(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 functions Φ(x) defined on[0,∞) such that Φ(0) = 0 and Φ(x)/x^p is nonin- creasing andΦ(x)/x^qis nondecreasing. 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).

(8)

S. Tikhonov

Title Page Contents

JJ II

J I

Go Back Close

Quit Page8of17

2. Results

The following theorems provide the sufficient conditions of belonging off(x) andg(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(nλn)<∞=⇒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. Let Φ(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

γn

n^2+qΦ(n²λ_n)<∞=⇒g(x)∈L(Φ, γ).

Remark 2.1. IfΦ(t) = t, then Theorem2.2implies Theorem1.5, and ifΦ(t) = t^p with 0 < p and {γ_n = 1, n ∈ N}, then Theorem2.1 is a generalization of Theorem 1.3. Also, ifΦ(t) = t^p with1 ≤ pand{γ_n = n^αS(n), n ∈ N} with corresponding conditions onαandS(n), then Theorems2.1and2.2imply the sufficiency parts (⇐=) of Theorems1.2and1.4.

(9)

S. Tikhonov

Title Page Contents

JJ II

J I

Go Back Close

Quit Page9of17

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)andt_j ≥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

t_j

!

≤

n

P

j=1

Φ^p¹^∗ (t_j)

!p^∗

, p^∗ = max(1, p).

Lemma 3.3. Let Φ ∈ 4(p, q) (0 ≤ q ≤ p). If λ_n > 0, a_n ≥ 0, and if there exists a constantK such that aν+j ≤Kaν 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

aν ≤

ξ−1

X

m=0 2^m+1−1

X

ν=2^m

aν +

k

X

ν=2^ξ

aν

≤C₁

ξ−1

X

m=0

2^ma₂^m + 2^ξa₂^ξ

!

≤C₁

ξ

X

m=0

2^ma₂^m.

(10)

S. Tikhonov

Title Page Contents

JJ II

J I

Go Back Close

Quit Page10of17

Lemma3.2implies

Φ

k

X

ν=1

a_ν

!

≤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 Lemma3.1, 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^∗ (ma_m) m

!p^∗

≤C

∞

X

k=1

Φ (ka_k)λ_k

P∞ ν=kλ_ν kλ_k

p^∗

.

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

(11)

S. Tikhonov

Title Page Contents

JJ II

J I

Go Back Close

Quit Page11of17

4. Proofs of Theorems

Proof of Theorem2.1. Let x ∈ (_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|Dk(x)|=O _x¹

andλn ∈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

λ_k+

∞

X

k=n+1

λ_ksinkx

≤

n

X

k=1

λ_k+

∞

X

k=n

(λ_k−λ_k+1)De_k(x)

≤C

n

X

k=1

λk+n

∞

X

k=n

|λk−λk+1|

!

≤C

n

X

k=1

λk+nλn

! ,

(12)

S. Tikhonov

Title Page Contents

JJ II

J I

Go Back Close

Quit Page12of17

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 that λ_n ≤ Kλ_k holds for any k ≤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 Lemmas3.2and3.3:

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

λ_k

!

≤C

∞

X

k=1

Φ (kλ_k)γ_k k²

k γ_k

∞

X

ν=k

γ_ν ν²

!p^∗

,

(13)

S. Tikhonov

Title Page Contents

JJ II

J I

Go Back Close

Quit Page13of17

wherep^∗ = max(1, p).Since there exists a constantε >0such 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 Theorem2.1is complete.

Proof of Theorem2.2. While proving Theorem 2.2 we will follow the idea of the proof of Theorem2.1.

Letx∈(_n+1^π ,^π_n]. Then

|g(x)| ≤

n

X

k=1

kxλ_k+

∞

X

k=n+1

λ_ksinkx (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

kλk+ 1 nλn

n

X

k=1

k

!

≤C1

1 n

n

X

k=1

kλk.

(14)

S. Tikhonov

Title Page Contents

JJ II

J I

Go Back Close

Quit Page14of17

Using Lemma3.2, Lemma3.3and the estimate (4.2), we can write

Z π

0

γ(x)Φ (|g(x)|)dx≤

∞

X

n=1

Φ C1

1 n

n

X

k=1

kλk

!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 k²λ_k 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 Theorem2.2is complete.

(15)

S. Tikhonov

Title Page Contents

JJ II

J I

Go Back Close

Quit Page15of17

References

[1] R.P. BOAS JR, Integrability of trigonometric series. III, Quart. J. Math.

Oxford Ser., 3(2) (1952), 217–221.

[2] P. HEYWOOD, On the integrability of functions defined by trigonometric series, Quart. J. Math. Oxford Ser., 5(2) (1954), 71–76.

[3] S. IGARI, Some integrability theorems of trigonometric series and mono- tone decreasing functions, Tohoku Math. J., 12(2) (1960), 139–146.

[4] L. LEINDLER, Generalization of inequalities of Hardy and Littlewood, Acta Sci. Math., 31 (1970), 279–285.

[5] L. LEINDLER, A new class of numerical sequences and its applications to sine and cosine series, Anal. Math., 28 (2002), 279–286.

[6] M. MATELJEVICANDM. PAVLOVIC,L^p-behavior of power series with positive coefficients and Hardy spaces, Proc. Amer. Math. Soc., 87 (1983), 309-316.

[7] J. NEMETH, Note on the converses of inequalities of Hardy and Little- wood, Acta Math. Acad. Paedagog. Nyhazi. (N.S.), 17 (2001), 101–105.

[8] J. NEMETH, Power-Monotone Sequences and integrability of trigono- metric series, J. Inequal. Pure and Appl. Math., 4(1) (2003). [ONLINE http://jipam.vu.edu.au]

[9] M.M. RAO AND Z.D. REN, Theory of Orlicz spaces, M. Dekker, Inc., New York, 1991.

(16)

S. Tikhonov

Title Page Contents

JJ II

J I

Go Back Close

Quit Page16of17

[10] T.M. VUKOLOVA AND M.I. DYACHENKO, On the properties of sums of trigonometric series with monotone coefficients, Moscow Univ. Math.

Bull., 50 (3), (1995), 19–27; translation from Vestnik Moskov. Univ. Ser. I.

Mat. Mekh., 3 (1995), 22–32.

[11] W. H. YOUNG, On the Fourier series of bounded variation, Proc. London Math. Soc., 12(2) (1913), 41–70.

[12] A. ZYGMUND, Trigonometric series, Volumes I and II, Cambridge Uni- versity Press, 1988.