We give necessary conditions for even or odd functions with Fourier series to belong to the classesΛm(p, r, α)

ON A GENERALIZATION OF LIPSCHITZ’S CLASSES

XH. Z. KRASNIQI

DEPARTMENT OFMATHEMATICS ANDCOMPUTERSCIENCES, AVENUE"MOTHERTERESA" 5,

PRISHTINË, 10000, REPUBLIC OFKOSOVO

xheki00@hotmail.com

Received 26 March, 2008; accepted 06 August, 2008 Communicated by H. Bor

ABSTRACT. In this paper we obtain a generalization of Lipschitz’s classesΛ^m(β, p, r)defined in [1]. We give necessary conditions for even or odd functions with Fourier series to belong to the classesΛ^m(p, r, α). We also give sufficient conditions for even or odd functions with Fourier series to belong to the same classes.

Key words and phrases: Lipschitz classes, Fourier series.

2000 Mathematics Subject Classification. 42A20, 42A32.

1. DEFINITIONS ANDUSEFULSTATEMENTS

We consider the series

(1.1) a₀

2 +

∞

X

n=1

ancosnx or

(1.2)

∞

X

n=1

a_nsinnx wherea_nare Fourier coefficients of integrable functionf.

Definition 1.1. We say that a functionf belongs toW A^p,(1< p <∞)if

∞

X

n=1

n^p−2

∞

X

k=n

|∆ak|

!p

<+∞

where∆a_k =a_k−a_k+1 (see [1]).

We say that any function α(t) is a function of type σ (see [4]) if it is measurable in [0,1], integrable in[δ,1]for each δ∈ (0,1), and there exist real numbersC_1,α >0, σandδ₀ ∈(0,1) such that

(1) α(t)≥C_1,α, for allt∈[0,1];

096-08

(2) Rδ

0 α(t)t^sdt <∞for eachs > σandδ∈(0, δ₀);

(3) Rδ

0 α(t)t^sdt =∞for eachs < σandδ∈(0, δ₀), and Z δ

0

α(t)t^σdt≤C₂δ^σ Z 2δ

δ

α(t)dt.

In [1], the classesΛ^m(β, p, r,)are defined in the following way Definition 1.2. f ∈Λ^m(β, p, r,), if

f

(m) β,p,r ≡

(Z 1

0

Z 2π 0

|∆mf(x, t)|^p t^βp dx

_p^r dt

t )¹_r

<+∞,

where1< p <+∞,1≤r <+∞, β >0, m∈Nand

∆_mf(x, t) =

m

X

i=1

(−1)ⁱC_mⁱ f[x+ (m−2i)t].

Now we define classesΛ^m(p, r, α)as follows:

Definition 1.3. We sayf ∈Λ^m(p, r, α), if f

(m) p,r,α ≡

(Z 1

0

α(t) Z 2π

0

|∆_mf(x, t)|^pdx _p^r

dt )¹_r

<+∞,

whereα(t)is function of the typeσ.

Forα(t) =t^−rβ−1, β > 0, we get the classesΛ^m(β, p, r), considered in [1]. Therefore classes Λ^m(p, r, α)are generalizations of classesΛ^m(β, p, r).

We need some auxiliary statements.

Lemma 1.1 ([2]). Leta_ν,b_νandβ_nbe numbers such thata_ν ≥0,b_ν ≥0andP∞

ν=na_ν =a_nβ_n: (1) For0< p≤1the following inequality is valid

∞

X

ν=1

a_ν

ν

X

µ=1

b_µ

!p

≥p^p

∞

X

ν=1

a_ν(b_νβ_ν)^p; (2) For1≤p < ∞we have

∞

X

ν=1

a_ν

ν

X

µ=1

b_µ

!p

≤p^p

∞

X

ν=1

a_ν(b_νβ_ν)^p.

Lemma 1.2 ([2]). Leta_ν,b_νandγ_nbe numbers such thata_ν ≥0,b_ν ≥0andPn

ν=1a_ν =b_nγ_n: (1) For0< p≤1, we have

∞

X

ν=1

a_ν

∞

X

µ=ν

b_µ

!p

≥p^p

∞

X

ν=1

a_ν(b_νγ_ν)^p; (2) For1≤p < ∞, we have

∞

X

ν=1

a_ν

∞

X

µ=ν

b_µ

!p

≤p^p

∞

X

ν=1

a_ν(b_νγ_ν)^p.

Lemma 1.3 ([3]). Letµ,τ anda_ν be numbers such that0< µ < τ <∞anda_ν ≥0. Then

∞

X

ν=1

a^τ_ν

!¹_τ

≤

∞

X

ν=1

a^µ_ν

!_µ¹ .

We denote by C a constant that depends only on m, p, r and may be different in different relations.

Theorem 1.4 ([1]). Iff ∈W A^p,1< p <+∞, then

ω^(m)_p (h;f) ^p ≤Ch^mp X

n≤[h¹]

n^(m+1)p−2

∞

X

k=n

|∆a_k|

!p

+C X

n>[h¹] n^p−2

∞

X

k=n

|∆a_k|

!p

,

whereω^(m)p (h;f)is the integral modulus of smoothness of orderm.

2. MAINRESULTS

Let us denote

A(n) :=

Z 1/n 1/(n+1)

α(t)dt,

b(n) := b1(n) +b2(n) =n^mr Z 1/n

0

α(t)t^mrdt+ Z 1

1/(n+1)

α(t)dt.

We have the following first main result.

Theorem 2.1. Letmbe any natural number and

f ∈AW^p, 1< p <+∞, 1≤r <+∞.

If for the coefficients of series(1.1)or(1.2)we haveP∞

k=1|∆a_k|<+∞, then:

(1) Forp≤rwe have f

(m) p,r,α ≤C

( _∞ X

n=1

n^{r 1−2p ∞}

X

k=n

∆a_k

!r

b(n)

b(n) A(n)

^r_p−1)¹_r

;

(2) Forp > rwe have f

(m) p,r,α ≤C

( _∞ X

n=1

n^{r 1−2p ∞}

X

k=n

∆a_k

!r

b(n) )¹_r

.

Proof. Using the characteristics of the integral modulus of smoothness we have n

f

(m) p,r,α

or

= Z 1

0

α(t) Z 2π

0

|∆_mf(x, t)|^pdx ^rp

dt

≤

∞

X

N=1

Z 1/N 1/(N+1)

α(t)

ω^(m)_p (f;t)r

dt

≤

∞

X

N=1

ω^(m)_p (f; 1/N)rZ 1/N 1/(N+1)

α(t)dt

=

∞

X

N=1

A(N)

ω_p^(m)(f; 1/N)^r .

According to the Theorem 1.4, we have n

f

(m) p,r,α

or

≤C

∞

X

N=1

A(N)N^−mr ( _N

X

n=1

n^(m+1)p−2

∞

X

k=n

|∆a_k|

!p)^r_p

+C

∞

X

N=1

A(N) ( _∞

X

n=N+1

n^p−2

∞

X

k=n

|∆a_k|

!p)^r_p

=I₁+I₂.

Now we estimateI₁ andI₂. Letr/p≥1. Then according to Lemma 1.1 we have I₁ ≤C

∞

X

N=1

A(N)N^−mr (

N^(m+1)p−2

∞

X

k=N

|∆a_k|

!p

β_N )^r_p

.

Now we estimate the quantityβ_N: A(N)N^−mrβ_N =

∞

X

i=N

A(i)i^−mr

=

∞

X

i=N

i+ 1 i

mr

· 1 (i+ 1)^mr

Z 1/i 1/(i+1)

α(t)dt

≤2^mr Z 1/N

0

α(t)t^mrdt, or

β_N ≤Cb₁(N) A(N). Consequently

(2.1) I₁ ≤C

∞

X

N=1

A(N)N^r(¹⁻²p)

∞

X

k=N

|∆a_k|

!r

b₁(N) A(N)

^r_p .

According to Lemma 1.2, forr/p≥1we have I2 ≤C

∞

X

N=1

A(N) (

N^p−2

∞

X

k=N

|∆ak|

!p

γN

)^r_p .

We estimate the quantityγN: A(N)γN =

N

X

i=1

A(i) = Z 1

1/(N+1)

α(t)dt⇒γN = b₂(N) A(N).

Consequently

(2.2) I₂ ≤C

∞

X

N=1

A(N)N^r(¹⁻²p)

∞

X

k=N

|∆a_k|

!r

b₂(N) A(N)

^r_p .

By (2.1) and (2.2) we get n

f

(m) p,r,α

or

≤C

∞

X

N=1

A(N)N^r(¹⁻²p)

∞

X

k=N

|∆a_k|

!r(

b₁(N) A(N)

^r_p +

b₂(N) A(N)

^r_p) .

Finally, according to Lemma 1.3, forr ≥pwe have f

(m) p,r,α ≤C

( _∞ X

N=1

A(N)N^r(¹⁻²p)

∞

X

k=N

|∆a_k|

!r

b(N) A(N)

^r_p)¹_r .

Now letr/p <1. Then, according to Lemma 1.3, we have I₁ ≤C

∞

X

N=1

A(N)N^−mr

N

X

n=1

n^(m+1)r−2rp

∞

X

k=n

|∆a_k|

!r

.

If we change the order of summation we get I₁ ≤C

∞

X

n=1

n^(m+1)r−2rp

∞

X

k=n

|∆a_k|

!r ∞

X

N=n

A(N)N^−mr

≤C

∞

X

n=1

n^r(¹⁻²p)

∞

X

k=n

|∆a_k|

!r

b₁(n).

(2.3)

Now we estimateI2. Using Lemma 1.3 and changing the order of summation we have:

I₂ ≤C

∞

X

N=1

A(N)

∞

X

n=N

n^r(¹⁻_p²)

∞

X

k=n

|∆a_k|

!r

=C

∞

X

n=1

n^r(¹⁻²p)

∞

X

k=n

|∆a_k|

!r n

X

N=1

A(N)

=C

∞

X

n=1

n^r(¹⁻²_p)

∞

X

k=n

|∆a_k|

!r

b₂(n).

(2.4)

From (2.3) and (2.4) we deduce f

(m) p,r,α ≤C

( _∞ X

n=1

n^{r 1−2p ∞}

X

k=n

∆a_k

!r

b(n) )¹_r

, which fully demonstrates Theorem 2.1.

Theorem 2.2. Letmbe any natural number and

1< p ≤2, 1≤r <+∞, 1/p+ 1/q= 1.

Ifanare the coefficients of series(1.1)or(1.2), then:

(1) Forr≤qwe have f

(m) p,r,α≥C

( _∞ X

n=1

n^−mr|a_n|^rb₁(n)

b₁(n) A(n)

^r_q−1)¹_r

;

(2) Forr > qwe have f

(m) p,r,α≥C

( _∞ X

n=1

n^−mr|a_n|^rb₁(n) )¹_r

.

Proof. Letf be an even function. Iff is an odd function then the proof of the theorem is anal- ogous to the even case. It is not difficult to see that the Fourier series of∆_mf(x, t)is

∆_mf(x, t)∼











(−1)^m22^m

∞

P

n=1

a_ncosnxsin^mnt, formeven

(−1)^{m−12 −1}2^m

∞

P

n=1

a_nsinnxsin^mnt, formodd.

According to the well-known Hausdorf-Young’s theorem we find C

Z 2π 0

|∆_mf(x, t)|^pdx ^r_p

≥

∞

X

n=1

|a_n|^q|sinnt|^mq

!^r_q , and then

n f

(m) p,r,α

or

≥C

∞

X

ν=1

Z 1/ν 1/(ν+1)

α(t)

ν

X

n=1

|an|^q|sinnt|^mq

!^r_q dt.

Using the well-known inequalitysinB ≥ _π²Bfor0≤B ≤ ^π₂, we get n

f

(m) p,r,α

or

≥C

∞

X

ν=1

A(ν)

ν

X

n=1

n^−mq|a_n|^q

!^r_q .

Letr ≤q, then according to Lemma 1.1 we have n

f

(m) p,r,α

or

≥C

∞

X

ν=1

A(ν)

ν^−mq|a_ν|^qβ_ν^r_q .

It is easy to prove thatβ_ν ≥ ^b_A(ν)^1(ν), from which we get

(2.5) n

f

(m) p,r,α

or

≥C

∞

X

ν=1

ν^−mr|a_ν|^rb₁(ν)

b₁(ν) A(ν)

^r_q−1

.

Let q < r, then according to Lemma 1.3 and with the change of the order of summation we have

n f

(m) p,r,α

or

≥C

∞

X

ν=1

A(ν)

ν

X

n=1

n^−mr|a_n|^r

=C

∞

X

n=1

n^−mr|a_n|^r

∞

X

ν=n

A(ν)

≥C

∞

X

n=1

n^−mr|a_n|^rb₁(n).

(2.6)

Relations (2.5) and (2.6) prove Theorem 2.2.

We can deduce three corollaries from Theorem 2.1 and Theorem 2.2.

Corollary 2.3. Under the conditions of Theorem 2.1 and withb(n)≤CA(n), we have f

(m) p,r,α ≤C

( _∞ X

n=1

n^{r 1−2p ∞}

X

k=n

∆a_k

!r

b(n) )¹_r

.

Corollary 2.4. Under the conditions of Theorem 2.2 and withb₁(n)≤CA(n), we have f

(m) p,r,α≥C

( _∞ X

n=1

n^−mr|a_n|^rb₁(n) )¹_r

. As a special case, forα(t) =t^−βr−1, it is easy to prove the estimates:

A(n)≤Cn^βr−1 and b(n)≤Cn^βr.

From Theorem 2.1 and the last estimates we can deduce the following result proved in [1].

Corollary 2.5 ([1]). Letmbe any natural number and

0< β≤m, 1< p <+∞, 1≤r <+∞, 1/p+ 1/q= 1.

If the coefficients of series (1.1) or (1.2) satisfyP∞

k=1|∆a_k|<+∞, then f

(m) β,p,r ≤C

( _∞ X

n=1

n^r(^β+1_q)⁻¹

∞

X

k=n

∆a_k

!r)¹_r .

REFERENCES

[1] T.Sh. TEVZADZE, Some classes of functions and trigonometric Fourier series, Some Questions of Function Theory, v. II, 31–92, Tbilisi University Press, 1981 (in Russian).

[2] M.K. POTAPOVANDM. BERISHA, Moduli of smoothnes and Fourier coefficients of functions of one variable, Publ. Inst. Math. (Beograd) (N.S.), 26(40) (1979), 215–228 (in Russian).

[3] B. HARDY, E. LITLEWOODAND G. POLYA, Inequalities, GIIL Moscow, 1948, 1–456 (in Rus- sian).

[4] M.K. POTAPOV, A certain imbedding theorem, Mathematica (Cluj), 14(37) (1972), 123–146.