Improved direct and converse theorems in weighted Lorentz spaces

Improved direct and converse theorems in weighted Lorentz spaces

Article  in  Bulletin of the Belgian Mathematical Society, Simon Stevin · April 2016 DOI: 10.36045/bbms/1464710117 CITATIONS 2 READS 31 2 authors: Ramazan Akgün Balikesir University 41PUBLICATIONS   303CITATIONS    SEE PROFILE

Yunus Emre Yıldırır Balikesir University

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weighted Lorentz spaces

Ramazan Akg¨

un

Yunus Emre Yıldırır

Abstract

In the present work we prove the equivalence of the fractional modulus of smoothness to the realization functional and to the Peetre K-functional in weighted Lorentz spaces. Using this equivalence we obtain an improvement of the direct approximation theorem. Furthermore we prove the improved converse theorem in this space.

1

Introduction and main results

In approximation theory improvements of direct and inverse theorems have been investigated by several authors in different function spaces [1, 9, 12, 18, 20, 21]. In this paper we deal with the improved direct and inverse approximation theorems

in the weighted Lorentz space Lωpq(T)with Muckenhoupt weights. To obtain the

improved direct theorem we need the realization and characterization theorem

in Lωpq(T). Therefore we will prove a realization result and an equivalence

rela-tion between the modulus of smoothness and the Peetre K-funcrela-tional in Lωpq(T).

Furthermore, the realization result has a lot of applications [6]. In particular, it is used to get Ul’yanov type inequalities [8]. First, we give some definitions and properties.

Let T := [−π, π] and ω : T → [0, ∞] be a weight function i.e., an almost

everywhere positive measurable function. We define the decreasing

rearrange-Received by the editors in June 2014 - In revised form in January 2016. Communicated by H. De Schepper.

2010 Mathematics Subject Classification : 41A25, 41A27, 42A10.

Key words and phrases : Fractional modulus of smoothness, realization, weighted Lorentz

spaces, Muckenhoupt weight, direct and converse theorem.

ment f_ω∗(t)[11] of f : T→R_{with respect to the Borel measure} ω(e) = Z eω(x)dx, by f_ω∗(t) = inf_{τ ≥0 : ω(x∈ T _{: |f(x)| >τ)≤t}.}

The weighted Lorentz space Lpqω(T)is defined [11] as

Lpqω(T) =  f ∈ M(T) : kfkpq,ω = Z T(f ∗∗_(t))q_tqpdt t 1/q <∞, 1<p, q <∞ ,

where M(T_{)is the set of 2π periodic integrable functions on T and}

f∗∗(t) = 1 t Z t 0 f ∗ ω(u)du.

If p=q, Lωpq(T)turns into the weighted Lebesgue space Lωp(T)[11, p.20].

A weight function ω : T → [0, ∞] belongs to the Muckenhoupt class Ap [17],

1< p<∞, if sup 1 |I| Z Iω(x)dx 1 |I| Z I ω1−p′(x)dx
p−1 =C_Ap <∞, p′ := p p−1

with a finite constant CAp independent of I, where the supremum is taken over

all intervals I with length≤2π and|I|denotes the length of I. The constant C_Ap

is called the Muckenhoupt constant of ω.

By the proof of [14, Prop. 3.3], we know that L_wpq(T_{) ⊂ L}1₍T_{). Let}

S[f] :=

∞

∑

k=−∞

c_keikx (1)

be the Fourier series of a function f ∈ L1(T_{). Assume that}

Z

T f(x)dx =0. (2)

For α∈ R+_{, we define the α-th fractional integral of f as [22, v.2, p.134]}

Iα(x, f) :=

∑

k∈Z∗ c_k(ik)−αeikx, with (ik)−α :_{= |}k|−αe(−1/2)πiαsign k as principal value.

We define the fractional derivative of a function f _∈ L1(T_{), satisfying ( 2), as}

f(α)(x) := d

[α]+1

whenever the right hand side exists.

For a function f ∈ Lωpq(T), 1 < p, q < ∞, w ∈ Ap, Steklov’s mean operator is

defined as σhf(x) := 1 2h Z x+h x−h f(u)du, x ∈ T_.

Whenever ω ∈ Ap, 1 < p, q < ∞, the Hardy-Littlewood maximal function of

f ∈ Lpqω(T)belongs to Lωpq(T)[5, Theorem 3]. Therefore the operator σhf belongs

to Lωpq(T). Using this fact and putting x, t ∈ T, r ∈ R+, ω ∈ Apand f ∈ Lωpq(T),

1< p, q<∞, we define σ_trf (x) := (I−σt)r f(x) = ∞

∑

k=0 (−1)k r k  1 (2t)k Z t −t· · · Z t −t f(x+u1+ · · · +uk)du1. . . duk,

where(_kr)are the binomial coefficients.

Since     α k     ≤ c kα+1, k∈ N (see [19, p.14, (1.51)]), we have ∞

∑

k=0     α k     <∞, and therefore kσ_tαfkpq,ω ≤ckfkpq,ω <∞, (3) for f ∈ Lωpq(T), 1< p, q <∞, ω∈ Ap.

For 1 < p, q < ∞, f ∈ L_ωpq(T_{)and r ∈} R+_{, we define the fractional modulus of}

smoothness of index r as Ω_{r(f , δ)pq,ω := sup} 0<_hi,t<_δ [r]

∏

i=1 (I−σ_hi)(I−σt)r−[r]f pq,ω, (4)

where [r] := max{n_∈ N _{: n≤r}. Since the operator σt is bounded in Lω}pq₍T_),

1< p, q<∞, ω ∈ A_pwe have by (3) that

Ω_{r(f , δ)pq,ω ≤ckfkpq,ω}

where the constant c>0 only depends on r, p , q and C_A

p.

Remark 1.1. Let r ∈ R+_{, 1} < p, q < ∞, ω ∈ A_p and f ∈ L_ωpq(T_{). For δ} > 0, the

modulus of smoothness Ωr(f , δ)pq,ω has the following properties.

(i) Ωr(f , δ)pq,ω is sub-additive in f , and a non-negative, non-decreasing

func-tion of δ≥0.

(ii) lim δ→0

By En(f)pq,ω we denote the best approximation of f ∈ Lωpq(T)by polynomials in

Tn, the set of trigonometric polynomials of degree≤n:

En(f)pq,ω = inf

Tn∈Tnk

f −Tnkpq,ω. In this paper we will use the following notations:

A(x) ≈ B(x) ⇔ ∃c1, c2>0 : c1B(x) ≤ A(x) ≤ c2B(x)

A(x)  B(x) ⇔ ∃c >0 : A(x) ≤cB(x).

Our main results are now the following.

Theorem 1.2. If r ∈ R+_{, 1}< p, q < ∞, ω ∈ A_pand f ∈ L_ωpq(T_{), then there exists a}

constant c>0 depending only on r, p, q and C_A

p such that En(f)pq,ω ≤c Ωr  f , 1 n+1  pq,ω (5) holds for n+1∈ N_.

The analogues of this direct approximation theorem were obtained in [10] for

r ∈ N_{, f ∈ Lω}p₍T_{), 1} < p < ∞, ω ∈ A_p and in [2] for r ∈ N, f ∈ L_ωpq(T),

1< p, q <∞, ω ∈ A_pwith the modulus of smoothness

Wr  f , 1 n  Lωpq := sup 0≤hi≤1/n r Π i=1 I−σhi
f Lωpq ,

and in [1] for r ∈R+_{, f ∈ Lω}p₍T_{), 1}< p <∞, ω∈ A_pwith the fractional modulus

of smoothness (4).

For f ∈ Lωpq(T), t, r >0 and 1< p, q <∞, the Peetre K-functional is defined as

Kr(f , t; Lωpq, Wpq,ωr ) := inf g∈Wrpq,w{k f ₋g_kLpq ω +t r_kg(r)_k Lpqω}. Here W_pq,ωr :=ng(x) ∈ Lpqω : g(r) ∈ Lωpq o .

We define the realization functional for f ∈ Lωpq(T)by

Rr(f , 1/n, Lωpq):= n kf −t∗_nk_Lpq ω + 1 nrk(t∗n) (r)_k Lωpq o ,

for r>0, 1< p, q <∞, n∈ N. Here t∗_n denotes the best approximating

trigono-metric polynomial for f . The following theorem holds.

Theorem 1.3. If R+, f ∈ Lωpq(T), 1 <p, q <∞ and ω ∈ Ap, then the equivalence

Ω_r_{f ,}1 n  pq,ω ≈R2r f , 1/n, L pq w
(6)

holds for n=1, 2, 3, ... Furthermore, we have, for δ≥0,

Ω_{r(f , δ)pq,ω ≈K2r(f , δ; Lw}pq_{, Wpq,w}r _{). (7)}

Corollary 1.4. Let r∈ R+_{, f ∈ Lw}pq₍T_{), 1}< p, q <∞, and ω∈ A_p. Then Ω_r(f , λδ)_pq,ω  (1+ [λ])2rΩ_r(f , δ)_pq,ω, δ, λ>0 and

Ω_{r(f , δ)pq,ωδ}−2r _Ω_{r(f , δ1)pq,ωδ}−2r

1 , 0<δ1≤δ.

An improvement of (5) is given by the following theorem.

Theorem 1.5. If r∈ R+_{, f ∈ Lω}pq₍T_{), 1} < p, q < ∞ and ω ∈ A_p, then there exists a

constant c>0 depending on r, p, q and C_A

p such that for n=1, 2, 3, . . .

 n

∏

j=1 E_j(f)pq,ω 1/n ≤c Ωr  f , 1 n  pq,ω. (8)

Remark 1.6. The inequality (8) is never worse than the classical Jackson inequality.

Since En(f)pq,ω →0 as n→∞ we obtain that

En(f)pq,ω ≤  n

∏

j=1 E_j(f)pq,ω 1/n ≤c Ωr  f , 1 n  pq,ω.

On the other hand, in some cases the inequality (8) gives better results than the

classical Jackson inequality. For example, if En(f)pq,ω = 2−n, then the

classical Jackson inequality implies Ωr

 f ,_n1

pq,ω ≥c2

−n_{but inequality (8) yields}

Ω_rf ,1_n

pq,ω ≥c2

−n/2_.

An analogue of Theorem 1.5 for the space L∞ was proved in [18]. In [2], the weak

converse of (5) Ω_r_{f ,} 1 n  Lωpq ≤ c n2r n

∑

ν=0 (ν+1)2r−1Eν(f)_Lpq ω, (9)

for r ∈N_{, f ∈ Lω}pq₍T_{), ω∈ Apand 1}< p, q <∞ was obtained.

Theorem 1.7. Let 1 < p < ∞ and 1 < q ≤ 2 or p > 2 and q ≥ 2, ω ∈ A_p,

f ∈ Lωpq(T). If n ∈ N, r ∈ R+ and γ := min{2, q}, then there is a constant c > 0

only depending on r, q, p and C_Ap such that

Ω_r_{f ,} 1 n  pq,ω ≤ c n2r  n

∑

ν=1 ν2γr−1Eγ_ν−1(f)_pq,ω1/γ. (10)

The analogues of this improved converse theorem were proven in [15] for r∈N_,

f ∈ Lpω(T), 1< p<∞, ω ∈ Apwith the modulus of smoothness Wr

 f ,1_n

Lωp

; in

[1] for r ∈ R+_{, f ∈ Lω}p₍T_{), 1} < p < ∞, ω ∈ A_p with the fractional modulus of

smoothness (4); in [14] for r ∈ N_{, f ∈ Lω}pq₍T_{), 1}< p <∞ and 1 <q ≤2 or p>2

and q_≥2, ω _∈ Ap with Wr

 f ,_n1

Lωpq

and 1<q ≤2 or p>2 and q≥2, ω ∈ A_pwith a modulus of smoothness defined by Ky [16].

The inequality (10) is better than (9). Indeed, using the fact that xγ is convex for

γ=min{2, p} we obtain that

 νν2r−1Eν(f)pq,ω γ −(ν−1)ν2r−1Eν(f)pq,ω γ ≤  ν

∑

µ=1 µ2r−1Eµ(f)pq,ω γ − ν−1

∑

µ=1 µ2r−1Eµ(f)pq,ω γ . Taking the summation over ν, we obtain that

n

∑

ν=1 n νν2r−1Eν(f)pq,ω γ −(ν−1)ν2r−1Eν(f)pq,ω γo ≤ n

∑

ν=1 n ν

∑

µ=1 µ2r−1Eµ(f)pq,ω  )γ₋ν

_∑

−1 µ=1 µ2r−1Eµ(f)pq,ω γo , hence we have the inequality

n n

∑

ν=1 ν2γr−1Eγ_ν−1(f)pq,ω o1/γ ≤2 n

∑

ν=1 ν2r−1Eν−1(f)pq,ω.

We give the Marcinkiewicz multiplier and Littlewood-Paley theorems in Lpqω (T)

which are used in the proofs of previous Theorems.

Theorem 1.8. Let λ0, λ1,· · · be a sequence of real numbers such that

|λ_l| ≤ M and

2l−1

∑

ν=2l−1

|λν−λν+1| ≤ M,

for all ν, l ∈ N_{. If 1} < p, q < ∞, ω ∈ A_p and f ∈ L_ωpq(T_{) with Fourier series}

∑∞_ν=0(aν(f)cos νx+bν(f)sin νx), then there exists h ∈ Lωpq(T) such that the series

∑∞_ν=0λν(aν(f)cos νx+bν(f)sin νx)is the Fourier series of h and

khkpq,ω ≤Ckfkpq,ω, (11)

where C does not depend on f .

Theorem 1.9. Let ν∈ N_{, 1}< p, q <∞, ω ∈ A_p and f ∈ L_ωpq(T_{)with Fourier series}

∑∞_ν=0(aν(f)cos νx+bν(f)sin νx), then there exist constants c1 and c2independent of

f such that c1  ∞

∑

µ=ν |∆_µ|21/2 pq,ω ≤ kfkpq,ω ≤c2  ∞

∑

µ=ν |∆_µ|21/2 pq,ω, (12) where ∆_{µ :=}∆_{µ(x, f):=} 2µ₋₁

∑

ν=2µ−1 (aν(f)cos νx+bν(f)sin νx).

2

Proof of the main results

From [5, 14] we recall four important properties of the spaces Lωpq(T).

Lemma A([5] or [13, prop. 5.1.2]) For 1< p, q <∞, there exists a c>0 such that

for every f ∈ Lωpq(T) c−1kfk_pq,ω ≤sup  Z T f(x)g(x)w(x)dx    ≤ckfkpq,ω,

where the supremum is taken over all functions g for whichkgk_p′_q′_,ω≤1.

Lemma B [14]. Let 1 < p < ∞ and 1 < q ≤ 2. Then for an arbitrary system of

functions ϕj(x) m j=1, ϕj ∈ L pq ω (T)we have  m

∑

j=1 ϕ_j21/2 pq,ω ≤c m

∑

j=1 kϕ_jkqpq,ω 1/q

with a constant c independent of ϕ_j and m.

Lemma C [14]. Let 2 < p < ∞ and q ≥ 2. For an arbitrary system  ϕ_j(x) m

j=1, ϕj ∈ Lpqω (T), we have  m

∑

j=1 ϕ_j21/2 pq,ω ≤c  m

∑

j=1 ϕ_j 2 pq,ω 1/2

with a constant c independent of ϕ_j and m.

Lemma D [14]. Let 1 < p, q < ∞, f ∈ Lpq_ω (T_{) and w ∈ Ap. If Bk,µ(x) =}

a_k(f)cos k+µπ₂
x +b_k(f)sin k+µπ₂
x, where ak, bk are the Fourier coefficients

of f , then 2i+1

∑

k=2i₊₁ kµB_k,µ pq,ω ≤c2 iµ_E 2i(f)_pq,ω,

where the constant c is independent of f and i.

Proof of Theorem 1.8. We define a linear operator

T f(x) :=

∞

∑

ν=0

λν(aν(f)cos νx+bν(f)sin νx)

for f ∈ Lωpq(T) which is bounded (in particular is of weak type(p, p)) in Lωp(T)

for every p > 1 by [4, Th. 4.4]. Therefore the hypothesis of the interpolation

theorem for Lorentz spaces [3, Th. 4.13] is fulfilled. Applying this theorem we get the desired result (11).

Proof of Theorem 1.9. Let us define a quasilinear operator

T f(x) := ∞

∑

µ=ν  ∆_µ(x, f)  2 1/2 .

This operator is bounded in Lωp(T) for every p >1 by [4, Th. 4.5]. Therefore the left hand side of the required result (12) is derived by means of the interpolation theorem for Lorentz spaces [3, Th. 4.13].

Using H ¨older’s inequality for f ∈ Lωpq(T)∩L2ω(T), g ∈ L

p′q′

ω (T)∩L2ω(T)and the

left hand side of (12) we obtain

Z T|f(x)g(x)|ω(x)dx = Z T      ∞

∑

µ=1 ∆_{µ(x, f)}∆_{µ(x, g)}      ω(x)dx ≤ Z T ∞

∑

µ=1 |∆_{µ(x, f)}∆_{µ(x, g)|ω(x)dx} ≤ Z T h ∞

∑

µ=1  ∆_µ(x, f)   2i1/2h ∞

∑

µ=1  ∆_µ(x, g)   2i1/2 ω(x)dx ≤ h ∞

∑

µ=1     ∆_µ(x, f)  2i1/2 pq,ω h ∞

∑

µ=1     ∆_µ(x, g)  2i1/2 p′_q′_,ω ≤ C h ∞

∑

µ=1  ∆_µ(x, f)  2 i1/2 pq,ωkgkp′q′,ω.

where p′ =p/(p−1), q′ =q/(q−1). Taking the supremum in the last

inequal-ity over all functions g _∈ Lpω′q′(T) satisfying kgk_p′_q′_,ω ≤ 1, we find, applying

Lemma A that kfk_pq,ω ≤C  ∞

∑

µ=1     ∆_µ  21/2 pq,ω . The density of Lωpq(T) ∩ L2ω(T) in L pq

ω (T) yields the last inequality for any

f ∈ Lωpq(T).

Lemma 2.1. If 0<α ≤β, ω ∈ A_p, 1< p, q <∞ and f ∈ L_ωpq(T_)then

Ω_β(f ,_{·)pq,ω ≤}c Ωα(f ,·)_pq,ω. (13)

Proof. The proof of Lemma 2.1 is similar to the proof of [1, Lemma 1].

Lemma 2.2. Let r∈ R+_{, 1}< p, q<∞, ω ∈ A_pand T_n ∈ T_n for n=1, 2,· · ·. Then

Ω_{r Tn,} 1 n
pq,ω  1 n2r T (2r) n pq,ω

holds with some constant only depending on r, p, q and C_Ap.

Proof. For all x_≥0, we have that

 1−sin x x  ∗ ≤ x2, where  1−sin x x  ∗ := ( 1−sin x_x if x ≥0; 0 if x =0.

For 0<t and h_i ≤ 1 n, we have that [r]

∏

i=1 I−σhi
(I−σt)r−[r]Tn pq,ω = n

∑

ν=0  1−sin νh1 νh1  ∗· · · 1−sin νh[r] νh[r] ! ∗  1−sin νt νt r−[r] ∗ Aν(Tn, x) pq,ω = n

∑

ν=1  1−sin νh1 νh1  (νh1)2 (νh1)2 · · ·  1−sin νh_νh[r][r] νh_[r]2  νh_[r]2 1−sin νt_νt (νt)2 !r−[r] (νt)2(r−[r])Aν(Tn, x) pq,ω ≤ n−2r n

∑

ν=1  1−sin νh1 νh₁  (νh1)2 ν2· · ·  1−sin νh[r] νh_[r]   νh[r] 2 ν 2 1− sin νtνt (νt)2 !r−[r] ν2(r−[r])Aν(Tn, x) pq,ω ≤ n−2r n

∑

ν=1 ν2r  1−sin νh1 νh1  (νh1)2 · · ·  1−sin νh[r] νh_[r]   νh[r] 2 1−sin νt_νt (νt)2 !r−[r] Aν(Tn, x) pq,ω.

Applying Theorem 1.8 we obtain that [r]

∏

i=1 I₋σhi
(I₋σt)r−[r]Tn pq,ω n −2r n

∑

ν=1 ν2rAν(Tn, x) pq,ω. For ν =1, 2, 3, ... we have Aν(Tn, x) = Aν(Tn, x+ rπ ν )cos rπ+Aν(T˜n, x+ rπ ν )sin rπ,

where ˜Tn is the Fourier conjugate of Tn. Therefore

[r]

∏

i=1 (I−σ_hi)(I−σt)r−[r]Tn pq,ω  n−2r n

∑

ν=1 ν2r(Aν(Tn, x+rπ ν )cos rπ+Aν(T˜n, x+ rπ ν )sin rπ) pq,ω  n−2r n

∑

ν=1 ν2rAν(Tn, x+rπ ν ) pq,ω+ n

∑

ν=1 ν2rAν(T˜n, x+rπ ν )  kpq,ω  . Since Aν(Tn(2r), x) = ν2rAν(Tn, x+ rπ ν ),

for ν=1, 2, 3,· · ·, we find Ω_rTn, 1 n  pq,ω  n−2r n

∑

ν=1 ν2rAν(Tn, x+rπ ν ) pq,ω+ n

∑

ν=1 ν2rAν(T˜n, x+rπ ν ) pq,ω   n−2r T  2r  n pq,ω+ T˜  2r  n pq,ω  n−2r T  2r  n pq,ω.

Lemma 2.3. Let r ∈ R_{+, 1}< p, q < ∞, ω ∈ A_p and T_n ∈ T_n. For n = 1, 2,· · ·, we

have that 1 n2r T_n(2r) pq,ω Ωr  Tn, 1 n  pq,ω

with some constant depending only on r, p, q and C_Ap.

Proof. n−2r T (2r) n pq,ω =n −2r n

∑

ν=1 ν2rAν(Tn, x+rπ ν ) pq,ω = n−2r n

∑

ν=1 ν2r Aν(Tn, x)cos rπ+Aν(T˜n, x)sin rπ
pq,ω ≤ n−2r n

∑

ν=1 ν2rAν(Tn, x)cos rπ pq,ω +n−2r n

∑

ν=1 ν2rAν(T˜n, x)sin rπ pq,ω = n

∑

ν=1 cos rπ   ν n
2 1− sinnν ν n   r  1−sin ν n ν n r Aν(Tn, x) pq,ω + n

∑

ν=1 sin rπ   ν n
2 1−sinνn ν n   r  1−sin ν n ν n r Aν(T˜n, x) pq,ω.

Applying Theorem 1.8 and the linearity of the conjugate operator we get n−2r T (2r) n pq,ω  n

∑

ν=1  1−sin ν n ν n r Aν(Tn, x) pq,ω + n

∑

ν=1  1−sin ν n ν n r Aν(T˜n, x) pq,ω = n

∑

ν=1  1−sin ν n ν n r Aν(Tn, x) pq,ω + n

∑

ν=1  1−sin ν n ν n r Aν(Tn, x) !˜ pq,ω.

From the boundedness of the conjugate operator [14] we have n−2r T (2r) n pq,ω  n

∑

ν=1  1−sin ν n ν n r Aν(Tn, x) pq,ω+ n

∑

ν=1  1−sin ν n ν n r Aν(Tn, x) pq,ω   I−σ1 n r Tn pq,ω ₀<_hsup_i,u<_1/n [r]

∏

i=1 I−σ_hi
(I−σu)r−[r]Tn pq,ω  Ω_rTn, 1 n  pq,ω.

Proof of Theorem 1.2. From Lemma 2.1 and [2, Th. 1.1] we have

En(f)pq,ω ≤c Ω[r]+1  f , 1 n+1  pq,ω ≤CΩr  f , 1 n+1  pq,ω

for n+1∈ N_{and the assertion (5) follows.}

Lemma 2.4. Let 1 < p, q < ∞, ω ∈ A_p, f ∈ L_ωpq(T_{) and γ} > 0 . Then for any

0<t<2, Ω_{γ(f , t)} pq,ω tγ f (γ) pq,ω.

Proof. There is some n=1, 2, 3, ... such that(1/n) < t_{≤ (}2/n). From Lemma 2.2

we get Ω_{γ(f , t)} pq,ω ≤Ωγ(f −Tn, t)pq,ω+Ωγ(Tn, t)pq,ω En(f)pq,ω+t2γ T (2γ) n pq,ω. On the other hand applying [20, (3.9) and Th. 1.3] and Theorem 1.2 we have

En(f)pq,ω  1 n2γEn(f (2γ)₎ pq,ω  1 n2γΩγ  f(2γ), 1 n  pq,ω  t2γ f (2γ) pq,ω. Using Theorem 1.2 and [20, Th. 1.3] the proof is completed.

Proof of Theorem 1.3. We have to show that (6) holds. Let Tn be the near best

approximating trigonometric polynomial to f . From Theorem 1.2

kf −Tnk_pq,ω En(f)pq,ω ≤c Ωr  f , 1 n+1  pq,ω. Applying Lemma 2.3, we find that

1 n2r T  2r  n pq,ω  Ω_r_Tn, 1 n  pq,ω  Ω_r_{Tn− f ,} 1 n  pq,ω+ Ω_r_{f ,}1 n  pq,ω  f −Tn pq,ω+ Ω_r_{f ,} 1 n  pq,ω  Ω_r_{f ,}1 n  pq,ω

and kf −Tnk_pq,ω+ 1 n2r T (2r) n pq,ω  Ω_r_{f ,}1 n  pq,ω. On the other hand using Lemma 2.2

Ω_rf ,1 n  pq,ω ≤ Ω_rf −Tn, 1 n  pq,ω+ Ω_rTn, 1 n  pq,ω  f −Tn pq,ω+ 1 n2r T (2r) n pq,ω =R2r  f , 1 n, L pq ω  .

This completes the proof of (6). Using Lemma 2.4, properties of the modulus of

smoothness and of the K−functional (7) are proven.

Proof of Theorem 1.5. By Corollary 1.4 we have for v ≤n

Ω_{r(f , 1/v)} pq,ω ≤ (1+n/v)2rΩr(f , 1/n)pq,ω and n

∏

v=1 Ω_{r(f , 1/v)} pq,ω ≤ n

∏

v=1 (1+n/v)2rΩ_{r(f , 1/n)} pq,ω n . For every n we have

n

∏

v=1 (1+n/v)2r ≤  2n n √ n! 2r . Using Stirling’s formula

n!≍√2πnnne−neθ(n)with |θ(n)| ≤1/(12n) we get n

∏

v=1 (1+n/v)2r ≤22re4r. Thus n

∏

v=1 Ω_{r(f , 1/v)} pq,ω !1/n ≤c Ωr(f , 1/n)_pq,ω.

From (5) and the property En(f)pq,ω →0 as n→∞ we find

n

∏

v=1 Ev(f)pq,ω !1/n ≤ n

∏

v=1 Ω_{r(f , 1/v)} pq,ω !1/n ≤c Ωr(f , 1/n)_pq,ω.

Proof of Theorem 1.7. For 1 < p, q < ∞, ω ∈ A_p, let f ∈ L_ωpq(T) be such that

R2π

0 f(x)dx =0. We assume that f has Fourier series (1). We choose m∈ Nsuch

that 2m ≤n <2m+1. Let us denote S_n(x) :=S_n(x, f):=∑n

k=0Ak(f , x), for x∈ T,

where A_k(f , x) = a_k(f)cos kx+b_k(f)sin kx. By [14, Prop. 3.4], we have that

It is well-known that σ_t,hr 1,h2,...,h[r]f := [r] ∏ i=1 I−σhi

(I−σt)r−[r] f has Fourier series σ_t,hr _1,h2,...,h [r]f (·) ∼ ∞

∑

ν=0  1−sin νt νt r−[r] ∗  1− sin νh1 νh1  ∗ . . . 1−sin νh[r] νh[r] ! ∗ Aν(f , x). Moreover σ_t,hr _1,h2,...,h [r]f (·) = σ_t,hr 1,h2,...,h[r](f (·) −S2m−1(·, f)) +σ r t,h1,h2,...,h[r]S2m−1(·, f).

From (14) and En(f)p,ω →0 we have

σ r t,h1,h2,...,h[r](f (·) −S2m−1(·, f)) pq,ω ≤ ckf (·) −S₂m−1(·, f)k_pq,ω ≤cE₂m−1(f)pq,ω ≤ c n2r ( n

∑

ν=1 ν2γr−1Eγ_ν−1(f)p,ω )1/γ . On the other hand, it follows from (12) that

σ r t,h1,h2,...,h[r]S2m−1(·, f) pq,ω ≤c n m

∑

µ=1  δ_µ  2 o1/2 pq,ω where δµ := 2µ−1

∑

ν=2µ−1  1−sin νt νt r−[r] 1−sin νh1 νh₁  . . . 1−sin νh[r] νh_[r] ! Aν(f , x). By Lemmas B and C n m

∑

µ=1   δµ    2o1/2 pq,ω ≤ n m

∑

µ=1 δµ γ pq,ω o1/γ . By Abel’s transformation we obtain

kδµkpq,ω ≤ 2µ₋₂

∑

ν=2µ−1     1−sin νt νt r−[r] 1−sin νh1 νh1  . . .1−sin νh[r] νh_[r]  −1₋sin(ν+1)t (ν+1)t r−[r] 1₋sin(ν+1)h1 (ν+1)h1  · · ·1−sin  ν+1h_[r] (ν+1)h_[r]    ν

∑

l=2µ−1 A_l(f , x) pq,ω

+   1−sin(2 µ_−1)t (2µ_−1)t r−[r] 1−sin(2 µ_−1)h 1 (2µ_−1)h 1 ) · · ·1−sin(2 µ_−1)h [r] (2µ_−1)h [r]    2µ−1

∑

l=2µ₋₁ Al(f , x) pq,ω and by Lemma D ν

∑

l=2µ−1 A_l(f , x) pq,ω ≤cE2µ−1−1(f)p,ω and 2µ₋₁

∑

l=2µ₋₁ A_l(f , x) pq,ω ≤CE2µ−1−1(f)pq,ω. Since xr 1−sin x_x
r

is non decreasing for positive x we have δ_µ pq,ω ≤c22µrt2 (r−[r])_h2 1h22. . . h2[r]E2µ₋₁₋₁(f)_pq,ω and hence σ r t,h1,h2,...,h[r]S2m−1  ·, f pq,ω ≤ ct2  r−[r]  h2₁h2₂. . . h2_[r]n m

∑

µ=1 2µrγEγ₂µ−1₋₁(f)pq,ω o1/γ ≤ ct2  r−[r]  h2₁h2₂. . . h2_[r]n2γrE₀γ(f)M o1/α + ct2  r−[r]  h2₁h2₂. . . h2_[r]n m

∑

µ=2 2µ−1−1

∑

ν=2µ₋₂ ν2γr−1Eγ_ν−1(f)pq,ω o1/γ ≤ ct2  r−[r]  h2₁h2₂. . . h2_[r]n 2m−1−1

∑

ν=1 ν2γr−1Eγ_ν−1(f)pq,ω o1/γ . Therefore we find Ω_rf , 1 n  pq,ω ≤ c n2r n n

∑

ν=1 ν2γr−1E_νγ₋₁(f)_pq,ωo1/γ finishing the proof of Theorem 1.7.

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Department of Mathematics, Faculty of Art-Science, Balikesir University

10100 Balikesir, Turkey

email: [email protected]

Department of Mathematics, Faculty of Education, Balikesir University

10100, Balikesir, Turkey

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