Pointwise strong and very strong approximation by matrix means of Fourier series

(1)

W lodzimierz Lenski, Bogdan Szal

Pointwise strong and very strong approximation by matrix means of Fourier series

Abstract. We generalize and extend the some results of the paper [6]. Considering a wider class of function and more general means we obtain the results of the V.

Totik type [8, 9].

2000 Mathematics Subject Classification: 42A24, 41A25.

Key words and phrases: Strong and very strong approximation, Rate of pointwise summability, Matrix means.

1. Introduction. Let L^p (1 < p < ∞) [resp.C] be the class of all 2π–periodic real–valued functions integrable in the Lebesgue sense with p–th power [continuous]

over Q = [−π, π] and let X = X^p where X^p = L^p when 1 < p < ∞ or X^p = C when p = ∞. Let us define the norm of f ∈ X^p as

kfkXp = kf(x)kXp =





R

Q | f(x) |^pdx1/p

when 1 < p <∞ , sup_x_∈Q| f(x) | when p =∞.

Consider the trigonometric Fourier series

Sf (x) =ao(f) 2 +X^∞

k=0

(ak(f) cos kx + bk(f) sin kx)

and denote by Skf the partial sums of Sf . Let

H_k^q₀_,k_rf (x) :=

( 1 r + 1

Xr ν=0

|S^k^νf (x)− f (x)|^q )¹_q

, (q > 0) .

(2)

where 0 ≤ k0< k1< k2< ... < kr (≥ r) , and

H_n,A^q f (x) :=

( _n X

k=0

ank|S^kf (x)− f (x)|^q )¹q

,

where A := (ank) (k, n = 0, 1, ...) be a lower triangular infinite matrix of real numbers satisfying the following conditions:

(1) ank≥ 0 (k, n = 0, 1, ...) , a^nk= 0, k > n and Xn k=0

ank= 1.

Now we define two classes of sequences (see [4]).

A sequence c := (cn) of nonnegative numbers tending to zero is called the Rest Bounded Variation Sequence, or briefly c ∈ RBV S, if it has the property

(2) X^∞

k=m

|cⁿ− cⁿ⁺¹| ≤ K (c) c^m

for all natural numbers m, where K (c) is a constant depending only on c.

A sequence c := (cn) of nonnegative numbers will be called the Head Bounded Variation Sequence, or briefly c ∈ HBV S, if it has the property

(3)

m−1X

k=0

|cⁿ− cⁿ⁺¹| ≤ K (c) c^m

for all natural numbers m, or only for all m ≤ N if the sequence c has only finite nonzero terms and the last nonzero terms is cN.

Therefore we assume that the sequence (K (αn))^∞_n=0 is bounded, that is, that there exists a constant K such that

0 ≤ K (αn) ≤ K

holds for all n, where K (αn) denotes the sequence of constants appearing in the in- equalities (2) or (3) for the sequence αn:= (ank)^∞_k=0.Now we can give the conditions to be used later on. We assume that for all n and 0 ≤ m ≤ n

(4)

X∞ k=m

|a^nk− a^nk+1| ≤ Ka^nm

and

(5)

mX−1 k=0

|a^nk− a^nk+1| ≤ Ka^nm

hold if αn:= (ank)^∞_k=0 belongs to RBV S or HBV S, respectively.

(3)

As a measure of approximation by the above quantities we use the pointwise characteristic

wx,β(δ; f)_Lp:=



 1 δ^1+βp

Zδ

0

|ϕ^x(t)|

sin t

2

β!p

dt





1 p

,

where β ≥ 0 and

ϕx(t) = f (x + t) + f (x − t) − 2f (x) .

The quantity wx,β(δ; f)_Lpis constructed on the base of the definition of the Lebesgue (L^p-points), introduced in [5] (cf.[1]) and every point x for which

1 δ^1+βp

Zδ

0

|f (x ± t) − f (x)|

sin t

2

β!^p

dt = o(1) as δ→ 0⁺

we will call the L^p_β-Lebesgue point.

We can observe that for f ∈ X^p^eand ep ≥ p ≥ 1, by the Minkowski inequality, kw·,βf (δ)L^pkX ep≤ ωX epf (δ) ,

where ωXf is the modulus of continuity of f in the space X = X^p^edefined by the formula

ωXf (δ) := sup

0<|h|≤δkϕ·(h)kX .

We shall write I1 I² if there exists a positive constant C such that I1≤ CI². Let us consider a function wx of modulus of continuity type on the interval [0, 2π], i.e. a nondecreasing continuous function having the following properties:

wx(0) = 0, wx(δ1+ δ2) ≤ wx(δ1) + wx(δ2) for any 0 ≤ δ1≤ δ²≤ δ¹+ δ2≤ 2π and let for β ≥ 0

L^p(wx)^?_β = {f ∈ L^p:

"

1 δ^1+βp

Z δ

0 |ϕ^x(t) − ϕx(t ± γ)|

sin t

2

β!p

dt

#^1/p

w^x(γ) and wx,βf (δ)_Lp w^x(δ) , where γ, δ > 0} .

Since, for β > α ≥ 0,

wx,β(δ; f)_Lp=



 δ^αp δ^βp

1 δ^1+αp

Zδ

0

|ϕ^x(t)|

sint

2

α^p sin₂^t^β sin^t₂^α

!^p dt





1 p

≤π^α 2^β



δ^p(α^−β) 1 δ^1+αp

Zδ

0

|ϕ^x(t)|

sin t

2

α^p

t^p(β^−α)dt





1 p

≤π^α

2^βwx,α(δ; f)_Lp

(4)

and

"

1 δ^1+βp

Z δ 0

|ϕ^x(t) − ϕx(t ± γ)|sins 2

βp

dt

#^1/p

≤ π^α 2^β

"

1 δ^1+αp

Z δ 0

|ϕ^x(t) − ϕx(t ± γ)|sins 2

α^p dt

#1/p

,

we have L^p(wx)^∗_α⊂ L^p(wx)^∗_β. Let also

L^p(ω) = {f ∈ L^p: ωL^pf (δ)≤ ω (δ)} , where ω is a modulus of continuity type function.

It is well-known that H_0,r^q f (x)− means tend to 0 (as r → ∞) at the L^p−point x of f ∈ L^p (1 < p ≤ ∞) . In [2] this fact was by G. H. Hardy, J. E. Littlewood proved as a generalization of the Fej´er classical result on the convergence of the (C, 1) -means of Fourier series. In [6] W. Lenski obtained an estimation of the H_k^q₀_,k_rf (x) means as an approximation version of the Totik type (see [8, Na1]) generalization of the result of G. H. Hardy, J. E. Littlewood, considered by N. L.

Pachulia in [7]. Namely, he proved the following theorem:

Theorem 1.1 [6] If f ∈ L^p(wx)^?₀ (1 < p ≤ 2) and 0 < k0 < k1 < k2 < ... < kr

(≥ r), then

(6) H_k^q₀_,k_rf (x) w^x

π

k0+ 1

log kr+ 1 r + 1/2, where _p¹+¹_q = 1.

In the present paper we shall generalize this result taking a wider class of functions and more general means. We also give some corollaries on norm approximation.

2. Main results. We start with the following theorem:

Theorem 2.1 If f ∈ L^p(wx)^?_β (p > 1, β ≥ 0) and 0 < k0 < k1 < k2 < ... < kr

(≥ r), then

(7) H_k^q₀_,k_rf (x) w^x

π

k0+ 1

1 + logkr+ 1/2 r + 1/2

,

where q is such that 1 < q (q − 1)⁻¹ ≤ p ≤ q.

Remark 2.2 If β = 0 and p = q (q− 1)⁻¹, then we have (6) as a corollary of (7).

Similarly, we have estimate for norms.

(5)

Theorem 2.3 If f ∈ L^p(ω) (p ≥ 2) and 0 < k0< k1< k2< ... < kr (≥ r), then

Hk^q0,krf (·)

L^p ω

π

k0+ 1

1 + logkr+ 1/2 r + 1/2

, where q is such that 0 < q ≤ p.

Basing on Theorem 2.1 and 2.3, we can formulate the next theorems.

Theorem 2.4 Let (1) and (4) hold. If f ∈ L^p(wx)^?_β (p > 1, β ≥ 0), then

H_n,A^q f (x) ( _n

X

k=0

ank

wx

π

k + 1

q)¹q

,

Theorem 2.5 Let (1) and (5) hold. If f ∈ L^p(wx)^?_β (p > 1, β ≥ 0), then

H_n,A^q f (x)





 [Xⁿ⁺¹4 ]

k=1

an,4k

wx

π k + 1

^q +

wx

π

n + 1

log ((n + 1) ann)^q





1 q

,

Theorem 2.6 Let (1) and (4) hold. If f ∈ L^p(ω) (p ≥ 2), then

Hn,A^q f (·)

L^p ( _n

X

k=0

ank

ω

π k + 1

^q)¹_q , where q is such that 0 < q ≤ p.

3. Lemmas. With the notation

Φxf (δ, γ) :=1 δ

Z γ+δ γ

ϕx(t) dt, wxf (δ, γ)_Lp:=

"

1 δ

Z γ+δ

γ |ϕ^x(t)|^pdt

#^1/p

we can formulate the following lemma:

Lemma 3.1 If f ∈ L^p(wx)^?_β (p ≥ 1) , then

|Φ^xf (δ, γ)| ≤ w^xf (δ, γ)_Lp ≤ π^β(1 + βp)^1/p(wx(δ) + wx(γ)) for any positive γ, δ.

(6)

Proof The first inequality is evident, then we prove the second one only. Using integration by parts we obtain

wxf (δ, γ)_Lp=

"

1 δ

Z δ

0 |ϕ^x(t + γ)|^pdt

#1/p

≤

"

1 δ

Z δ

0 |ϕ^x(t + γ) − ϕx(t)|^pdt

#1/p

+

"

1 δ

Z δ

0 |ϕ^x(t)|^pdt

#1/p

=



1 δ

Z δ 0

|ϕ^x(t + γ) − ϕx(t)|sin^t₂^βp

sin₂^t^βp dt





1/p

+



1 δ

Z δ 0

|ϕ^x(t)|sin₂^t^βp

sin^t₂^βp dt





1/p

≤ π^β





"

1 δ

Z δ 0

1 t^βp

d dt

Z t 0

|ϕ^x(s + γ) − ϕx(s)|sins 2 ^β

p

ds

dt

#1/p

"

1 δ

Z δ 0

1 t^βp

d dt

Z t 0

|ϕ^x(s)|sins 2 ^β

p

ds

dt

#1/p





= π^β ("

1 δ

1 t^βp

Z t 0

|ϕ^x(s + γ) − ϕx(s)|sins 2 ^β

^p ds

^δ

0

βp δ

Z δ 0

1 t^1+βp

Z t 0

|ϕ^x(s + γ) − ϕx(s)|sins 2

βp

dsdt

#^1/p

+

"

1 δ

1 t^βp

Z t 0

|ϕ^x(s)|sins 2

βp

ds

^δ

0

βp δ

Z δ 0

1 t^1+βp

Z t 0

|ϕ^x(s)|sins 2 ^β

p

dsdt

#1/p



. Since f ∈ L^p(wx)^?_β, we have

wxf (δ, γ)_Lp≤ π^β





"

(wx(γ))^p+βp δ

Z δ 0

(wx(γ))^pdt

#1/p

"

(wx(δ))^p+βp δ

Z δ 0

(wx(t))^pdt

#1/p



≤ π^β(1 + βp)^1/p(wx(δ) + wx(γ))

and this ends of our proof.

(7)

Lemma 3.2 If f ∈ L^p(ω) (p ≥ 1), then

kΦ·f (δ, γ)kL^p≤ kw·f (δ, γ)_LpkL^p≤ 2 (ω (δ) + ω (γ)) and

"

1 δ^1+βp

Z δ

0 |ϕ·(t) − ϕ·(t ± γ)|sin t 2

β!p

dt

#^1/p _Lp

≤ 2ω (γ)

for any γ, δ > 0 and β ≥ 0.

Proof The first inequality is proved in [6]. Therefore we prove the second one, only. If we change the order of integration, then we obtain

"

1 δ^1+βp

Z δ

0 |ϕ·(t) − ϕ·(t ± γ)|sin t 2

β!^p dt

#^1/p

L^p

≤

"

1 δ

Z δ

0 |ϕ·(t) − ϕ·(t ± γ)|^pdt

#1/p _L_p

≤ (1

δ Z δ

0

Z π

−π|ϕ^x(t) − ϕx(t ± γ)|^pdx

dt

)1/p

≤ (1

δ Z δ

0

[2ωL^pf (γ)] dt )1/p

= 2ωL^pf (γ)

and our lemma follows.

Lemma 3.3 [N1, Theorem 5.20 II, Ch. XII] Suppose that 1 < q (q − 1)⁻¹ ≤ p ≤ q and ξ = _p¹+¹_q − 1. Ift^−ξg (t)

∈ L^p then

(8)

(|a⁰(g)|^q 2 +X^∞

k=0

(|ak(g)|^q+ |bk(g)|^q) )¹q



 Zπ

−π

t^−ξg (t)^pdt





1 p

.

4. Proofs of the Results. We prove the theorems on pointwise approximation only. The estimations of the norm of our means are an immediate consequences of these which will be proved.

4.1. Proof of Theorem 2.1. Let as usual

H_k^q₀_,k_r(x) = ( 1

r + 1 Xr ν=0

1

π Z π

0

ϕx(t) Dkν(t) dt

q)1/q

≤ A^r+ Br+ Cr,

(8)

where

Ar= ( 1

r + 1 Xr ν=0

1 π

Z 2δν

0

ϕx(t) Dkν(t) dt

q)^1/q

Br= ( 1

r + 1 Xr ν=0

1

π Z 2γr

2δν

ϕx(t) Dkν(t) dt

q)1/q

Cr= ( 1

r + 1 Xr ν=0

1

π Z π

2γr

ϕx(t) Dkν(t) dt

q)1/q

,

with Dkν(t) = ^sin((^k^ν⁺¹2)^t)

2 sin₂^t , δν= _k ^π

ν+1/2 and γr= _r+1/2^π , and

Ar≤ π 2

( 1 r + 1

Xr ν=0

"

kν+ 1/2 π

Z 2δν

0 |ϕ^x(t)| dt

#^q)^1/q .

Using integration by parts we obtain Z 2δν

0 |ϕ^x(t)| dt = Z 2δν

0

|ϕ^x(t)|sin₂^t^β sin₂^t^β dt

Z 2δν

0

1 t^β

d dt

Z t

0 |ϕ^x(s)|sins 2 ^βds

dt

= 1 t^β

Z t

0 |ϕ^x(s)|sins 2

β

ds

^2δ^ν

0

+βZ 2δν

0

1 t^1+β

Z t

0 |ϕ^x(s)|sins 2

β

dsdt

= 2δνwx,βf (2δν)_L1+ βZ 2δν

0

wx,βf (t)_L1dt

≤ 2δ^νwx,βf (2δν)_Lp+ βZ 2δν

0

wx,βf (t)_Lpdt

(9) ≤ 2δ^νwx(2δν) + βZ 2δν

0

wx(t) dt δνwx(2δν) , whence

Ar ( 1

r + 1 Xr ν=0

[wx(2δν)]^q )^1/q

w^x(δ0).

The quantities Brand Cr we will estimate by the Totik method [9].

(9)

At the begin, we divide the term Brinto the two parts and use (9). Thus

Br= ( 1

r + 1 Xr ν=0

1 π

Z 2γr

2δν

ϕx(t) Dkν(t) dt

q)1/q

≤ ( 1

r + 1

νX0−1 ν=0

+ Xr ν=ν₀

! 1 π

Z 2γr

2δν

ϕx(t) Dkν(t) dt

q)^1/q

≤ ( 1

r + 1

νX0−1 ν=0

1 π

Z 2γr

2δν

ϕx(t) Dkν(t) dt

q)^1/q

+ ( 1

r + 1 Xr ν=ν0

1

π Z 2γr

2δν

ϕx(t) Dkν(t) dt

q)^1/q

( 1

r + 1

νX0−1 ν=0

"

kν+ 1/2 π

Z 2δν

2γr

|ϕ^x(t)| dt

#q)^1/q + Br,ν₀

w^x(δ0) + Br,ν0,

where the index ν0 is such that kν₀−1 < r≤ k^ν0. Next, we divide the term Br,ν₀

into the three following parts

Br,ν0. = ( 1

r + 1 Xr ν=ν0

1

π Z 2γr

2δν

ϕx(t) Dkν(t) dt

q)^1/q

= 1

2 ( 1

r + 1 Xr ν=ν0

1 π

Z 2γr

2δν

+Z 2γr−δν

δν

+Z 2γr

2γr−δν

− Z 2δν

δν

!

ϕx(t) Dkν(t) dt

q)^1/q

≤ Br,ν¹ 0+ B_r,ν² ₀+ B_r,ν³ ₀, where the first term

B¹_r,ν₀ = 1 2

( 1 r + 1

Xr ν=ν0

1 π

Z 2γr

2δν

+Z 2γr−δν

δν

!

ϕx(t) Dkν(t) dt

q)^1/q

= 1 2

( 1 r + 1

Xr ν=ν0

1 π

Z 2γr

2δν

[ϕx(t) Dkν(t) + ϕx(t − δν) Dkν(t − δν)] dt

q)1/q

≤ 1 2

( 1 r + 1

Xr ν=ν0

1 π

Z 2γr

2δν

(ϕx(t) − ϕx(t − δν)) Dkν(t) dt

q)1/q

(10)

+1 2

( 1 r + 1

Xr ν=ν0

1

π Z 2γr

2δν

ϕx(t − δν) (Dkν(t) + Dkν(t − δν)) dt

q)^1/q

≤ 1 2

( 1 r + 1

Xr ν=ν₀

Z 2γr

2δν

1

t |ϕ^x(t) − ϕ^x(t − δ^ν)| dt

q)1/q

+1 2

( 1 r + 1

Xr ν=ν0

1 π

Z 2γr

2δν

ϕx(t − δν) 1

2 sin₂^t − 1 2 sin^t^−δ₂^ν

!

sin(2kν+ 1) t

2 dt

q)^1/q

( 1

r + 1 Xr ν=ν0

Z 2γr

2δν

1

t |ϕ^x(t) − ϕx(t − δν)| dt

q)1/q

+ ( 1

r + 1 Xr ν=ν0

δ^ν

Z 2γr

2δν

|ϕ^x(t − δν)|

t (t− δ^ν) dt

q)^1/q

≤ ( 1

r + 1 Xr ν=ν₀

Z 2γr

2δν

1

t |ϕ^x(t) − ϕ^x(t − δ^ν)| dt

q)1/q

+ ( 1

r + 1 Xr ν=ν0

δ^ν

Z 2γr

δν

|ϕ^x(t)|

t² dt

q)1/q

. Integrating by parts and applying our assumption we get

B¹_r,ν₀ ( 1

r + 1 Xr ν=ν0

Z 2γr

2δν

1 t^1+β

d dt

Z t

0 |ϕ^x(s) − ϕx(s − δν)|sins 2 ^βds

dt

q)1/q

+ ( 1

r + 1 Xr ν=ν0

δ^ν

Z 2γr

δν

1 t^2+β

d dt

Z t

dt

q)1/q

≤ ( 1

r + 1 Xr ν=ν0

"

1 (2γr)^1+β

Z 2γr

0 |ϕ^x(s) − ϕx(s − δν)|sins 2

β

ds

#q)^1/q

+ ( 1

r + 1 Xr ν=ν0

"

1 (2δν)^1+β

Z 2δν

β

ds

#q)^1/q

+ ( 1

r + 1 Xr ν=ν0

Z 2γr

2δν

1 t^2+β

Z t

β

dsdt

^q)^1/q

+ ( 1

r + 1 Xr ν=ν0

"

δν

1 (2γr)^2+β

Z 2γr

#q)^1/q

(11)

+ ( 1

r + 1 Xr ν=ν0

"

1 (δν)^1+β

Z δν

0 |ϕ^x(s)|sins 2

β

ds

#q)^1/q

+ ( 1

r + 1 Xr ν=ν0

δν

Z 2γr

δν

1 t^3+β

Z t

0 |ϕ^x(s)|sins 2

β

dsdt

^q)1/q

≤ w^x(δ0) + ( 1

r + 1 Xr ν=ν0

wx(δν)Z 2γr

2δν

1 tdt

^q)1/q

( 1 r + 1

Xr ν=ν0

δν

wx(2γr) 2γr

^q)1/q

+ wx(δ0) + ( 1

r + 1 Xr ν=ν0

δν

Z 2γr

δν

wx(t) t² dt

^q)1/q

w^x(δ0)

1 + logkr+ 1/2 r + 1/2

+

( 1 r + 1

Xr ν=ν₀

[wx(δν)]^q )1/q

+ ( 1

r + 1 Xr ν=ν0

wx(δν)Z 2γr

δν

1 tdt

^q)1/q

w^x(δ0)

1 + logkr+ 1/2 r + 1/2

. Consequently, by Lemma 3.1,

B_r,ν² ₀ =1 2

( 1 r + 1

Xr ν=ν0

1 π

Z 2γr

2γr−δν

ϕx(t) Dkν(t) dt

q)1/q

≤1 4

( 1 r + 1

Xr ν=ν0

Z 2γr

2γr−δν

1

t |ϕ^x(t)| dt

q)^1/q

≤1 4

( 1 r + 1

Xr ν=ν0

1

2γr− δ^ν Z 2γr

2γr−δν

|ϕ^x(t)| dt^q)^1/q

≤ 1 4

( 1 r + 1

Xr ν=ν0

"

δν

2γr− δ^ν 1 δν

Z 2γr−δν+δν

2γr−δν

|ϕ^x(t)| dt

#q)^1/q

≤1 4

( 1 r + 1

Xr ν=ν0

δν

2γr− δ^ν[wx(2γr− δ^ν) + wx(δν)]^q)1/q

≤ 1 4

( 1 r + 1

Xr ν=ν₀

δνwx(2γr− δ^ν)

2γr− δ^ν + wx(δν)

q)1/q

( 1

r + 1 Xr ν=ν0

δνwx(δν) δν

+ wx(δν)q)1/q

w^x(δ0)

(12)

and

B³_r,ν₀ = 1 2

( 1 r + 1

Xr ν=ν0

1 π

Z 2δν

δν

ϕx(t) Dkν(t) dt

q)^1/q

≤1 4

( 1 r + 1

Xr ν=ν0

Z 2δν

δν

|ϕ^x(t)|1 tdt

q)^1/q

( 1

r + 1 Xr ν=ν0

[wx(δν)]^q )1/q

w^x(δ0) . Thus

Br w^x(δ0)

1 + logkr+ 1/2 r + 1/2

. For the estimate of Cr we use our notation in the form Φxf (δ0, t) := _δ¹₀ Rt+δ0

t ϕx(u) du with δ0= _k₀_+1/2^π . Then

Cr= ( 1

r + 1 Xr ν=0

1 π

Z π 2γr

ϕx(t)

2 sint

2

−1

sin

kν+1 2

t

dt

q)^1/q

≤ ( 1

r + 1 Xr ν=0

1 π

Z π 2γr

Φxf (δ0, t)− ϕ^x(t) 2 sin^t₂

sin

kν+1 2

t

dt

q)1/q

+ ( 1

r + 1 Xr ν=0

1

π Z π

2γr

Φxf (δ0, t) 2 sin₂^t sin

kν+1 2

t

dt

q)1/q

= C_r¹+ C_r² and, by inequality (8),

C_r¹ 1 (1 + r)^1/q

(1 π

Z π 2γr

[Φxf (δ0, t)− ϕ^x(t)] 2 sin t

2

−1

t¹^−1/p−1/q

p

dt )^1/p

1

(1 + r)^1/q

Z π 2γr

|Φ^xf (δ0, t)− ϕ^x(t)|^p

t^1+p/q dt

1/p

= 1

(1 + r)^1/q (Z π

2γr

1 δ0t^1/p+1/q

Z t+δ0

t

[ϕx(u) − ϕx(t)] du

p

dt )^1/p

≤ 1

(1 + r)^1/q (Z π

2γr

"

1 δ0t^1/p+1/q

Z δ0

0 |ϕ^x(u + t) − ϕx(t)| du

#p

dt )^1/p

. Applying the generalized Minkowski inequality and the partial integration we obtain

C_r¹ 1 δ0(1 + r)^1/q

Z δ0

0

Z π 2γr

|ϕ^x(u + t) − ϕx(t)|^p t^1+p/q dt

^1/p

du 1

δ0(1 + r)^1/q

(13)

· Z δ0

0

Z π 2γr

1 t^1+p/q+βp

d dt

Z t 0

|ϕ^x(u + v) − ϕx(v)|sinv 2

βp

dv

dt

^1/p du

= 1

(1 + r)^1/q 1 δ0

Z δ0

0

( 1

t^1+p/q+βp Z t

0

βp

dv

^π

t=2γr

+ (1 + p/q + βp)Z π 2γr

1

t^2+p/q+βp Z t

0

|ϕ^x(u + v) − ϕx(v)|sinv 2 ^β

^p dv

dt

^1/p du.

Hence, since f ∈ L^p(wx)^?_β,we have

C_r¹ 1 (1 + r)^1/q

1 δ0

Z δ0

0

1

π^1+p/q+βp Z π

0

βp

dv

+

"

1 (2γr)^1+p/q+βp

Z 2γr

0

|ϕ^x(u + v) − ϕx(v)|sinv 2 ^β

^p dv

#

+Z π 2γr

1

t^1+p/q(wx(u))^pdt

1/p

du

≤ 1

(1 + r)^1/q 1 δ0

Z δ0

0

(

[wx(u)]^p 1

π^p/q + 1

(2γr)^p/q +Z π 2γr

1 t^1+p/qdt

!)1/p

du

1

(1 + r)^1/qwx(δ0) (

1 + 1

(γr)^p/q +Z π 2γr

t^−(1+p/q)dt )1/p

1

(1 + r)^1/qwx(δ0)

( 1

(γr)^p/q +t^−p/q

−p/q

^π

t=2γr

)1/p

≤ 1

(1 + r)^1/qwx(δ0)

( 1

(γr)^p/q

1 +q

p

)^1/p

w^x(δ0) .

For the estimate of C_r² we will consider the two cases r ≤ k0 and r ≥ k0 . In the first case, using the partial integration and inequality (8), we obtain

C_r²= 1 2 (r + 1)^1/q

( _r X

ν=0

1 π

Z π 2γr

Φxf (δ0, t) sin₂^t

d dt

cos kν+¹₂ t kν+¹₂

! dt

q)^1/q

= 1

2π (r + 1)^1/q



 Xr ν=0

"

Φxf (δ0, t) sin^t₂

#π

2γr

− Z π

2γr

d dt

cos kν+¹₂ t kν+¹₂ dt

q)^1/q

(14)

1 (r + 1)^1/q

( _r X

ν=0

"

Φxf (δ0, 2γr) sin γr

cos kν+¹₂2γr kν+¹₂

+

Z π 2γr

d dt

#^q)^1/q

and analogously

C_r² 1 (r + 1)^1/q

((r + 1)^1/q|Φ^xf (δ0, 2γr)|

γr(k0+ 1)

+ 1

k0+ 1

Z π 2γr

d

dt

t^{1−1/p−1/q}

p

dt

^1/p)

1

γr(k0+ 1)wx(δ0) + 1 k0+ 1

wx(2γr) 2γr

+ 1

(k0+ 1) (r + 1)^1/q

Z π 2γr

d

dt

t¹^−1/p−1/q

p

dt

^1/p

w^x(δ0) + 1 k0+ 1

wx(2δ0) 2δ0

+ 1

(k0+ 1) (r + 1)^1/q

"Z π 2γr

d

dtΦxf (δ0, t)

sin₂^t −Φxf (δ0, t) cos₂^t 2 sin₂^t²

!

t¹^−1/p−1/q

p

dt

#^1/p

w^x(δ0)

+ 1

(k0+ 1) (r + 1)^1/q

"Z π 2γr

1

δ₀|ϕ^x(δ0+ t) − ϕx(t)|

t^1/p+1/q +wx(δ0) + wx(t) t^1+1/p+1/q

p

dt

#^1/p

w^x(δ0) + 1

(k0+ 1) (r + 1)^1/q

· 1 δ₀^p

Z π 2γr

t^{−1−p/q−βp}d dt

Z t 0

|ϕ^x(δ0+ u) − ϕ^x(u)|sinu 2 ^β

p

du

dt

^1/p

+ 1

(k0+ 1) (r + 1)^1/q (Z π

2γr

wx(δ0) t^1+1/p+1/q

p

dt

^1/p +Z π

2γr

wx(t) t^1+1/p+1/q

p

dt

^1/p)

w^x(δ0)

+ 1

(k0+ 1) (r + 1)^1/qδ0

1

π^1+p/q+βp Z π

0

|ϕ^x(δ0+ u) − ϕx(u)|sinu 2

βp

du

− 1

(2γr)^1+p/q+βp Z 2γr

0

βp

du

(15)

+ (1 + p/q + βp)Z π 2γr

t^{−2−p/q−βp} Z t

0

βp

dudt

^1/p

+ wx(δ0) (k0+ 1) (r + 1)^1/q

Z π 2γr

t^{−p−1−p/q}dt

1/p

+ 1

(k0+ 1) (r + 1)^1/q

Z π 2γr

wx(t) /t t^1/p+1/q

p

dt

^1/p

w^x(δ0)+ wx(δ0) (k0+ 1) (r + 1)^1/qδ0

"

1 π^p/q + 1

γr^p/q

+ (1 + p/q + βp)Z π 2γr

t^−1−p/qdt

#1/p

+(γr)^−1−1/qwx(δ0)

(k0+ 1) (r + 1)^1/q + (γr)^−1/qwx(γr) (k0+ 1) (r + 1)^1/qγr

w^x(δ0) + wx(δ0) (k0+ 1) (r + 1)^1/qδ0

"

1 γr^p/q

+ (1 + p/q + βp)(2γr)^−p/q p/q

#1/p

+ wx(δ0)

(k0+ 1) (r + 1)^1/q(γr)^1+1/q + wx(γr) /γr

(k0+ 1) (r + 1)^1/qγr^1/q

w^x(δ0) + wx(δ0)

(k0+ 1) δ0(r + 1)^1/qγr^1/q

+ wx(δ0)

(k0+ 1) (r + 1)^1/qγr^1+1/q

w^x(δ0) . In the case r ≥ k0we will divide the interval of integrability into two parts. Thus, by inequality (8), Lemma 3.1 and the partial integration,

C_r²= 1 2 (r + 1)^1/q

( _r X

ν=0

1

π Z π

2γr

Φxf (δ0, t) sin₂^t sin

kν+1 2

t

dt

q)1/q

1

(r + 1)^1/q (Z 2δ0

2γr

Φxf (δ0, t)

sin^t₂ t¹^−1/p−1/q

p

dt )1/p

+ 1

2 (r + 1)^1/q ( _r

X

ν=0

1 π

Z π 2δ₀

d dt

! dt

q)^1/q

≤ 1

(r + 1)^1/q (Z 2δ0

2γr

Φxf (δ0, t)

sin₂^t t¹^−1/p−1/q

p

dt )1/p

+ 1

2π (r + 1)^1/q ( _r

X

ν=0

"

#^π

2δ0

− Z π

2δ0

d dt

q)^1/q

(16)

1 (r + 1)^1/q

(Z 2δ0

2γr

wx(δ0) + wx(t) t^1/p+1/q

p

dt )^1/p

+Φxf (δ0, 2δ0) sin δ0

1 k0+¹₂

+ 1

(k0+ 1) (r + 1)^1/q ( _r

X

ν=0

Z π 2δ0

d dt

cos

kν+1 2

t

dt

q)1/q

. Using once more inequality (8) and Lemma 3.1

C_r² wx(δ0) (r + 1)^1/q

(Z 2δ0

2γr

t^−1−p/qdt )^1/p

+wx(δ0) + wx(2δ0) (k0+ 1) δ0

+ 1

(k0+ 1) (r + 1)^1/q

Z π 2δ0

d

dt

t¹^−1/p−1/q

p

dt

^1/p

w^x(δ0)

+ 1

(k0+ 1)^1+1/q

"Z π 2δ₀

d

dtΦxf (δ0, t)

sin^t₂ −Φxf (δ0, t) cos₂^t sin₂^t2

!

t^{1−1/p−1/q}

p

dt

#^1/p . Similarly to the first case

C_r² w^x(δ0) + 1 (k0+ 1)^1+1/q

· 1 δ^p₀

Z π 2δ0

t^{−1−p/q−βp}d dt

Z t 0

βp

du

dt

^1/p

+ 1

(k0+ 1)^1+1/q (Z π

2δ0

wx(δ0) t^1+1/p+1/q

p

dt

^1/p +Z π

2δ0

wx(t) t^1+1/p+1/q

p

dt

^1/p)

w^x(δ0) + 1 (k0+ 1)^1+1/qδ0

1

π^1+p/q+βp Z π

0

|ϕ^x(δ0+ u) − ϕx(u)|sinu 2 ^p

du

+ 1

(2δ0)^1+p/q+βp Z 2δ0

0

βp

du

+ (1 + p/q + βp)Z π 2δ0

t^{−2−p/q−βp}

Z t 0

|ϕ^x(δ0+ u) − ϕ^x(u)|sinu 2 ^β

p

du

dt

^1/p

+ wx(δ0) (k0+ 1)^1+1/q

Z π 2δ0

t^{−p−1−p/q}dt

1/p

+ 1

(k0+ 1)^1+1/q

Z π 2δ0

wx(t) /t t^1/p+1/q

p

dt

^1/p

w^x(δ0) + wx(δ0) (k0+ 1)^1+1/qδ0

"

1 π^p/q + 1

δ^p/q₀ + (1 + p/q + βp)Z π 2δ0

t^−1−p/qdt

#1/p

+(δ0)^−1−1/qwx(δ0)

(k0+ 1)^1+1/q +(δ0)^−1/qwx(δ0) (k0+ 1)^1+1/qδ0