Jackson and Stechkin type inequalities of trigonometric approximation in A(w,v)(p,q)((.))

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doi:10.3906/mat-1712-84 h t t p : / / j o u r n a l s . t u b i t a k . g o v . t r / m a t h /

Research Article

Jackson and Stechkin type inequalities of trigonometric approximation in A

p,q(·)_w,ϑ Ahmet Hamdi AVŞAR∗ _{and Hüseyin KOÇ}_,

Department of Mathematics, Balıkesir University, Balıkesir, Turkey

Received: 29.12.2017 • Accepted/Published Online: 19.09.2018 • Final Version: 27.11.2018

Abstract: In this paper, we study Jackson and Stechkin type theorems of trigonometric polynomial approximation in

the space Ap,q(_w,ϑ·) by considering a modulus of smoothness defined by virtue of the Steklov operator.

Key words: Modulus of smoothness, trigonometric polynomials, Muckenhoupt weight, direct and inverse theorems

1. Introduction and main results

Let T := [−π, π] . The space Ap,q(·)

w,ϑ (T) is the intersection of the well-known weighted Lebesgue space L p w(T)

and the variable exponent weighted Lebesgue space Lq(_ϑ·)(T) , i.e. Ap,q(_w,ϑ·)(T) = Lp_w(T) ∩ Lq(_ϑ·)(T). In the literature, the variable exponent Lebesgue spaces were first investigated by Orlicz [23]. In the space Lp

w(T) the

direct and inverse theorems of trigonometric approximation were proved in the papers [1, 2, 5, 6,19, 20, 29]. In the space Lq(_ϑ·)(T), similar theorems were obtained in the papers [3, 7, 10,18, 23, 25]. Variable exponent weighted Lebesgue spaces have important applications in the theory of elasticity, fluid dynamics, and differential equations [15,24]. Also, in weighted Lorentz spaces Lp,q

w (T) , which is a generalization of Lpw(T) , direct and

inverse theorems of approximation theory were proved in [8,9,21,28]. In this paper, we prove the direct and inverse theorems of trigonometric approximation in Ap,q(_w,ϑ·)(T) . First, we give some definitions and notations.

Let ℘(T) be the class of all measurable functions p(·) : T→ [1, ∞] called the variable exponent on T. In this paper, the notation p(·) is used always for a variable exponent. Let p(·) ∈ ℘(T) and we set

p−= ess

x∈Tinf p(x), p

+_{= ess}

x∈Tsup p(x).

A measurable function w :T → [0, ∞] is called a weight function if the preimage set w−1₍_{{0, ∞}) has}

the Lebesgue measure zero.

Let 1≤ p < ∞ and f be a measurable function on T. The weighted Lebesgue space Lp

w is defined as

the set of all measurable functions f such that

∥f∥p,w := (∫ T|f(x)| p_w(x)dx )1 p <∞. It is known that Lp

w is a Banach space with respect to the norm ∥f∥p,w.

∗_{Correspondence: ahmet.avsar@balikesir.edu.tr}

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

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Weighted variable exponent Lebesgue space Lp(_ϑ·) is defined as the set of all measurable functions f on T such that ∥f∥p(·),ϑ= fϑp(·)1 p(·) <∞ where ∥f∥p(·):= inf { α > 0 : ∫ T f (x)α p(x)dx≤ 1 } .

It is a Banach space with respect to the norm ∥f∥p(_·),ϑ.

Let 1 < p <∞, q(·) ∈ ℘(T). We set Ap,q(_w,ϑ·):=

{

f : f∈ Lp_w∩ Lq(_ϑ·)

} and equip this Banach space with the norm

∥f∥p,q(·)

w,ϑ =∥f∥p,w+∥f∥q(·),ϑ

for any f ∈ Ap,q(·) w,ϑ .

We denote by Cc(T) the space of all continuous, complex valued functions on T. C∞(T) consists of

functions that are continuously differentiable arbitrarily many times. The set C₀∞(T) is the subset of C∞(T) of functions that have compact support. C(T) is the set of all continuous functions on T.

Some properties of Ap,q(_w,ϑ·) were given in [26]:

I) (Ap,q(_w,ϑ·),∥·∥p,q(_w,ϑ·)

)

is a Banach space with q+_<_∞,

II) The space Ap,q(_w,ϑ·) is a Banach function space on T. The set C(T) is dense subset of Ap,q(·)

w,ϑ (T).

Indeed: The set C0∞(T) is dense subset of A p,q(·)

w,ϑ (T) [26, Corollary 2] , so for every f∈ A p,q(·)

w,ϑ (T) there

exists g∈ C∞

0 (T) such that ∥f − g∥Ap,q(·)_w,ϑ < ε. On the other hand, as the embedding C0∞(T) ,→ C(T) is valid

[13, p. 15], we have g∈ C(T). Thus, the set C(T) is a dense subset of Ap,q(_·) w,ϑ (T).

The weight functions used in this paper satisfy Muckenhoupt condition Ap [22] defined as

sup ( 1 |I| ∫ I w(x)dx ) ( 1 |I| ∫ I w1−p′(x)dx )p−1 = CAp<∞, p ′ = p p− 1, 1 < p <∞,

where the supremum is taken with respect to all intervals I , which are any subintervals of T. The constant

CAp is called the Muckenhoupt constant of w . For weights w, the class Aq(_·) is defined as

∥w∥q(_·):= sup I_∈ς|I| −qI∥w∥ L1(I) 1 w L q′ (·) q(·)(I) <∞,

where ς denotes the set of all intervals of T, qI =

( 1 |I| ∫ I 1 q(x)dx ) and 1 q(x)+ 1 q′(x) = 1 [11,14].

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We define the modulus of smoothness of a function f ∈ Ap,q(_w,ϑ·) as Ωr(f, δ)_Ap,q(·) w,ϑ := sup 0≤h≤δ ∥(I − σh)rf∥_Ap,q(·) w,ϑ , r∈ N (1.1) where σhf (x) := 1 2h x+h∫ x_−h f (u)du, x∈ T. If the condition |q(x) − q(y)| ≤ _{− ln |x − y|}C , x, y∈ Ω, |x − y| ≤1 2

holds, then we say that the exponent q(x) satisfies the Log–Hölder continuous function where Ω ⊂ T. We

denote the family of all Log–Hölder continuous functions by the symbol Plog₍_T).

Denote by M f the Hardy–Littlewood maximal operator, defined for f ∈ L1₍_{T) by}

M f (x) := sup I 1 |I| ∫ I |f(y)| dy, x ∈ T

where the supremum is taken with respect to all the intervals I , which are any subintervals of T. Let 1 < p < ∞, q ∈ Plog _{and 1 < q}− _{≤ q}+ _< _{∞. The maximal operator M : A}p,q(·)

w,ϑ → A p,q(·) w,ϑ is

bounded iff w∈ Ap and ϑ∈ Aq(_·) [26, Theorem 10]. Therefore, the shift operator σh(f ) belongs to A p,q(·) w,ϑ and

thus Ωr(f, δ)_Ap,q(·) w,ϑ

makes sense under these conditions. The best approximation En(f )_Ap,q(·)

w,ϑ of f ∈ Ap,q(·) w,ϑ is given by En(f )A p,q(·) w,ϑ =_Tinf n∈Tn ∥f − Tn∥_Ap,q(·) w,ϑ ,

where Tn is the set of trigonometric polynomials Tn of degree not more than n.

For an f ∈ Ap,q(·)

w,ϑ the K -functional is defined as

K ( f, t; Ap,q(_w,ϑ·), W_p,q(r _·),w,ϑ ) := inf g∈Wr p,q(·),w,ϑ { ∥f − g∥Ap,q(·)_w,ϑ + t r _dxdrrg Ap,q(·)_w,ϑ } , t > 0 where W_p,q(r _·),w,ϑ := { g∈ Ap,q(_w,ϑ·): d r dxrg∈ A p,q(_·) w,ϑ } .

The notation ≲ indicates that ‘‘A ≲ B iff there exists a positive constant C, independent of essential parameters, such that A≤ CB”.

If A≲ B and B ≲ A, simultaneously, we will write A ≈ B.

The aim of this paper is to prove the Jackson and Stechkin type theorems of polynomial approximation in Ap,q(_w,ϑ·) by considering modulus of smoothness (1.1).

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Theorem 1.1 Let f ∈ Ap,q(_w,ϑ·), w∈ Ap, ϑ∈ Aq(·), 1 < p <∞, q ∈ Plog, 1 < q− ≤ q+ <∞, and n, r ∈ N. Then En(f )_Ap,q(·) w,ϑ ≲ Ωr ( f,1 n ) Ap,q(·)_w,ϑ (1.2)

holds for some constant depending only on r, p, q(·) and CAp.

Theorem 1.2 Let f ∈ Ap,q(·) w,ϑ , 1 < p <∞, q ∈ P log_{, 1 < q}₋ _{≤ q}+ _<_{∞, w ∈ A} p, ϑ∈ Aq(_·), and n, r∈ N. Then Ωr ( f,1 n ) Ap,q(·)_w,ϑ ≲ 1 n2r n ∑ k=0 (k + 1)2r−1Ek(f )_Ap,q(·) w,ϑ

holds with some constant depending only on r, p, q(·) and CAp and CAq(·) are constants corresponding to Lp and Lq(·), respectively.

From Theorem 1.1 and Theorem 1.2, we obtain the following Marchaud type inequality.

Corollary 1.3 Let f ∈ Ap,q(·) w,ϑ , 1 < p <∞, q ∈ P log_{, 1 < q}₋_{≤ q}+_<_{∞, w ∈ A} p, ϑ∈ Aq(_·). Then we have Ωr(f, δ)_Ap,q(·) w,ϑ ≲ δ 2r 1 ∫ δ u−2r−1Ωk(f, u)_Ap,q(·) w,ϑ du, 0 < δ < 1, for r, k∈ N with r < k. Corollary 1.4 Let f ∈ Ap,q(·) w,ϑ , w∈ Ap, ϑ∈ Aq(_·), 1 < p <∞, q ∈ Plog, 1 < q− ≤ q+<∞. If En(f )_Ap,q(·) w,ϑ ≲ n −α_, _n_{∈ N}

for some α > 0, then, for a given r∈ N, we have the estimations

Ωr(f, δ)_Ap,q(·) w,ϑ ≲    δα , r > α/2; δ2rlog1 δ , r = α/2; δ2r , r < α/2.

If we define the generalized Lipschitz class Lip(α, Ap,q(_w,ϑ·)

) for α > 0 as Lip ( α, Ap,q(_w,ϑ·) ) := { f ∈ Ap,q(_w,ϑ·): Ωk(f, δ)_Ap,q(·) w,ϑ ≲ δ α_{, δ > 0}}_,

then by virtue of Theorem 1.1 and Corollary 1.3 we obtain the following result, which gives a constructive characterization of the Lipschitz classes Lip(α, Ap,q(_w,ϑ·)

)

in the case of k := [α/2]+1, [x] := max{n ∈ Z : n ≤ x} , α > 0.

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Corollary 1.5 Let f ∈ Ap,q(_w,ϑ·), w∈ Ap, ϑ∈ Aq(·), 1≤ α < ∞, 1 < p < ∞, q ∈ Plog, and 1 < q−≤ q+<∞. The following conditions are equivalent:

(i) f ∈ Lip ( α, Ap,q(_w,ϑ·) ) ; (ii) En(f )_Ap,q(·) w,ϑ ≲ n −α_{, n}_{∈ N.} 2. Auxiliary results

To prove the above theorems we need the following auxiliary results.

Lemma 2.1 Let f ∈ Ap,q(·) w,ϑ , 1 < p <∞, q ∈ P log_{, 1 < q}− _{≤ q}+_<_{∞, w ∈ A} p, ϑ∈ Aq(_·), and t, k > 0. Then Ω1(f, t)_Ap,q(·) w,ϑ ≈ K ( f, t; Ap,q(_w,ϑ·), W_p,q(2 _·),w,ϑ ) and Ω1(f, kt)_Ap,q(·) w,ϑ ≲ (1 + [k]) 2 Ω1(f, t)_Ap,q(·) w,ϑ

hold for some constants depending only on p, q(·), and CAp and CAq(·) are constants corresponding to Lp and Lq(·), respectively.

Proof Let t > 0. Then there exists an n∈ N such that (1/n) < t ≤ (2/n) .

For this n∈ N, we shall consider an operator Ln on A p,q(_·) w,ϑ as follows: (Lnf ) (x) := 3n3 1/n∫ 0 t ∫ 0 u ∫ −u f (x + s) dsdudt, x∈ T, f ∈ Ap,q(_w,ϑ·).

Then from [16] and [3, p. 14], with some constant c∈ R, d2 dx2(Lnf ) (x) = cn 2(( I− σ1/n )) f (x) (2.1)

is valid for a.e. x∈ T.

The operator Ln is bounded in A p,q(_·)

w,ϑ . In fact, using the uniform boundedness of the operator f → σ1/nf

in Ap,q(_w,ϑ·) we obtain that

d2

dx2Lnf (x)∈ A p,q(·) w,ϑ

and so Lnf ∈ W_p,q(2 _·),w,ϑ. From the generalized Minkowski inequality and the boundedness of σ1/n we obtain

∥Lnf∥_Ap,q(·) w,ϑ = 3n3 1/n∫ 0 t ∫ 0 u ∫ −u f (x + s) dsdudt Ap,q(·)_w,ϑ ≲ 3n3 1/n∫ 0 t ∫ 0 2u∥σuf∥_Ap,q(·) w,ϑ dudt ≲ 3n3_∥f∥ Ap,q(·)_w,ϑ 1/n_∫ 0 t ∫ 0 2ududt =∥f∥_Ap,q(·) w,ϑ . (2.2)

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By (2.2) we have f− Lnf ∈ A p,q(·) w,ϑ for f∈ A p,q(·) w,ϑ and K ( f, t; Ap,q(_w,ϑ·), W_p,q(2 _·),w,ϑ ) ≤ 4K(f, 1/n; Ap,q(_w,ϑ·), W_p,q(2 _·),w,ϑ ) (2.1) ≲ ∥f − Lnf∥_Ap,q(·) w,ϑ + n−2 d 2 dx2Lnf (x) Ap,q(·)_w,ϑ = : I1+ I2.

Using the generalized Minkowski inequality, we have

I1 = ∥f − Lnf∥_Ap,q(·) w,ϑ ≲ n 3 1/n∫ 0 t ∫ 0 2u∥(I − σu) f∥_Ap,q(·) w,ϑ dudt ≲ sup 0≤u≤1/n ∥(I − σu) f∥_Ap,q(·) w,ϑ 3n3 1/n∫ 0 t ∫ 0 2ududt ≲ sup 0≤u≤1/n ∥(I − σu) f∥_Ap,q(·) w,ϑ = Ω1 ( f,1 n ) Ap,q(·)_w,ϑ . (2.2) Using (2.1), we have I2= n−2 dxd22Lnf (x) Ap,q(·)_w,ϑ = n−2 d 2 dx2Lnf (x) Ap,q(·)_w,ϑ = C(I− σ1/n ) f Ap,q(·)_w,ϑ ≲ sup 0≤u≤1/n ∥(I − σu) f∥_Ap,q(·) w,ϑ = Ω1 ( f,1 n ) Ap,q(·)_w,ϑ . (2.5) From (2.1)–(2.5), K ( f, t; Ap,q(_w,ϑ·), W_p,q(2 _·),w,ϑ ) ≲ Ω1 ( f,1 n ) Ap,q(·)_w,ϑ ≲ Ω1(f, t)_Ap,q(·) w,ϑ .

Now we obtain the converse estimate of the last inequality. For g∈ W2 p,q(·),w,ϑ, (I− σh) g(x) = 1 2h h ∫ −h (g(x)− g (x + t)) dt = − 1 8h h ∫ 0 t ∫ 0 u ∫ −u ( d2 dx2g ) (x + s) dsdudt.

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Then ∥(I − σh) g∥_Ap,q(·) w,ϑ ≤ 1 8h h ∫ 0 t ∫ 0 2u 2u1 u ∫ −u ( d2 dx2g ) (x + s) ds Ap,q(·)_w,ϑ dudt ≲ 1 8h h ∫ 0 t ∫ 0 2u d 2 dx2g (x) Ap,q(_w,ϑ·) dudt = h2 d 2 dx2g (x) Ap,q(_w,ϑ·) (2.3) and we get Ω1(g, t)_Ap,q(·) w,ϑ ≲ t 2 _dxd22g (x) Ap,q(·)_w,ϑ for g∈ W2 p,q(·),w,ϑ. Then for g∈ W2 p,q(·),w,ϑ, Ω1(f, t)_Ap,q(·) w,ϑ ≲ ∥f − g∥A p,q(·) w,ϑ + t2 d 2 dx2g (x) Ap,q(·)_w,ϑ .

If we take the infimum on g∈ W2

p,q(·),w,ϑ in the last inequality,

Ω1(f, t)_Ap,q(·) w,ϑ ≲ K ( f, t; Ap,q(_w,ϑ·), W_p,q(2 _·),w,ϑ ) . Consequently, Ω1(f, t)_Ap,q(·) w,ϑ ≈ K ( f, t; Ap,q(_w,ϑ·), W_p,q(2 _·),w,ϑ ) .

Using the last equivalence we have

Ω1(f, kt)_Ap,q(·) w,ϑ ≲ inf g∈W2 p,q(·),w,ϑ { ∥f − g∥_Ap,q(·) w,ϑ + (kt)2 d 2 dx2g (x) Ap,q(·)_w,ϑ } ≲ (1 + [k])2 inf g_∈W2 p,q(·),w,ϑ { ∥f − g∥Ap,q(·)_w,ϑ + t 2 _dxd22g (x) Ap,q(·)_w,ϑ } ≲ (1 + [k])2 Ω1(f, t)_Ap,q(·) w,ϑ

and the lemma is proved. 2

Lemma 2.2 (a) Let F ∈ C(T), w ∈ Ap, ϑ∈ Aq(_·), 1 < p <∞, q ∈ Plog, 1 < q− ≤ q+ <∞, and n, m, r∈ N. Then there exists a number δ ∈ (0, 1) , depending only on p, q (·) and CAp, such that

Ωr(F, t)_Ap,q(·) w,ϑ ≤ C1

δmr∥F ∥_C(_T)+ C2Ωr+1(F, t)_Ap,q(·) w,ϑ

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holds for any t∈ (0, 1/n) , where the constant C1> 0 depends only on r , p, q (·) and CAp, and the constant

C2> 0 depends only on r, m, p, q(·) and CAp.

(b) Let f ∈ Ap,q(_w,ϑ·). Then there exists F ∈ C(T) so that

Ωr(f, t)_Ap,q(·) w,ϑ ≤ C1

δmr∥F ∥_C(_T)+ C2Ωr+1(f, t)_Ap,q(·) w,ϑ

where the constant C1 > 0 depends only on r , p, q (·) and CAp, and the constant C2 > 0 depends only on

r, m, p, q(·) and CAp.

Proof (a) We will follow as in Lemma 3.2 of [4]. There exists a constant C > 1, such that boundedness of

σhF in A p,q(·) w,ϑ implies ∥σhF∥_Ap,q(·) w,ϑ ≤ C ∥F ∥A p,q(·) w,ϑ for any h > 0. We define δ := C/ (C + 1) .

We can write the equalities

I− σh= 1 2(I− σh) (I + σh) + 1 2(I− σh) 2 and σh(I− σh) = 1 2(I− σh) (I + σh)− 1 2(I− σh) 2 , h > 0. Hence, ∥(I − σh) g∥_Ap,q(·) w,ϑ +∥σh(I− σh) g∥_Ap,q(·) w,ϑ ≤ ∥(I − σh) (I + σh) g∥_Ap,q(·) w,ϑ + (I − σh)2g Ap,q(·)_w,ϑ , (2.4)

for g∈ C(T) and h > 0. On the other hand, ∥(I − σh) r F∥ Ap,q(·)_w,ϑ = δ ( (1/C)∥(I − σh) r F∥_Ap,q(·) w,ϑ +∥(I − σh) r F∥_Ap,q(·) w,ϑ ) ≤ δ(∥(I − σh) r F∥_Ap,q(·) w,ϑ +∥(I − σh) r F∥_Ap,q(·) w,ϑ ) = δ ( (I − σh) (I− σh) r₋₁ F Ap,q(·)_w,ϑ +∥(I − σh) r F∥_Ap,q(·) w,ϑ ) = δ ( (σh(I− σh) + (I− σh) 2) (I− σh) r−1 F Ap,q(·)_w,ϑ +∥(I − σh) r F∥_Ap,q(·) w,ϑ ) ≤ δ( σh(I− σh) (I− σh) r−1 F Ap,q(·)_w,ϑ + (I − σh) 2 (I− σh) r−1 F Ap,q(·)_w,ϑ ) +δ∥(I − σh) r F∥_Ap,q(·) w,ϑ ≤ δ ( ∥σh(I− σh)rF∥_Ap,q(·) w,ϑ + (I − σh)r+1F Ap,q(·)_w,ϑ +∥(I − σh)rF∥_Ap,q(·) w,ϑ ) . (2.5)

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If we take g := (I− σh) r₋₁ F in (2.5), then ∥σh(I− σh) r F∥_Ap,q(·) w,ϑ +∥(I − σh) r F∥_Ap,q(·) w,ϑ ≤ ∥(I − σh) r (I + σh) F∥_Ap,q(·) w,ϑ + (I − σh) r+1 F Ap,q(·)_w,ϑ . Applying this in (2.4), ∥(I − σh)rF∥_Ap,q(·) w,ϑ ≤ δ ( ∥σh(I− σh) r F∥_Ap,q(·) w,ϑ + (I − σh) r+1 F Ap,q(_w,ϑ·) +∥(I − σh) r F∥_Ap,q(·) w,ϑ ) ≤ δ ( ∥(I − σh) r (I + σh) F∥_Ap,q(·) w,ϑ + (I − σh) r+1 F Ap,q(·)_w,ϑ ) +δ (I − σh) r+1 F Ap,q(·)_w,ϑ ≤ δ ∥(I − σh)r(I + σh) F∥_Ap,q(·) w,ϑ + 2δ (I − σh)r+1F Ap,q(·)_w,ϑ . (2.6)

If we repeat r times the last inequality, then

∥(I − σh) r F∥_Ap,q(·) w,ϑ ≤ δ ∥(I − σh )r(I + σh) F∥_Ap,q(·) w,ϑ + 2δ (I − σh) r+1 F Ap,q(·)_w,ϑ ≤ δ2 _{(I − σ} h)r(I + σh)2F Ap,q(·)_w,ϑ +2δ2 (I − σh) r+1 (I + σh) F Ap,q(·)_w,ϑ + 2δ (I − σh) r+1 F Ap,q(·)_w,ϑ ≤ ... ≤ δr_{∥(I − σ} h) r (I + σh) r F∥_Ap,q(·) w,ϑ +2 r ∑ k=1 δk (I − σh)r+1(I + σh)k−1F Ap,q(·)_w,ϑ = δr (I− σ2_h)rF Ap,q(·)_w,ϑ + 2 r ∑ k=1 δk (I − σh) r+1 (I + σh) k₋₁ F Ap,q(·)_w,ϑ .

Hence, the last inequality gives

∥(I − σh)rF∥_Ap,q(·) w,ϑ ≤ δ r (_I_{− σ}2 h )r F Ap,q(·)_w,ϑ + C ( r, Ap,q(_w,ϑ·) ) Ωr+1(F, h)_Ap,q(·) w,ϑ (2.10)

for 0 < h≤ 1/n. From (2.10) and applying

∥F ∥_Ap,q(·)

w,ϑ ≲ ∥F ∥C(T)

,

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∥(I − σh) r F∥_Ap,q(·) w,ϑ ≲ ≲ δr (_I_{− σ}2 h )r F Ap,q(·)_w,ϑ + C ( r, Ap,q(_w,ϑ·) ) Ωr+1(F, h)_Ap,q(·) w,ϑ ≲ δ2r (_I_{− σ}4 h )r F Ap,q(·)_w,ϑ + (δr+ 1) C ( r, Ap,q(_w,ϑ·) ) Ωr+1(F, h)_Ap,q(·) w,ϑ ≲ ... ≲ δmr (_I_{− σ}2m h )r F Ap,q(·)_w,ϑ + C ( r, Ap,q(_w,ϑ·), m ) Ωr+1(F, h)_Ap,q(·) w,ϑ ≲ δmr (_I_{− σ}2m h )r F C(_T)+ C ( r, Ap,q(_w,ϑ·), m ) Ωr+1(F, h)_Ap,q(·) w,ϑ ≲ δmr_{∥F ∥} C(_T)+ C ( r, Ap,q(_w,ϑ·), m ) Ωr+1(F, h)_Ap,q(·) w,ϑ . (2.11)

Taking the supremum, from the last inequality we obtain Ωr(F, t)_Ap,q(·) w,ϑ ≲ δ mr_{∥F ∥} C(_T)+ Ωr+1(F, t)_Ap,q(·) w,ϑ , since (I− σ_h2m)rF C(T)≤ 2 r_{∥F ∥} C(T). (b) Let f ∈ Ap,q(_·)

w,ϑ . For any ε > 0, there is F ∈ C(T) such that ∥f − F ∥C(T)< ε . From (a) and density

of C(T) in Ap,q(_·) w,ϑ we get Ωr(f, t)_Ap,q(·) w,ϑ ≤ C1 δmr∥F ∥_C(_T)+ C2Ωr+1(f, t)_Ap,q(·) w,ϑ . 2 Lemma 2.3 ( [27, p. 344]) Let σ∗ be the Fejer operator, M (x) be a maximal function, and f ∈ L1_{. Then}

there exists an absolute constant c such that

σ∗(x, f )≤ cM(x).

Lemma 2.4 Let 1 < p <∞, q ∈ Plog_{, 1 < q}−_{≤ q}+_<_{∞. If w ∈ A}

p, ϑ∈ Aq(·), then for any trigonometric polynomial Tn of degree n the following inequality holds:

∥T′

n∥Ap,q(·)_w,ϑ ≤ cn ∥Tn∥Ap,q(·)_w,ϑ ,

with a constant c independent of n. Proof It is known that |T′

n(x)| ≤ 2nσ∗(|Tn′(x)|) .

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3. Proofs of main results

Proof of Theorem 1.1. The case r = 1. Let n∈ N and f ∈ Ap,q(_w,ϑ·) be fixed. We will use the operator Lnf.

We know that the relations Lp

w⊂ L1 and L q(·) ϑ ⊂ L 1 _{hold, so A}p,q(·) w,ϑ ⊂ L 1_{. Let S}

n(f ) be the n th partial sum

of the Fourier series of f. Using (2.2) , (2.5) , and the uniformly boundedness (in n ) of the operator f → Sn

on Ap,q(_w,ϑ·) ( [3,17]), we get En(f )_Ap,q(·) w,ϑ = En(f− Lnf + Lnf )_Ap,q(·) w,ϑ ≤ En (f− Lnf )_Ap,q(·) w,ϑ + En(Lnf )_Ap,q(·) w,ϑ ≲ ∥f − Lnf∥_Ap,q(·) w,ϑ + n−2 d 2 dx2Lnf (x) Ap,q(·)_w,ϑ ≲ Ω1 ( f,1 n ) Ap,q(·)_w,ϑ . (3.1)

The case r≥ 2. We will use induction on r as in [12] . We obtained above that the estimate (1.2) holds for r = 1. We suppose that the inequality (1.2) holds for some r = 2, 3, 4, .... We will show that the inequality (1.2) holds for r + 1. We set g(·) := f(·) − Snf (·). Then Sn(g) = Sn(f− Sn(f )) = Sn(f )− Sn(Sn(f )) = Sn(f )− Sn(f ) = 0 and ∥Snf∥_Ap,q(·) w,ϑ ≲ ∥Sn f∥_p,w+∥Snf∥q(·),ϑ≲ ∥f∥p,w+∥f∥q(·),ϑ≲ ∥f∥Ap,q(·)_w,ϑ .

Using the induction hypothesis, [17], and [3],

∥g∥_Ap,q(·) w,ϑ =∥g − Sn(g)∥_Ap,q(·) w,ϑ ≤ CEn (g)_Ap,q(·) w,ϑ ≤ CΩr ( g,1 n ) Ap,q(·)_w,ϑ .

From Lemma 2.2 we have that Ωr ( f,1 n ) Ap,q(·)_w,ϑ ≤ C1δmr∥f∥_Ap,q(·) w,ϑ ∥F ∥C(T) + C2Ωr+1 ( f,1 n ) Ap,q(·)_w,ϑ (2.13) for any f ∈ Ap,q(·) w,ϑ . Indeed: If ∥f∥_Ap,q(·) w,ϑ = 1, then Ωr ( f,1 n ) Ap,q(·)_w,ϑ ≤ C1δmr∥F ∥C(T)+ C2Ωr+1 ( f,1 n ) Ap,q(·)_w,ϑ (3.2) = C1δmr∥F ∥C(_T)∥f∥Ap,q(·)_w,ϑ + C2Ωr+1 ( f, 1 n ) Ap,q(·)_w,ϑ . If ∥f∥_Ap,q(·) w,ϑ = 0, then Ωr ( f,1 n ) Ap,q(·)_w,ϑ = C1δmr∥f∥_Ap,q(·) w,ϑ ∥F ∥C(T) + C2Ωr+1 ( f,1 n ) Ap,q(·)_w,ϑ .

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In the case of ∥f∥_Ap,q(·) w,ϑ > 0 and ∥f∥_Ap,q(·) w,ϑ ̸= 1, let f ∗ ₌ f ∥f∥_Ap,q(·) w,ϑ . Then ∥f∗∥_Ap,q(·) w,ϑ = 1. Thus, from (3.2) we get Ωr ( f∗, 1 n ) Ap,q(_w,ϑ·) ≤ C1δmr∥F ∥C(_T)∥f∗∥Ap,q(_w,ϑ·) + C2Ωr+1 ( f∗,1 n ) Ap,q(_w,ϑ·) . Hence, 1 ∥f∥_Ap,q(·) w,ϑ Ωr ( f,1 n ) Ap,q(·)_w,ϑ ≤ C1δmr∥F ∥C(T) 1 ∥f∥_Ap,q(·) w,ϑ ∥f∥Ap,q(·)_w,ϑ + C2 ∥f∥_Ap,q(·) w,ϑ Ωr+1 ( f,1 n ) Ap,q(·)_w,ϑ and (2.13) follows.

If we choose m so big that CC1δmr∥F ∥_C(_T)< 1/2, we obtain

∥g∥Ap,q(·)_w,ϑ ≤ CΩr ( g,1 n ) Ap,q(·)_w,ϑ ≤ CC1δmr∥F ∥C(T)∥g∥Ap,q(·)_w,ϑ + CC2Ωr+1 ( g,1 n ) Ap,q(·)_w,ϑ . Then ∥g∥Ap,q(·)_w,ϑ ≲ Ωr+1 ( g,1 n ) Ap,q(·)_w,ϑ .

From [17] and [3] and using

Ωr ( Snf ; 1 n ) Ap,q(·)_w,ϑ ≤ CΩr ( f ;1 n ) Ap,q(·)_w,ϑ we have Ωr+1 ( f− Snf, 1 n ) Ap,q(·)_w,ϑ ≲ Ωr+1 ( f,1 n ) Ap,q(·)_w,ϑ . Consequently, En(f )_Ap,q(·) w,ϑ ≤ ∥f − S n(f )∥_Ap,q(·) w,ϑ =∥g∥_Ap,q(·) w,ϑ ≲ Ω r+1 ( f,1 n ) Ap,q(_w,ϑ·) .

This completes the proof. 2

Proof of Theorem 1.2. Using (2.3) and the equalities [3]

d2 dx2((I− σh) g) (x) = (I− σh) d2 dx2g (x) (I− σh) m+n f (x) = (I− σh) m (I− σh) n f (x), m, n∈ N we obtain Ωr(g, t)_Ap,q(·) w,ϑ ≲ δ 2r _g(2r) Ap,q(·)_w,ϑ , r∈ N, (2.15) for g(2r) _{∈ A}p,q(·)

w,ϑ and δ > 0. On the other hand, for any m∈ N

Ωr(f, δ)_Ap,q(·) w,ϑ ≤ Ωr

(f− S2m+1(f ) , δ)

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and

Ωr(f− S2m+1(f ) , δ)

Ap,q(·)_w,ϑ ≲ ∥f − S2m+1(f )∥A_w,ϑp,q(·) ≲ E2m+1(f )Ap,q(·)_w,ϑ . (2.17)

Then by (2.15) and the weighted version of Bernstein’s inequality in Ap,q(_w,ϑ·) [28, Lemma 2.2],

Ωr(S2m+1(f ) , δ) Ap,q(·)_w,ϑ ≲ δ 2r _S(2r) 2m+1(f ) Ap,q(·)_w,ϑ ≲ δ2r{ S(2r) 1 (f )− S (2r) 0 (f ) Ap,q(·)_w,ϑ + m ∑ i=1 S(2r) 2i+1(f )− S (2r) 2i (f ) Ap,q(·)_w,ϑ } ≲ δ2r { E0(f )_Ap,q(·) w,ϑ + m ∑ i=1 2(i+1)2rE2i(f ) Ap,q(·)_w,ϑ } ≲ δ2r { E0(f )_Ap,q(·) w,ϑ + 22rE1(f )_Ap,q(·) w,ϑ + m ∑ i=1 2(i+1)2rE2i(f ) Ap,q(·)_w,ϑ } .

Applying here the inequality and choosing m as 2m_{≤ n < 2}m+1_{, from (}_2.16₎_{− (}_2.19_{) , we obtain the following}

result: 2(i+1)2rE2i(f )_Ap,q(·) w,ϑ ≲ 2m ∑ k=2i−1+1 k2r−1Ek(f )_Ap,q(·) w,ϑ , i≥ 1, (2.18) and we obtain Ωr(S2m+1(f ) , δ) Ap,q(·)_w,ϑ ≲ δ 2r { E0(f )_Ap,q(·) w,ϑ + 22rE1(f )_Ap,q(·) w,ϑ + 2m ∑ k=2 k2r−1Ek(f )_Ap,q(·) w,ϑ } ≲ δ2r { E0(f )_Ap,q(·) w,ϑ + 2m ∑ k=1 k2r−1Ek(f )_Ap,q(·) w,ϑ } , (2.19)

using the estimate

E2m+1(f )_Ap,q(·) w,ϑ ≲ 1 n2r 2m ∑ k=2m−1+1 k2r−1Ek(f )_Ap,q(·) w,ϑ . Acknowledgment

The authors sincerely thank the anonymous reviewers for their careful reading and constructive comments.

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