Approximation Order for Multivariate Durrmeyer Operators with Jacobi Weights

Abstract

Using the equivalence relation between 𝐾-functional and modulus of smoothness, we establish a strong direct theorem and an inverse theorem of weak type for multivariate Bernstein-Durrmeyer operators with Jacobi weights on a simplex in this paper. We also obtain a characterization for multivariate Bernstein-Durrmeyer operators with Jacobi weights on a simplex. The obtained results not only generalize the corresponding ones for Bernstein-Durrmeyer operators, but also give approximation order of Bernstein-Durrmeyer operators.

1. Introduction

Let 𝑆=𝑆_𝑑(𝑑=1,2,…) be a simplex in 𝑅^𝑑 defined by 𝑆=𝑥=𝑥₁,𝑥₂,…,𝑥_𝑑∶𝑥_𝑖≥0,𝑖=1,2,…,𝑑,|𝑥|=^𝑑_𝑖=0𝑥_𝑖≤1. (1.1) For 𝑝≥1, we denote by 𝐿^𝑝(𝑆) the space of 𝑝-order Lebesgue integrable functions on 𝑆 with𝜔𝑓_𝑝=_𝑆𝜔(𝑥)𝑓(𝑥)^𝑝𝑑𝑥^1/𝑝<∞1≤𝑝<+∞,max_𝑥∈𝑆𝜔(𝑥)𝑓(𝑥)𝑝=+∞, (1.2) where 𝐿^∞(𝑆)=𝐶(𝑆) denote the space of continuous functions on 𝑆. For 𝑓∈𝐿(𝑆), the multivariate Bernstein-Durrmeyer Operators with 𝑑 variables on 𝑆 are given by 𝑀_𝑛,𝑑𝑓;𝑥=_{|𝑘|≤𝑛}𝑃_𝑛⋅𝑘(𝑥)(𝑛+𝑑)!𝑛!_𝑆𝑃_𝑛,𝑘(𝑢)𝑓(𝑢)𝑑𝑢, (1.3) where 𝑃_𝑛,𝑘(𝑥)=(𝑛!/(𝑘!(𝑛−|𝑘|)!))𝑥^𝑘(1−|𝑥|)^{𝑛−|𝑘|}(𝑥∈𝑆) and 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑑)∈𝑅^𝑑, 𝑘=(𝑘₁,𝑘₂,…,𝑘_𝑑)∈𝑁^𝑑₀, with the convention |𝑥|=^𝑑_𝑖=1𝑥_𝑖,𝑥^𝑘=𝑥^𝑘1₁𝑥^𝑘2₂⋯𝑥^𝑘𝑑_𝑑,|𝑘|=^𝑑_𝑖=1𝑘_𝑖,𝑘!=𝑘₁!𝑘₂!⋯𝑘_𝑑!. (1.4)

For multivariate Jacobi weights 𝜔(𝑥)=𝑥^𝛼(1−|𝑥|)^𝛽, (𝑥∈𝑆, 𝛼=(𝛼₁,…,𝛼_𝑑)∈𝑅^𝑑, 0<𝛼_𝑖, 𝛽<1, 𝑖=1,2,…,𝑑, 𝑥^𝛼=𝑥^𝛼1₁𝑥^𝛼2₂⋯𝑥^𝛼𝑑_𝑑). We give some further notations, for 𝑥∈𝑆, and we write 𝜑_𝑖(𝑥)=𝜑_𝑖𝑖(𝑥)=𝑥_𝑖(1−|𝑥|)(1≤𝑖≤𝑑), 𝜑_𝑖𝑗(𝑥)=𝑥_𝑖𝑥_𝑗, (1≤𝑖<𝑗≤𝑑) and 𝐷_𝑖=𝐷_𝑖𝑖=𝜕𝜕𝑥_𝑖,1≤𝑖≤𝑑,𝐷_𝑖𝑗=𝐷_𝑖−𝐷_𝑗𝐷,1≤𝑖<𝑗≤𝑑,_𝑟𝑖𝑗=𝐷_𝑖𝑗𝐷^𝑟−1_𝑖𝑗,1≤𝑖≤𝑗≤𝑑,𝑟∈𝑁,𝐷^𝑘=𝐷^𝑘1₁𝐷^𝑘2₂⋯𝐷^𝑘𝑑_𝑑,𝑘∈𝑁^𝑑₀. (1.5)

For 𝑓∈𝐿^𝑝(𝑆), the weighted Sobolev space is given by𝑊_𝜙^𝑟,𝑝(𝑆)=𝑓∈𝐿^𝑝(𝑆)∶𝜔𝑓∈𝐿^𝑝(𝑆),𝐷^𝑘𝑓∈𝐿𝑆_loc⁰,𝜔𝜑^𝑟_𝑖𝑗𝐷^𝑟_𝑖𝑗𝑓∈𝐿^𝑝(𝑆),|𝑘|,𝑊≤𝑟,1≤𝑖≤𝑗≤𝑑,𝑟∈𝑁_𝜙𝑟,∞(𝑆)=𝑓∈𝐶(𝑆)∶𝜔𝑓∈𝐶(𝑆),𝑓∈𝐶𝑆^𝑟0,𝜔𝜑^𝑟_𝑖𝑗𝐷^𝑟_𝑖𝑗𝑓∈𝐶(𝑆),1≤𝑖≤𝑗≤𝑑,𝑟∈𝑁, (1.6) where 𝑆⁰ is the interior of 𝑆. To characterize the approximation capability of multivariate Bernstein-Durrmeyer operators, we introduce the weighted 𝐾-functional 𝐾^𝑟_𝜑𝑓,𝑡^𝑟_𝜔=inf_{𝑔∈𝑊𝜙𝑟,𝑝}𝜔(𝑓−𝑔)_𝑝+𝑡^𝑟_{1≤𝑖≤𝑗≤𝑑}𝜔𝜑^𝑟_𝑖𝑗𝐷^𝑟_𝑖𝑗𝑔_𝑝 (1.7) and a measure of smoothness of 𝑓𝜔^𝑟_𝜑𝑓,𝑡_𝜔=sup_{0<ℎ≤𝑡1≤𝑖≤𝑗≤𝑑}𝜔Δ^𝑟_{ℎ𝜑𝑖𝑗𝑒𝑖𝑗}𝑓_𝑝. (1.8)

Since 1967, Durrmeyer introduced Bernstein-Durrmeyer operators, and there are many papers which studied theirproperties [1–7]. In 1991, Zhang studied the characterization of convergence for 𝑀_𝑛,1(𝑓;𝑥) with Jacobi weights. In 1992, Zhou [5] considered multivariate Bernstein-Durrmeyer operators 𝑀_𝑛,𝑑(𝑓;𝑥) and obtained a characterization of convergence. In 2002, Xuan et al. studied the equivalent characterization of convergence for 𝑀_𝑛,𝑑(𝑓;𝑥) with Jacobi weights and obtained the following result.

Theorem 1.1. For 𝜔𝑓∈𝐿^𝑝(𝑆), 0<𝑟<1, the following results are equivalent:(i)‖𝜔(𝑀_𝑛,𝑑𝑓−𝑓)‖_𝑝=𝑂(𝑛^−𝑟); (ii)𝐾²_𝜑(𝑓,𝑡)_𝜔=𝑂(𝑡^𝑟).

In this paper, using the Ditzian-Totik modulus of smoothness, we will give the upper bound and lower bound of approximation function by 𝑀_𝑛,𝑑(𝑓;𝑥) on simplex. The main results are as follows.

Theorem 1.2. If 𝜔𝑓∈𝐿^𝑝(𝑆), then 𝜔(𝑀_𝑛,𝑑𝑓−𝑓)_𝑝≤𝐶𝜔²_𝜑1𝑓,√𝑛_𝜔+𝜔𝑓_𝑝𝑛. (1.9) And there exists a positive number 𝛿(0<𝛿<1) such that the following inequality is satisfied: 𝜔²_𝜑1𝑓,√𝑛_𝜔≤𝐶𝑛^𝑛_𝑘=1𝑛𝑘^𝛿𝜔(𝑀_𝑛,𝑑𝑓−𝑓)_𝜔. (1.10)

Throughout the paper, the letter 𝐶, appearing in various formulas, denotes a positive constant independent of 𝑛, 𝑥, and 𝑓. Its value may be different at different occurrences, even within the same formula.

From Theorem 1.2, we can easily obtain the following corollary.

Corollary 1.3. If 𝜔𝑓∈𝐿^𝑝(𝑆), 0<𝑟<1, we has the following equivalent results:(i)‖𝜔(𝑀_𝑛,𝑑𝑓−𝑓)‖_𝑝=𝑂(𝑛^−𝑟);(ii)𝐾²_𝜑(𝑓,𝑡)_𝜔=𝑂(𝑡^𝑟);(iii)𝜔²_𝜑(𝑓,𝑡)_𝜔=𝑂(𝑡^2𝑟).

2. Some Lemmas

To prove Theorem 1.2, we will show some lemmas in this section. For the simplex 𝑆, the transformation 𝑇: 𝑆→𝑆^[10] defined by 𝑇𝑥₁,𝑥₂,…,𝑥_𝑑=𝑢₁,𝑢₂,…,𝑢_𝑑,𝑢_𝑙=𝑥_𝑗|𝑥|𝑙=𝑗,1−𝑙≠𝑗 (2.1) satisfies 𝑇²=𝐼, and 𝐼 is the identity operator. So we have 𝜕𝜕𝑢_𝑙=𝜕𝜕𝑥_𝑙−𝜕𝜕𝑥_𝑗,𝜕𝑙≠𝑗𝜕𝑢_𝑗𝜕=−𝜕𝑥_𝑗,𝑀_𝑛,𝑑𝑓;𝑥=𝑀_𝑛,𝑑𝑓_𝑇;𝑇𝑥;𝑀_𝑛,𝑑𝑓;𝑇𝑥=𝑀_𝑛,𝑑𝑓_𝑇;𝑥, (2.2) where 𝑓_𝑇(𝑢)=𝑓(𝑇𝑥).

Lemma 2.1. If 𝜔𝑓∈𝐿^𝑝(𝑆), then 𝜔𝑀_𝑛,𝑑𝑓_𝑝≤𝜔𝑓_𝑝,𝜔(𝑀_𝑛,𝑑𝑓−𝑓)_𝑝≤𝐶𝑛𝜔𝑓_𝑝+_{1≤𝑖≤𝑗≤𝑑}𝜔𝜑²_𝑖𝑗𝐷²_𝑖𝑗𝑓_𝑝,𝑓∈𝑊_𝜙^𝑟,𝑝(𝑆). (2.3)

Proof. Letting 𝑆^′={𝑥∶(𝑥₁,𝑥)∈𝑆_𝑑}, 𝑥=(𝑥₂,𝑥₃,…,𝑥_𝑑), 𝑘=(𝑘₂,𝑘₃…,𝑘_𝑑), 𝑘=(𝑘₁,𝑘), 𝑃_𝑛,𝑘1(𝑥₁)=(𝑛!/𝑘₁!(𝑛−𝑘₁)!)𝑥^𝑘1₁(1−𝑥₁)^𝑛−𝑘1, then 𝑀_𝑛,𝑑𝑓;𝑥=^𝑛_𝑘1=0𝑃_𝑛,𝑘1𝑥_{1𝑘≤𝑛−𝑘1}𝑃_{𝑛−𝑘1,𝑘}𝑥1−𝑥₁(𝑛+𝑑)!×𝑛!₁₀𝑃_𝑛,𝑘1𝑢_1𝑆′𝑃_{𝑛−𝑘1,𝑘}𝑢1−𝑢₁𝑓(𝑢)𝑑𝑢𝑑𝑢₁=^𝑛_𝑘1=0𝑃_𝑛,𝑘1𝑥₁(𝑛+𝑑)!𝑛!¹₀𝑃_𝑛,𝑘1𝑢₁1−𝑢₁^𝑑−1_{𝑘≤𝑛−𝑘1}𝑃_{𝑛−𝑘1,𝑘}𝑥1−𝑥₁×_𝑆𝑑−1𝑃_{𝑛−𝑘1,𝑘}(𝑡)𝑓𝑢₁,1−𝑢₁𝑡𝑑𝑡𝑑𝑢₁=^𝑛_𝑘1=0𝑃_𝑛,𝑘1𝑥₁(𝑛+𝑑)¹₀𝑃_{𝑛+𝑑−1,𝑘1}𝑢₁𝑀_{𝑛−𝑘1,𝑑−1}𝑓𝑢₁,1−𝑢₁⋅;𝑥1−𝑥₁𝑑𝑢₁. (2.4)
Using the transformation 𝑇, (2.2), (2.4), the method of [7], we can easily get (2.3).

Lemma 2.2 (see [8]). If 𝑓∈𝐿^𝑝(𝑆), then 𝐶⁻¹𝜔^𝑟_𝜑𝑓,𝑡_𝜔≤𝐾^𝑟_𝜑𝑓,𝑡^𝑟_𝜔≤𝐶𝜔^𝑟_𝜑𝑓,𝑡_𝜔. (2.5)

Proof. Lemma 2.2 is proved when 𝑓∈𝐶(𝑆) in [8]. Similarly, we can prove 𝑓∈𝐿^𝑝(𝑆).

Lemma 2.3. If 0<𝑎<1, 𝑏>0, 𝑥∈(0,1), 𝑃_𝑛,𝑘(𝑥)=𝐶^𝑘_𝑛𝑥^𝑘(1−𝑥)^𝑛−𝑘 is basis function of the classical Bernstein operators, then ^𝑛−1_𝑘=1𝑃_𝑛,𝑘(𝑥𝑛)𝑘^𝑎≤𝐶𝑥^−𝑎,_{𝑛−1𝑘=1}𝑃_𝑛,𝑘(𝑥)𝑛𝑛−𝑘^𝑏≤𝐶(1−𝑥)^−𝑏. (2.6)

Proof. The first inequality can be inferred by Hölder inequality. In the following we prove the second inequality.(i)If 0<𝑏<1, using Hölder inequality, we can easily obtain the result.(ii)If 𝑏≥1, let 𝑏=𝑚+𝑟, 𝑚∈𝑁, 0≤𝑟<1, then ^𝑛−1_𝑘=1𝑃_𝑛,𝑘(𝑥𝑛)𝑛−𝑘^𝑏=^𝑛−1_𝑘=1𝑃_𝑛,𝑘(𝑥𝑛)𝑛−𝑘^𝑚𝑛𝑛−𝑘^𝑟≤𝐶(1−𝑥)^{−𝑚𝑛−1}_𝑘=1𝑃_{𝑛+𝑚,𝑘}(𝑥)𝑛+𝑚𝑛+𝑚−𝑘^𝑟≤𝐶(1−𝑥)^{−𝑚−𝑟}=𝐶(1−𝑥)^−𝑏. (2.7)Lemma 2.3 is completed.

Lemma 2.4. If 𝑓∈𝐿^𝑝(𝑆), 1≤𝑝≤∞, then 𝜔𝜑²_𝑖𝑗𝐷²_𝑖𝑗𝑀_𝑛,𝑑𝑓_𝑝≤𝐶𝑛𝜔𝑓_𝑝1≤𝑖≤𝑗≤𝑑. (2.8)

Proof. In the following we use the induction on the dimension number 𝑑 to prove the result. The case 𝑑=1 was proved by Lemma  4 of [6]. Next, suppose that Lemma 2.4 is valid for 𝑑=𝑟(𝑟≥1); we prove it is also true for 𝑑=𝑟+1. To observe this, we use a decomposition formula (2.4), and we have 𝜔(𝑥)𝜑²₂₂(𝑥)𝐷²₂₂𝑀_𝑛,𝑑𝑓;𝑥=𝑥^𝛼1₁1−𝑥₁^{|𝑛𝛼|+𝛽}_𝑘1=0𝑃_𝑛,𝑘1𝑥₁(𝑛+𝑑)𝑧^𝛼2₁𝑧^𝛼3₂⋯𝑧^𝛼𝑑_𝑑−1×(1−|𝑧|)^𝛽𝜑²₁₁(𝑧)¹₀𝑃_{𝑛+𝑑−1,𝑘1}𝑢₁𝐷²₁₁𝑀_{𝑛−𝑘1,𝑑−1}𝑓𝑢₁,1−𝑢₁⋅;𝑧𝑑𝑢₁, (2.9) where 𝑧=(𝑧₁,𝑧₂,…,𝑧_𝑑−1)=(𝑥₂/(1−𝑥₁),𝑥₃/(1−𝑥₁),…,𝑥_𝑑/(1−𝑥₁)). Thus we have _𝑆𝜔(𝑥)𝜑²₂₂(𝑥)𝐷²₂₂𝑀_𝑛,𝑑𝑓;𝑥𝑑𝑠≤𝐶¹₀𝑥^𝛼1₁1−𝑥₁^{|𝑛𝛼|+𝛽}_𝑘1=0𝑃_𝑛,𝑘1𝑥₁(𝑛+𝑑)¹₀𝑃_{𝑛+𝑑−1,𝑘1}𝑢₁𝑛−𝑘₁×_{𝑧∈𝑆𝑑−1}𝜔(𝑧)𝑓𝑢₁,1−𝑢₁𝑧𝑑𝑧𝑑𝑥₁𝑑𝑢₁≤𝐶𝑛+𝑑𝑛𝑛+1¹₀^𝑛_𝑘1=0𝑘₁+1𝑛+1^𝛼1𝑛−𝑘₁+1𝑛+1^|𝛼|+𝛽𝑃_{𝑛+𝑑−1,𝑘1}𝑢₁×_{𝑧∈𝑆𝑑−1}𝜔(𝑧)𝑓𝑢₁,1−𝑢₁𝑧𝑑𝑧𝑑𝑢₁≤𝐶𝑛¹₀𝑢^𝛼1₁1−𝑢₁^|𝛼|+𝛽1𝑢₁^𝛼11−𝑢₁^{−|𝛼|−𝛽}_{z∈𝑆𝑑−1}𝜔𝑓𝑢₁,1−𝑢₁𝑧𝑑𝑧𝑑𝑢₁=𝐶𝑛𝜔𝑓₁. (2.10)
In the above derivation, we have used the formula [6] ¹₀𝑥^𝛼1₁1−𝑥₁^|𝛼|+𝛽𝑃_𝑛,𝑘1𝑥₁𝑑𝑥₁1≤𝐶𝑘𝑛+1¹+1𝑛+1^𝛼1𝑛−𝑘₁+1𝑛+1^|𝛼|+𝛽 (2.11) and the inequality ^𝑛_𝑘1=0𝑘₁+1𝑛+1^𝛼1𝑛−𝑘₁+1𝑛+1^|𝛼|+𝛽𝑃_{𝑛+𝑑−1,𝑘1}𝑢₁≤𝐶𝑢^𝛼1₁1−𝑢₁^|𝛼|+𝛽. (2.12) When 𝑝=∞, we have 𝜔(𝑥)𝜑²₂₂(𝑥)𝐷²₂₂𝑀_𝑛,𝑑𝑓;𝑥=𝑥^𝛼1₁1−𝑥₁^{|𝑛𝛼|+𝛽}_𝑘1=0𝑃_𝑛,𝑘1𝑥₁(𝑛+𝑑)𝑧^𝛼2₁𝑧^𝛼3₂⋯𝑧^𝛼𝑑_𝑑−1×(1−|𝑧|)^𝛽𝜑²₁₁(𝑧)¹₀𝑃_{𝑛+𝑑−1,𝑘1}𝑢₁𝐷²₁₁𝑀_{𝑛−𝑘1,𝑑−1}𝑓𝑢₁,1−𝑢₁⋅;𝑧𝑑𝑢₁, (2.13) where 𝑧=(𝑧₁,𝑧₂,…,𝑧_𝑑−1)=(𝑥₂/(1−𝑥₁),𝑥₃/(1−𝑥₁),…,𝑥_𝑑/(1−𝑥₁)).
From the Cauchy-Swartz inequality, Hölder inequality, and Lemma 2.3, we have 𝜔(𝑥)𝜑²₂₂(𝑥)𝐷²₂₂𝑀_𝑛,𝑑𝑓;𝑥≤𝐶𝑥^𝛼1₁1−𝑥₁^{|𝑛𝛼|+𝛽}_𝑘1=0𝑃_𝑛,𝑘1𝑥₁(𝑛+𝑑)¹₀𝑃_{𝑛+𝑑−1,𝑘1}𝑢₁𝑛−𝑘₁×max_{𝑧∈𝑆𝑑−1}𝑧^𝛼2₁𝑧^𝛼3₂⋯𝑧^𝛼𝑑_𝑑−1(1−|𝑧|)^𝛽𝑓𝑢₁,1−𝑢₁𝑧𝑑𝑢₁≤𝐶𝑛𝜔𝑓_∞𝑥^𝛼1₁1−𝑥₁^{|𝑛𝛼|+𝛽}_𝑘1=0𝑃_𝑛,𝑘1𝑥₁(𝑛+𝑑)¹₀𝑃_{𝑛+𝑑−1,𝑘1}𝑢₁1𝑢₁^𝛼11−𝑢₁^{−|𝛼|−𝛽}𝑑𝑢₁≤𝐶𝑛𝜔𝑓_∞𝑥^𝛼1₁1−𝑥₁^{|𝑛𝛼|+𝛽}_𝑘1=0𝑃_𝑛,𝑘1𝑥₁(𝑛+𝑑)¹₀𝑃_{𝑛+𝑑−1,𝑘1}𝑢₁𝑢^−2𝛼1₁𝑑𝑢₁^1/2×¹₀𝑃_{𝑛+𝑑−1,𝑘1}𝑢₁1−𝑢₁^{−2|𝛼|−2𝛽}𝑑𝑢₁^1/2≤𝐶𝑛𝜔𝑓_∞𝑥^𝛼1₁1−𝑥₁^{|𝑛𝛼|+𝛽}_𝑘1=0𝑃_𝑛,𝑘1𝑥₁(𝑛+𝑑)¹₀𝑃_{𝑛+𝑑−1,𝑘1}𝑢₁𝑢₁⁻²𝑑𝑢₁^𝛼1/2×¹₀𝑃_{𝑛+𝑑−1,𝑘1}𝑢₁𝑑𝑢₁^{(1−𝛼1)/21}₀𝑃_{𝑛+𝑑−1,𝑘1}𝑢₁1−𝑢₁^−2𝑑𝑑𝑢₁^{(|𝛼|+𝛽)/2𝑑}×¹₀𝑃_{𝑛+𝑑−1,𝑘1}𝑢₁𝑑𝑢₁^{1/2−(|𝛼|+𝛽)/2𝑑}≤𝐶𝑛𝜔𝑓_∞𝑥^𝛼1₁1−𝑥₁^{|𝑛𝛼|+𝛽}_𝑘1=2𝑃_𝑛,𝑘1𝑥₁(𝑛+𝑑)𝑛+𝑑−1𝑘₁𝑘₁−1^𝛼1/2×(𝑛+𝑑−1)!(𝑛−𝑑)!𝑛−𝑑−𝑘₁−1!𝑛+𝑑−𝑘₁−1!^{(|𝛼|+𝛽)/2𝑑}(𝑛+𝑑)^{1−(|𝛼|+𝛽)/2𝑑−𝛼1/2−1}≤𝐶𝑛𝜔𝑓_∞𝑥^𝛼1₁1−𝑥₁^{|𝛼|+𝛽𝑛−1}_𝑘1=1𝑃_𝑛,𝑘1𝑥₁𝑛𝑘₁^𝛼1𝑛𝑛−𝑘₁^|𝛼|+𝛽≤𝐶𝑛𝜔𝑓_∞. (2.14) By Riesz interpolation theorem, we get 𝜔𝜑²₂₂𝐷²₂₂𝑀_𝑛,𝑑𝑓_𝑝≤𝐶𝑛𝜔𝑓_𝑝. (2.15) Similarly, the other cases for 𝑖=1,3,4,…,𝑑(=𝑗) can be proved. For 𝑖≠𝑗, by the transformation 𝑇, we have 𝜔𝜑²_𝑖𝑗𝐷²_𝑖𝑗𝑀_𝑛,𝑑𝑓_𝑝=𝜔_𝑇𝜑²_𝑗𝑗𝐷²_𝑗𝑗𝑀_𝑛,𝑑𝑓_𝑇𝑝≤𝐶𝑛𝜔_𝑇𝑓_𝑇𝑝=𝐶𝑛𝜔𝑓_𝑝. (2.16) Lemma 2.4 is completed.

Lemma 2.5. If 𝑓∈𝑊_𝜙^𝑟,𝑝(𝑆)⊂𝐿^𝑝(𝑆), 1≤𝑝≤∞, then 𝜔𝜑²_𝑖𝑗𝐷²_𝑖𝑗𝑀_𝑛,𝑑𝑓_𝑝≤𝐶𝜔𝜑²_𝑖𝑗𝐷²_𝑖𝑗𝑓_𝑝1≤𝑖≤𝑗≤𝑑. (2.17)

Proof. We use the induction on the dimension number 𝑑 to prove Lemma 2.5. The case 𝑑=1 was proved by Lemma  3 of [6], that is, 𝜔𝜑²𝐷²𝑀_𝑛,1𝑓_𝑝≤𝐶𝜔𝜑²𝐷²𝑓_𝑝. (2.18) Next, suppose that Lemma 2.5 is valid for 𝑑=𝑟(𝑟≥1), and we prove it is also true for 𝑑=𝑟+1. Noticing formula (2.4), we have 𝜔(𝑥)𝜑²₂₂(𝑥)𝐷²₂₂𝑀_𝑛,𝑑𝑓;𝑥=𝑥^𝛼1₁1−𝑥₁^{|𝑛𝛼|+𝛽}_𝑘1=0𝑃_𝑛,𝑘1𝑥₁(𝑛+𝑑)𝑧^𝛼2₁𝑧^𝛼3₂⋯𝑧^𝛼𝑑_𝑑−1×(1−|𝑧|)^𝛽𝜑²₁₁(𝑧)¹₀𝑃_{𝑛+𝑑−1,𝑘1}𝑢₁𝐷²₁₁𝑀_{𝑛−𝑘1,𝑑−1}𝑓𝑢₁,1−𝑢₁⋅;𝑧𝑑𝑢₁, (2.19) where 𝑧=(𝑧₁,𝑧₂,…,𝑧_𝑑−1)=(𝑥₂/(1−𝑥₁),𝑥₃/(1−𝑥₁),…,𝑥_𝑑/(1−𝑥₁)). When 𝑝=1, from the inductive assumption of 𝑝=1, we have _𝑆𝜔(𝑥)𝜑²₂₂(𝑥)𝐷²₂₂𝑀_𝑛,𝑑𝑓;𝑥𝑑𝑠≤𝐶¹₀𝑥^𝛼1₁1−𝑥₁^{|𝑛𝛼|+𝛽}_𝑘1=0𝑃_𝑛,𝑘1𝑥₁(𝑛+𝑑)¹₀𝑃_{𝑛+𝑑−1,𝑘1}𝑢₁×_{𝑧∈𝑆𝑑−1}𝜔(𝑧)𝜑²₁₁(𝑧)𝐷²₁₁𝑓𝑢₁,1−𝑢₁𝑧𝑑𝑧𝑑𝑥₁𝑑𝑢₁≤𝐶𝑛+𝑑𝑛+1¹₀^𝑛_𝑘1=0𝑘₁+1𝑛+1^𝛼1𝑛−𝑘₁+1𝑛+1^|𝛼|+𝛽𝑃_{𝑛+𝑑−1,𝑘1}𝑢₁×_{𝑧∈𝑆𝑑−1}𝜔(𝑧)𝜑²₁₁(𝑧)𝐷²₁₁𝑓𝑢₁,1−𝑢₁𝑧𝑑𝑧𝑑𝑢₁≤𝐶¹₀𝑢^𝛼1₁1−𝑢₁^|𝛼|+𝛽1𝑢₁^𝛼11−𝑢₁^{−|𝛼|−𝛽}_{𝑧∈𝑆𝑑−1}𝜔𝜑²₂₂𝐷²₂₂𝑓𝑢₁,1−𝑢₁𝑧𝑑𝑧𝑑𝑢₁≤𝐶𝜔𝜑²₂₂𝐷²₂₂𝑓₁. (2.20) When 𝑝=∞, we have 𝜔(𝑥)𝜑²₂₂(𝑥)𝐷²₂₂𝑀_𝑛,𝑑𝑓;𝑥=𝑥^𝛼1₁1−𝑥₁^{|𝑛𝛼|+𝛽}_𝑘1=0𝑃_𝑛,𝑘1𝑥₁(𝑛+𝑑)𝑧^𝛼2₁𝑧^𝛼3₂⋯𝑧^𝛼𝑑_𝑑−1×(1−|𝑧|)^𝛽𝜑²₁₁(𝑧)¹₀𝑃_{𝑛+𝑑−1,𝑘1}𝑢₁𝐷²₁₁𝑀_{𝑛−𝑘1,𝑑−1}𝑓𝑢₁,1−𝑢₁⋅;𝑧𝑑𝑢₁, (2.21) where 𝑧=(𝑧₁,𝑧₂,…,𝑧_𝑑−1)=(𝑥₂/(1−𝑥₁),𝑥₃/(1−𝑥₁),…,𝑥_𝑑/(1−𝑥₁)). From the inductive assumption, the Cauchy-Swartz inequality, Holder inequality, and Lemma 2.4, we get 𝜔(𝑥)𝜑²₂₂(𝑥)𝐷²₂₂𝑀_𝑛,𝑑𝑓;𝑥≤𝐶𝑥^𝛼1₁1−𝑥₁^{|𝑛𝛼|+𝛽}_𝑘1=0𝑃_𝑛,𝑘1𝑥₁(𝑛+𝑑)¹₀𝑃_{𝑛+𝑑−1,𝑘1}𝑢₁×max_{𝑧∈𝑆𝑑−1}𝑧^𝛼2₁𝑧^𝛼3₂⋯𝑧^𝛼𝑑_𝑑−1(1−|𝑧|)^𝛽𝜑²_𝑧𝐷²_𝑧𝑓𝑢₁,1−𝑢₁𝑧𝑑𝑢₁≤𝐶𝑥^𝛼1₁1−𝑥₁^{|𝑛𝛼|+𝛽}_𝑘1=0𝑃_𝑛,𝑘1𝑥₁(𝑛+𝑑)¹₀𝑃_{𝑛+𝑑−1,𝑘1}𝑢₁𝑥^−𝛼1₁1−𝑥₁^{−|𝛼|−𝛽}×𝜔𝜑²₂₂𝐷²₂₂𝑓_∞𝑑𝑢₁≤𝐶𝜔𝜑²₂₂𝐷²₂₂𝑓_∞. (2.22) By Riesz interpolation theorem, we get 𝜔𝜑²₂₂𝐷²₂₂𝑀_𝑛,𝑑𝑓_𝑝≤𝐶𝜔𝜑²₂₂𝐷²₂₂𝑓_𝑝. (2.23) Similarly, the other cases for 𝑖=1,3,4,…,𝑑(=𝑗) can be proved. For 𝑖≠𝑗, by the transformation 𝑇, we have 𝜔𝜑²_𝑖𝑗𝐷²_𝑖𝑗𝑀_𝑛,𝑑𝑓_𝑝=𝜔_𝑇𝜑²_𝑗𝑗𝐷²_𝑗𝑗𝑀_𝑛,𝑑𝑓_𝑇𝑝≤𝐶𝜔_𝑇𝜑²_𝑗𝑗𝐷²_𝑗𝑗𝑓_𝑇𝑝≤𝐶𝜔𝜑²_𝑖𝑗𝐷²_𝑖𝑗𝑓_𝑝 (2.24) Lemma 2.5 is completed.

Lemma 2.6 (see [9]). Let {𝜎_𝑛}, {𝜙_𝑛} be nonnegative sequences (𝜎₁=0, 𝑛∈𝑁). For 𝑙>0, if the sequences {𝜎_𝑛}, {𝜙_𝑛} satisfy 𝜎_𝑛𝑘≤𝑄𝑛^𝑙𝜎_𝑘+𝜙_𝑘(𝑄≥1,1≤𝑘≤𝑛,𝑛∈𝑁), (2.25) one has 𝜎_𝑛≤𝑀𝑛^𝑛−𝑠_𝑘=1𝑘^𝑠−1𝜙_𝑘. (2.26) If 𝑄=1, then 𝑙=𝑠. If 𝑄>1, then 0<𝑠<𝑙.

3. The Proof of Theorems

Now we prove (1.9) of Theorem 1.2. By using Lemma 2.1, for arbitrary 𝑔∈𝑊_𝜙^𝑟,𝑝(𝑆)⊂𝐿^𝑝(𝑆), we have 𝜔(𝑀_𝑛,𝑑𝑓−𝑓)_𝑝≤𝐶𝜔𝑀_𝑛,𝑑𝑓−𝑔_𝑝+𝜔𝑀_𝑛,𝑑𝑔−𝜔𝑔_𝑝+𝜔𝑓−𝑔_𝑝≤𝐶𝜔𝑓−𝑔_𝑝+1𝑛_{1≤𝑖≤𝑗≤𝑑}𝜔𝜑²_𝑖𝑗𝐷²_𝑖𝑗𝑔_𝑝+𝜔𝑔_𝑝≤𝐶𝜔𝑓−𝑔_𝑝+1𝑛_{1≤𝑖≤𝑗≤𝑑}𝜔𝜑²_𝑖𝑗𝐷²_𝑖𝑗𝑔_𝑝+1𝑛𝜔𝑓_𝑝. (3.1) Hence, from Lemma 2.2, we obtain 𝜔(𝑀_𝑛,𝑑𝑓−𝑓)_𝑝≤𝐶𝐾²_𝜑1𝑓,𝑛_𝜔+1𝑛𝜔𝑓_𝑝≤𝐶𝜔²_𝜑𝑓,𝑡_𝜔+1𝑛𝜔𝑓_𝑝. (3.2) Next, we prove (1.10) of Theorem 1.2. Leting 𝜎_𝑛=𝐶(1/𝑛)‖𝜔𝜑²_𝑖𝑗𝐷²_𝑖𝑗𝑀_𝑛,𝑑(𝑓)‖_𝑝 (1≤𝑖≤𝑗≤𝑑), 𝜙_𝑛=𝐶‖𝜔(𝑀_𝑛,𝑑(𝑓)−𝑓)‖_𝑝, then 𝜎₁=0. By Lemmas 2.4 and 2.5, we have 𝜎_𝑛1≤𝐶𝑛𝜔𝜑²_𝑖𝑗𝐷²_𝑖𝑗𝑀_𝑛,𝑑𝑓−𝑀_𝑘,𝑑𝑓_𝑝1+𝐶𝑛𝜔𝜑²_𝑖𝑗𝐷²_𝑖𝑗𝑀_𝑛,𝑑𝑀_𝑘,𝑑𝑓_𝑝≤𝐶𝜔𝑓−𝑀_𝑘,𝑑𝑓_𝑝1+𝐶𝑛𝜔𝜑²_𝑖𝑗𝐷²_𝑖𝑗𝑀_𝑘,𝑑𝑓_𝑝𝑘=𝐶𝑛𝜎_𝑘+𝜙_𝑘(𝐶>1). (3.3) Using Lemma 2.6, we get 𝜎_𝑛≤𝐶(1/𝑛)∑^𝑛_𝑘=1(𝑛/𝑘)^𝛿𝜙_𝑘(0<𝛿<1). That is, 𝜔𝜑²_𝑖𝑗𝐷²_𝑖𝑗𝑀_𝑛,𝑑(𝑓)_𝑝≤𝐶^𝑛_𝑘=1𝑛𝑘^𝛿𝜔𝑀_𝑘,𝑑𝑓−𝑓_𝑝. (3.4) When 𝑛≥2, there exists (𝑚∈𝑁) such that 𝑛/2≤𝑚≤𝑛 and satisfies the equation 𝜔(𝑀_𝑚,𝑑𝑓−𝑓)_𝑝=min_{𝑛/2≤𝑘≤𝑛}𝜔(𝑀_𝑘,𝑑𝑓−𝑓)_𝑝. (3.5) Thus, 𝜔(𝑀_𝑚,𝑑𝑓−𝑓)_𝑝≤2𝑛_{𝑛/2≤𝑘≤𝑛}𝜔(𝑀_𝑘,𝑑𝑓−𝑓)_𝑝. (3.6) Using Lemma 2.2, we have 𝜔²_𝜑1𝑓,√𝑛_𝜔≤𝐶𝐾²_𝜑1𝑓,𝑛≤𝐶𝜔𝑀_𝑚,𝑑𝑓−𝑓_𝑝+1𝑛_{1≤𝑖≤𝑗≤𝑑}𝜔𝜑²_𝑖𝑗𝐷²_𝑖𝑗𝑀_𝑚,𝑑𝑓_𝑝1≤𝐶𝑛^𝑛_𝑘=1𝑛𝑘^𝛿𝜔(𝑀_𝑘,𝑑𝑓−𝑓)_𝑝. (3.7) Theorem 1.2 is completed.

Acknowledgments

This paper was supported by the Natural Science Foundation of China (nos. 11001227, 60972155), Natural Science Foundation Project of CQ CSTC (no. CSTC, 2009BB2306, CSTC2009BB2305), and the Fundamental Research Funds for the Central Universities (no. XDJK2010B005, XDJK2010C023).