Derivatives of Multivariate Bernstein Operators and Smoothness with Jacobi Weights

Abstract

Using the modulus of smoothness, directional derivatives of multivariate Bernstein operators with weights are characterized. The obtained results partly generalize the corresponding ones for multivariate Bernstein operators without weights.

1. Introduction

For the simplex 𝑆=𝑆_𝑑 in 𝑅^𝑑(𝑑=1,2,…),𝑆=𝐱=𝑥₁,𝑥₂,…,𝑥_𝑑;𝑥_𝑖≥0,𝑖=1,2,…,𝑑,|𝐱|=^𝑑_𝑖=0𝑥_𝑖≤1, (1.1)

we denote 𝐶(𝑆) the space of continuous functions on 𝑆 equipped with the norm𝑓=sup_𝐱∈𝑆𝑓(𝐱). (1.2) Let 𝑓∈𝐶(𝑆), for each 𝑛∈𝑁₀ (𝑁₀=𝑁∪{0},𝑁^𝑑₀=𝑁₀×𝑁₀×⋯×𝑁₀∈𝑅^𝑑), the multivariate Bernstein polynomial of 𝑓 is defined by𝐵_𝑛,𝑑𝑓;𝐱=_{|𝐤|≤𝑛}𝑃_𝑛,𝐤(𝐱)𝑓𝐤𝑛,𝐱∈𝑆, (1.3) where 𝐤=(𝑘₁,𝑘₂,…,𝑘_𝑑) with nonnegative integers 𝑘_𝑖(𝑖=0,1,2,…,𝑛), and𝑃_𝑛,𝐤(𝑥)=𝑛!(|𝐤!𝐱𝑛−|)^𝐤(1−|𝐱|)^{𝑛−|𝐤|},|𝐱|=∑^𝑑_𝑖=0𝑥_𝑖,|𝐤|=∑^𝑑_𝑖=0𝑘_𝑖, (1.4)

with the convention𝑘!=𝑘₁!𝑘₂!⋯𝑘_𝑑!,𝐱^𝐤=𝑥^𝑘1₁𝑥^𝑘2₂⋯𝑥^𝑘𝑑_𝑑. (1.5)

Obviously, the multivariate Bernstein operators given in (1.3) can be reduced as the classical Bernstein polynomials in case 𝑑=1, that is,𝐵_𝑛𝑓,𝑥∶=𝐵_𝑛,1𝑓,𝑥∶=^𝑛_𝑘=0𝑓𝑘𝑛𝑃_𝑛,𝑘(𝑥),𝑥∈[0,1]. (1.6)

Here introduce the crucial notations of our investigation. First, with the simplex 𝑆, we denote 𝑉_𝑆 the set of unit vectors in the directions of the edges of 𝑆 where 𝑒_𝑖 and −𝑒_𝑖 are considered to be the same vectors. That is, 𝑒_𝑖=(0,0,…,1^ith,0,…,0)(1≤𝑖≤𝑑) and 𝑒_𝑖𝑗=𝑒_𝑖−𝑒_𝑗(1≤𝑖<𝑗≤𝑑). With a direction 𝜉∈𝑉_𝑆 and a point 𝐱∈𝑆, we define the step-weight function𝜑²_𝜉(𝐱)=inf_{𝐱+𝜆𝜉∉𝑆,𝜆>0}𝑑(𝐱,𝐱+𝜆𝜉)inf_{𝐱−𝜆𝜉∉𝑆,𝜆>0}𝑑(𝐱,𝐱−𝜆𝜉), (1.7)

where 𝑑(𝐱,𝐲) is the Euclidean distance between 𝐱 and 𝐲 in 𝑅^𝑑. Obviously, as 𝐱∈𝑆, the 𝜑²_𝜉(𝑥) can further be expressed as:𝜑²_𝜉(𝐱)=𝑥_𝑖(1−|𝑥|),if𝜉=𝑒_𝑖,1≤𝑖≤𝑑,2𝑥_𝑖𝑥_𝑗𝑒if𝜉=^𝑖−𝑒_𝑗√2,1≤𝑖<𝑗≤𝑑. (1.8)It is clear that 𝜑²_𝜉(𝑥) can be reduced as the classical Bernstein polynomials’ step-weight function 𝜑²(𝑥)=𝜑_𝜉(𝑥)²=𝑥(1−𝑥)(𝑥∈[0,1]) in case 𝑑=1.

The multivariate Jacobi weight function in this paper is denoted as follows:𝜔(𝐱)=𝐱^𝛼(1−|𝐱|)^𝛽,𝐱∈𝑆, (1.9)

where 𝛼=(𝛼₁,𝛼₂,…,𝛼_𝑑)∈𝑅^𝑑, 0<𝛼_𝑖,𝛽<1,𝑖=1,2,…,𝑑,  𝐱^𝛼=𝑥^𝛼1₁𝑥^𝛼2₂,…,𝑥^𝛼𝑑_𝑑.

The rth symmetric difference of function 𝑓 with the direction 𝑒 is given byΔ^𝑟_ℎ𝑒𝑓(𝑥)=^𝑟_𝑖=0𝐶^𝑖_𝑟(−1)^𝑖𝑓𝑟𝑥+2−𝑖ℎ𝑒,if𝑥±𝑟ℎ𝑒2∈𝑆,0,otherwise. (1.10)

Using the above notation, the weighted Sobolev space in 𝑆 is then defined by𝑊_𝜙^𝑟,∞(𝑆)=𝑓∈𝐶(𝑆)∶𝜔𝑓∈𝐶(𝑆),𝑓∈𝐶𝑆^𝑟∘,𝜔𝜑^𝑟_𝑖𝑗𝐷^𝑟_𝑖𝑗𝑓∈𝐶(𝑆),1≤𝑖≤𝑗≤𝑑,𝑟∈𝑁, (1.11) where 𝑆^∘ is the inner of 𝑆.

Furthermore, the weighted 𝐾-functional is defined by𝐾^𝑟_𝜑(𝑓,𝑡^𝑟)_𝜔=inf_{𝑔∈𝑊𝜙𝑟,𝑝}𝜔𝑓−𝑔+𝑡^𝑟_{1≤𝑖≤𝑗≤𝑑}𝜔𝜑^𝑟_𝑖𝑗𝐷^𝑟_𝑖𝑗𝑔, (1.12)

and the weighted modulus isΩ^𝑟_𝜑(𝑓,𝑡)_𝜔=sup_{0<ℎ≤𝑡1≤𝑖≤𝑗≤𝑑}𝜔Δ^𝑟_{ℎ𝜑𝑖𝑗𝑒𝑖𝑗}𝑓, (1.13)

where ‖𝜔𝑓‖=max_𝑥∈𝑆|𝜔(𝑥)𝑓(𝑥)|is the weighted form. From [1], there exists a positive constant 𝐶,𝐶⁻¹𝐾^𝑟_𝜑𝑓,𝑡^𝑟_𝜔≤Ω^𝑟_𝜑𝑓,𝑡_𝜔≤𝐶𝐾^𝑟_𝜑𝑓,𝑡^𝑟_𝜔. (1.14)

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.

The close connection between the derivatives of Bernstein-type operators and the smoothness of functions has been well investigated by Ditzian, Totik, Ivanov and some other mathematicians (see [2–6], etc.) In [2], Ditzian has studied the relations between the derivatives of classical Bernstein operators 𝐵_𝑛,1(𝑓,𝑥) and the smoothness of the function 𝑓. In [7], we have presented the relation between the derivatives of classical Bernstein operators and the smoothness of function 𝑓 with Jacobi weights. Zhou has considered the approximation problems of higher-dimensional Bernstein operators with Jacobi weights, and has pointed out the unboundedness of Bernstein operators with Jacobi weights in the usual norm [8]. Because of the unboundedness of 𝐵_𝑛,𝑑(𝑓,𝐱) operators with weights in 𝐶(𝑆), he used the method of space reduction, that is,𝐶₀(𝑆)=𝑓∈𝐶(𝑆)∶𝑓(𝐱)|_{𝐱∈𝜕𝑆}=0 (1.15)

has been taken instead of 𝐶(𝑆)(𝜕𝑆 is the boundary of 𝑆). He then has shown the characteristic of the two dimensional Bernstein operators with Jacobi weights. In [1], Cao has yielded the order of approximation of d-dimensional Bernstein Operators with Jacobi weights by using the equivalence relation (1.14). In [6], Cao has evaluated extensively derivatives of the multivariate Bernstein operators on a simplex, and he proved the following.

Theorem 1.1. Let𝑓∈𝐶(𝑆),0<𝛼≤𝑟,0≤𝜆≤1,𝑟∈𝑁,and𝜉∈𝑉_𝑆, and suppose Ω^𝜉_𝑟(𝑓,𝑡)=𝑂(𝑡^𝛼),then 𝜑_𝜉^𝑟𝜆𝜕𝜕𝜉^𝑟𝐵_𝑛,𝑑𝑓,𝑥=𝑂min𝑛^2−𝜆,𝑛𝜑_𝜉^{2(1−𝜆)(𝑟−𝛼)/2}. (1.16)

In this paper, we study the characterization of derivatives of multivariate Bernstein polynomials with Jacobi weights by using the measure of smoothness in the space 𝐶₀(𝑆). The main result is expressed as follows.

Theorem 1.2. Let 𝑓∈𝐶₀(𝑆), 0<𝛼≤𝑟, 𝑟∈𝑁, and 𝜉∈𝑉_𝑆, and suppose Ω^𝜉_𝑟(𝑓,𝑡)_𝜔=𝑂(𝑡^𝛼), one has 𝜔𝜑_𝜉^2𝑟𝜕𝜕𝜉^2𝑟𝐵_𝑛,𝑑𝑓,𝑥=𝑂𝑛^𝑟−𝛼. (1.17)

Remark 1.3. Theorem 1.2 shows that the characterization of derivatives for multivariate bernstein operators with jacobi weight by using the measure of smoothness Ω^𝜉_𝑟(𝑓,𝑡)_𝜔. conversely, we conjecture that the inverse theorem is also correct, that is, 𝜔𝜑_𝜉^2𝑟𝜕𝜕𝜉^2𝑟𝑀_𝑛,𝑑𝑓,𝑥=𝑂𝑛^𝑟−𝛼⟺Ω^𝜉_𝑟(𝑓,𝑡)_𝜔=𝑂𝑡^𝛼. (1.18) The above equivalent relation without Jacobi weight has been proved in [6] when 𝜆=1. In fact, the proof of Theorem 1.2 shows that the direct part holds true, we leave the inverse part as an open problem.

2. Lemmas

To prove Theorem 1.2, some lemmas will be shown in this section.

Lemma 2.1. Consider the following; _{|𝑘|≤𝑛}𝑃_𝑛,𝑘(𝑥)𝜔⁻¹𝑘𝑛≤𝐶𝜔⁻¹(𝑥). (2.1)

Proof. When 𝑑=1, one has ^𝑛_𝑘=0𝑃_𝑛,𝑘(𝑥𝑛)𝑘+1^𝛼𝑛𝑛−𝑘+1^𝛽≤^𝑛_𝑘=0𝑃_𝑛,𝑘(𝑥𝑛)𝑘+1^{2𝛼1/2𝑛𝑘=0}𝑃_𝑛,𝑘(𝑥𝑛)𝑛−𝑘+1^2𝛽1/2=∶𝐼^1/2𝐽^1/2. (2.2) Consider different conditions,(1) if 0<2𝛼<1, 𝐼≤^𝑛_𝑘=0𝑃_𝑛,𝑘(𝑥)𝑛𝑘+1^{2𝛼𝑛𝑘=0}𝑃_𝑛,𝑘(𝑥)^1−2𝛽≤𝐶𝑥^−2𝛼, (2.3)(2) if 1<2𝛼<2,let2𝛼=1+𝑟,0≤𝑟<1, 𝐼=^𝑛_𝑘=0𝑃_𝑛,𝑘(𝑥𝑛)𝑛𝑘+1𝑘+1^𝑟≤2𝑥^𝑛𝑘=0𝑃_{𝑛+1,𝑘+1}(𝑥𝑛)𝑘+1^𝑟≤𝐶𝑥^−(1+𝑟)=𝐶𝑥^−2𝛼. (2.4)By the same methods 𝐽≤𝐶(1−𝑥)^−2𝛽 can also be given.
Suppose the lemma is correct when 𝑑−1. We prove the lemma is also correct when 𝑑. Through a simple computation, the following results can be easily obtained 𝑃_𝑛,𝑘(𝑥)=𝑃_𝑛,𝑘1𝑥₁𝑃_{𝑛−𝑘1,𝑘}𝑥1−𝑥₁, (2.5)where 𝑘=(𝑘₂,𝑘₃,…,𝑘_𝑑)𝑥=(𝑥₂,𝑥₃,…,𝑥_𝑑), 𝜔(𝑥)_{|𝑘|≤𝑛}𝑃_𝑛,𝑘(𝑥)𝜔⁻¹𝑘𝑛=𝜔(𝑥)_{|𝑘|≤𝑛}𝑃_𝑛,𝑘1𝑥₁𝑃_{𝑛−𝑘1,𝑘}𝑥1−𝑥₁𝑘₁𝑛^−𝛼1𝑘₂𝑛^−𝛼2⋯𝑘_𝑑𝑛^−𝛼𝑑|𝑘|1−𝑛^−𝛽=𝑥^𝛼1₁^𝑛_𝑘1=0𝑃_𝑛,𝑘1𝑥₁𝑥^𝛼2₂𝑥^𝛼3₃⋯𝑥^𝛼𝑑_𝑑(1−|𝑥|)^𝛽𝑘₁𝑛^−𝛼1𝑛−𝑘₁𝑛^{−|𝛼|−𝛽}×_{𝑘≤𝑛−𝑘1}𝑃_{𝑛−𝑘1,𝑘}𝑥1−𝑥₁⋅𝑘₂𝑛−𝑘₁^−𝛼2⋯𝑘_𝑑𝑛−𝑘₁^−𝛼𝑑1−𝑘𝑛−𝑘₁^−𝛽=𝑥^𝛼1₁^𝑛_𝑘1=0𝑃_𝑛,𝑘1𝑥₁𝑥^𝛼2₂𝑥^𝛼3₃⋯𝑥^𝛼𝑑_𝑑(1−|𝑥|)^𝛽𝑘₁𝑛^−𝛼1𝑛−𝑘₁𝑛^{−|𝛼|−𝛽}×_{𝑘≤𝑛−𝑘1}𝑃_{𝑛−𝑘1,𝑘}𝑥1−𝑥₁𝜔⁻¹𝑘𝑛−𝑘₁≤𝐶𝑥^𝛼1₁^𝑛_𝑘1=0𝑃_𝑛,𝑘1𝑥₁𝑥^𝛼2₂𝑥^𝛼3₃⋯𝑥^𝛼𝑑_𝑑(1−|𝑥|)^𝛽𝑘₁𝑛^−𝛼1𝑛−𝑘₁𝑛^{−|𝛼|−𝛽}𝜔⁻¹𝑥1−𝑥₁≤𝐶𝑥^𝛼1₁^𝑛_𝑘1=0𝑃_𝑛,𝑘1𝑥₁1−𝑥₁^|𝛼|+𝛽𝑘₁𝑛^−𝛼1𝑛−𝑘₁𝑛^{−|𝛼|−𝛽}≤𝐶. (2.6)

Lemma 2.2. Let𝑓∈𝐶₀(𝑆), 𝑟∈𝑁,and𝜉∈𝑉_𝑆, then 𝜔𝜑_𝜉(𝐱)^2𝑟𝜕𝜕𝜉^2𝑟𝐵_𝑛,𝑑𝑓,𝑥≤𝐶𝑛^𝑟𝜔𝑓𝑓∈𝐶₀(𝑆). (2.7)

Proof. First, we recall the discussion of theorem 4.1 of [9] that will allow us to consider lemma 1 with 𝜉=𝑒₂. it is clear that if 𝜉=𝑒_𝑖,𝑖=1,3,4,…,𝑑, we may just rename the coordinates. the following transformation will help us to complete the other case of 𝜉. the transformation 𝑇∶𝑆→𝑆 is defined by [9]𝑇𝑥𝑇∶¹,𝑥₂,…,𝑥_𝑑=𝑢₁,𝑢₂,…,𝑢_𝑑,𝑇²=𝐼, (2.8) where 𝑢_𝑖=𝑥_𝑖(𝑖≠𝑗); 𝑢_𝑗=1−|𝑥| and 𝐼 is the identity operator.
Obviously, 𝜕𝜕𝑢_𝑖=𝜕𝜕𝑥_𝑖−𝜕𝜕𝑥_𝑗𝜕,𝑖≠𝑗,𝜕𝑢_𝑗𝜕=−𝜕𝑥_𝑗, (2.9)𝐵_𝑛,𝑑𝑓;𝐱=𝐵_𝑛,𝑑𝑓_𝑇;𝑇𝐱,𝐵_𝑛,𝑑𝑓;𝑇𝐱=𝐵_𝑛,𝑑𝑓_𝑇;𝐱, (2.10) where 𝑓_𝑇(𝐮)=𝑓(𝐱) and 𝐮=𝑇𝐱. So, for 𝜉=𝑒_𝑖𝑗/√2, 1≤𝑖<𝑗≤𝑑, we have 𝜔𝜑_𝜉^2𝑟𝜕𝜕𝜉^2𝑟𝐵_𝑛,𝑑𝑓=𝜔_𝑇𝜑_𝑒^2𝑟_𝑖𝜕𝜕𝑢_𝑖^𝑟𝐵_𝑛,𝑑𝑓_𝑇≤𝐶𝑛^𝑟𝜔_𝑇𝑓_𝑇≤𝐶𝑛^𝑟𝜔𝑓. (2.11)Secondly, we prove 𝜔𝜑_𝑒^2𝑟₂𝜕𝜕𝑥₂^2𝑟𝐵_𝑛,𝑑𝑓≤𝐶𝑛^𝑟𝜔𝑓. (2.12) In The following we use mathematical induction on the dimension number 𝑑 to prove (2.12). When 𝑑=1, Lemma 3.2 in [10] proved the above inequality for 𝑟=1, for 𝑟>1, from the expression of derivatives of Bernstein operator in [4] (page125,(9.4.3)), we can easily prove it. Next, suppose that (2.12) is valid for 𝑑−1(𝑑>1); we prove (2.12) is also true for 𝑑. Assume 𝑆^′=𝐱∶𝑥₁,𝐱∈𝑆_𝑑,𝐱=𝑥₂,𝑥₃,…,𝑥_𝑑,𝐤=𝑘₂,𝑘₃…,𝑘_𝑑,𝐤=𝑘₁,𝐤. (2.13)
Let 𝐳=𝐱/(1−𝑥₁)=(𝑥₂/(1−𝑥₁),𝑥₃/(1−𝑥₁),…,𝑥_𝑑/(1−𝑥₁)).𝜔(𝐱) can therefore be rewritten as 𝜔(𝐱)=𝑥^𝛼1₁1−𝑥₁^|𝛼|+𝛽𝜔(𝐳), (2.14) and 𝐵_𝑛,𝑑(𝑓,𝐱) can be decomposed as 𝐵_𝑛,𝑑𝑓,𝐱=^𝑛_𝑘1=0𝑝_𝑛,𝑘1𝑥₁𝐵_{𝑛−𝑘1,𝑑−1}(𝐻,𝐳), (2.15)where 𝐻(𝐮)=𝑓(𝑘₁/𝑛,(1−𝑘₁/𝑛)𝑢). Using the inductive assumption, we have 𝜔𝜑_𝑒^2𝑟₂𝜕𝜕𝑥₂^2𝑟𝐵_𝑛,𝑑𝑓=𝑥^𝛼1₁1−𝑥₁^{|𝛼|+𝛽𝑛}_𝑘1=0𝑝_𝑛,𝑘1𝑥₁𝑧^𝛼2₁⋯𝑧^𝛼𝑑_𝑑−1(1−|𝑧|)^𝛽𝜑_𝑒^2𝑟₁(𝑧)𝜕𝜕𝑧₁^2𝑟𝐵_{𝑛−𝑘1,𝑑−1}(𝐻,𝐳)=𝑥^𝛼1₁1−𝑥₁^{|𝛼|+𝛽𝑛}_𝑘1=0𝑝_𝑛,𝑘1𝑥₁𝜔(𝐳)𝜑_𝑒^2𝑟₁(𝑧)𝜕𝜕𝑧₁^2𝑟𝐵_{𝑛−𝑘1,𝑑−1}(𝐻,𝐳)≤𝑥^𝛼1₁1−𝑥₁^{|𝛼|+𝛽𝑛}_𝑘1=0𝑝_𝑛,𝑘1𝑥₁𝐶(𝑛−𝑘₁)^𝑟max_{𝑍∈𝑆𝑑−1}|𝜔(𝑧)𝐻(𝑧)|≤𝐶𝑛^𝑟𝜔𝑓𝑥^𝛼1₁1−𝑥₁^{|𝛼|+𝛽𝑛}_𝑘1=0𝑝_𝑛,𝑘1𝑥₁𝑘₁𝑛^−𝛼1𝑛−𝑘₁𝑛^{−|𝛼|−𝛽}≤𝐶𝑛^𝑟𝜔𝑓. (2.16)Here, the equality 𝜔(𝑧)𝐻(𝑧)=𝑘₁𝑛^−𝛼1𝑛−𝑘₁𝑛^{−|𝛼|−𝛽}𝑘𝜔𝑓¹𝑛,𝑘1−¹𝑛𝑧, (2.17) and the inequality 𝑥^𝛼1₁(1−𝑥₁)^{|𝑛𝛼|+𝛽}_𝑘1=0𝑝_𝑛,𝑘1𝑥₁𝑘₁𝑛^−𝛼1𝑛−𝑘₁𝑛^{−|𝛼|−𝛽}≤𝐶 (2.18) have been used in the proof of (2.16). The proof of Lemma 2.2 is complete.

Lemma 2.3. Let 𝑟∈𝑁and𝜉∈𝑉_𝑆, then 𝜔𝜑_𝜉(𝐱)^2𝑟𝜕𝜕𝜉^2𝑟𝐵_𝑛,𝑑𝑓,𝑥≤𝐶𝜔𝜑_𝜉^2𝑟𝜕𝜕𝜉^2𝑟𝑓𝑓∈𝐶^𝑟₀(𝑆). (2.19)

Proof. By (2.10), for 𝜂=𝑒_𝑖,𝑢=𝑇𝑥and𝜉=𝑒_𝑖𝑗/√2, 1≤𝑖<𝑗≤𝑑, we have𝜔𝜑_𝜉^2𝑟𝜕𝜕𝜉^2𝑟𝐵_𝑛,𝑑𝑓,𝐱=𝜔_𝑇𝜑_𝜂^2𝑟𝜕𝜕𝜂^2𝑟𝐵_𝑛,𝑑𝑓_𝑇𝜔,𝑇𝐱≤𝐶_𝑇𝜑_𝜂^2𝑟𝜕𝜕𝜂^2𝑟𝑓_𝑇(𝐮)≤𝐶𝜔𝜑_𝜉^2𝑟𝜕𝜕𝜉^2𝑟𝑓_𝑝. (2.20) Similar to the discussion in the proof of Lemma 2.2., we need only to prove the case of 𝜉=𝑒₂, that is,𝜔𝜑_𝑒^2𝑟₂𝜕𝜕𝑥₂^2𝑟𝐵_𝑛,𝑑𝑓,𝐱≤𝐶𝜔𝜑_𝑒^2𝑟₂𝜕𝜕𝑥₂^2𝑟𝑓. (2.21)
The steps to prove (2.21) are similar to those to prove the inequality (2.12). Hence, the proof of Lemma 2.3 is complete

3. Proof of Theorem

We will prove Theorem 1.2 in the followings. For 𝜉=𝑒₂ and for all 𝑔∈𝑊_𝜙^𝑟,∞(𝑆), it follows from Lemmas 2.2 and 2.3 that𝜔𝜑_𝑒^2𝑟₂𝜕𝜕𝑥₂^2𝑟𝐵_𝑛,𝑑≤𝑓,𝑥𝜔𝜑_𝑒^2𝑟₂𝜕𝜕𝑥₂^2𝑟𝐵_𝑛,𝑑𝑓−𝑔,𝑥+𝜔𝜑_𝑒^2𝑟₂𝜕𝜕𝑥₁^2𝑟𝐵_𝑛,𝑑𝑛𝑔,𝑥≤𝐶^𝑟𝜔𝑓−𝑔+𝜔𝜑_𝑒^2𝑟₂𝜕𝜕𝑥₂^2𝑟𝑔. (3.1)

From the definition of 𝐾-functional and (1.14), we obtain𝜔𝜑_𝑒^2𝑟₂𝜕𝜕𝑥₂^2𝑟𝐵_𝑛,𝑑𝑓,𝑥≤𝐶𝑛^𝑟𝐾^𝑒2_𝑟𝑓;𝑛^−𝑟_𝜔≤𝐶𝑛^𝑟Ω^𝑒2_𝑟1𝑓,𝑛_𝜔≤𝐶𝑛^𝑟−𝛼. (3.2)

Similarly, the case of 𝜉=𝑒_𝑖, 𝑖=1,3,4,…,𝑑 can also be proved. If 𝜉=((𝑒_𝑖−𝑒_𝑗)/√2)1≤𝑖<𝑗≤𝑑, it is not difficult to obtain𝜔𝜑_𝜉^2𝑟𝜕𝜕𝜉^2𝑟𝐵_𝑛,𝑑𝑓,𝑥=𝜔_𝑇𝜑_𝜂^2𝑟𝜕𝜕𝜂^2𝑟𝐵_𝑛,𝑑𝑓_𝑇,𝑢≤𝐶𝑛^𝑟−𝛼, (3.3) by assuming 𝜂=𝑒_𝑖,𝑢=𝑇𝑥. The proof of Theorem 1.2 is complete.

Acknowledgment

This paper was supported by Natural Science Foundation of China (nos. 11001227, 11171275, 60972155, and 61105041), Natural Science Foundation Project of Chongqing (nos. CSTC, 2009BB2306, CSTC, 2009BB2305), and the Fundamental Research Funds for the Central Universities (nos. XDJK2010B005 and XDJK2010C023).