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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 610649, 9 pages
http://dx.doi.org/10.1155/2012/610649
Research Article

Boundedness of Weighted Hardy Operator and Its Adjoint on Triebel-Lizorkin-Type Spaces

Department of Mathematics, Dalian Maritime University, Liaoning, Dalian, 116026, China

Received 26 October 2010; Accepted 18 November 2010

Academic Editor: Hans Triebel

Copyright © 2012 Canqin Tang and Ruohong Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝑝[1,], 𝑞[1,), 𝜏(0,), and 𝛼(0,1) such that 𝜏>1/𝑝1/𝑞 and 𝛼𝑛(1/𝑝𝜏), let 𝑈𝜓 be the weighted Hardy operator and 𝑉𝜓 its adjoint operator with respect to the weight function 𝜓. In this paper, the authors establish a sufficient and necessary condition on weight function 𝜓 to ensure the boundedness of 𝑈𝜓 and 𝑉𝜓 on the Triebel-Lizorkin-type spaces ̇𝐹𝛼,𝜏𝑝,𝑞(𝑛) and their predual spaces, Triebel-Lizorkin-Hausdorff spaces, which unify and generalize the known results on 𝑄-type spaces.

1. Introduction

This paper focuses on the boundedness of the weighted Hardy operator 𝑈𝜓 and its adjoint operator 𝑉𝜓 on Triebel-Lizorkin-type spaces and their predual spaces. Recall that, for a fixed function 𝜓[0,1][0,), the weighted Hardy operator 𝑈𝜓 is defined by 𝑈𝜓𝑓(𝑥)=10𝑓(𝑡𝑥)𝜓(𝑡)𝑑𝑡,𝑥𝑛,(1.1) see Carton-Lebrun and Fosset [1]. Accordingly, the adjoint operator 𝑉𝜓 of 𝑈𝜓, named by the weighted Cesàro average operator, is defined by 𝑉𝜓𝑓(𝑥)=10𝑓𝑥𝑡𝑡𝑛𝜓(𝑡)𝑑𝑡,𝑥𝑛.(1.2)

This paper is motivated by the following facts. On one hand, 𝑈𝜓 is related to the Hardy-Littlewood maximal operators in harmonic analysis. If 𝜓1 and 𝑛=1, then 𝑈𝜓 goes back to the classical Hardy-Littlewood average 𝑈: 1𝑈𝑓(𝑥)=𝑥𝑥0𝑓(𝑥)𝑑𝑦,𝑥0.(1.3) Its adjoint operator 𝑉𝜓 is the classical Cesàro average operator: 𝑉𝑓(𝑥)=𝑥𝑓(𝑦)𝑦𝑑𝑦,𝑥>0,𝑥𝑓(𝑦)𝑦𝑑𝑦,𝑥<0.(1.4)

Xiao in [2] obtained the boundedness of 𝑈𝜓 on BMO(𝑛), the boundedness of 𝑉𝜓 on 𝐻1(𝑛), and their corresponding operator norms. Recall that 𝑄𝛼(𝑛) can be viewed as a generalization of BMO(𝑛) (cf. [3]). Recently Tang and Zhai in [4] obtained the boundedness and norm of 𝑈𝜓 on 𝑄𝛼(𝑛). They even work to more general spaces 𝑄𝑝𝛼,𝑞(𝑛). The boundedness of 𝑈𝜓 and 𝑉𝜓 on 𝑄𝑝𝛼,𝑞(𝑛) and its dual space are worked out. Recall that 𝑄 spaces (𝑄𝛼 spaces) were originally defined by Aulaskari et al. [5] in 1995 as spaces of holomorphic functions on the unit disk, which are geometric in the sense that they transform naturally under conformal mappings, and later, in 2000, were extended to the 𝑛-dimensional Euclidean spaces 𝑛 by Essén et al. By [3, Theorem 2.3], 𝑄𝛼(𝑛) is always a subspace of BMO(𝑛). As a generation of 𝑄 spaces, the spaces 𝑄𝑝𝛼,𝑞(𝑛) when 𝛼(0,1) and 2𝑞<𝑝< were first introduced by Cui and Yang [6] and later extended to 𝛼(0,1), 𝑝(0,], and 𝑞[1,] in [7]. We also refer to [4, 811] for more studies on these spaces.

On the other hand, it is well known that Besov spaces ̇𝐵𝛼𝑝,𝑞(𝑛) and Triebel-Lizorkin spaces ̇𝐹𝛼𝑝,𝑞(𝑛) allow a unified approach to various types of function spaces, such as Hölder-Zygmund spaces, Soblev spaces, Hardy spaces, BMO, and Bessel-potential spaces. Over hundreds of papers and books are focused on these spaces and their applications. Very recently, Yang and Yuan [7, 12] introduced the scales of homogeneous Besov-Triebel-Lizorkin-type spaces ̇𝐵𝛼,𝜏𝑝,𝑞(𝑛) and ̇𝐹𝛼,𝜏𝑝,𝑞(𝑛) for all 𝛼, 𝜏[0,), and 𝑝,𝑞(0,]. These spaces unify and generalize the homogeneous Besov spaces ̇𝐵𝛼𝑝,𝑞(𝑛), Triebel-Lizorkin spaces ̇𝐹𝛼𝑝,𝑞(𝑛), and the spaces 𝑄𝛼(𝑛) simultaneously. The following equivalent definition of the spaces ̇𝐹𝛼,𝜏𝑝,𝑞(𝑛) were given in [7, Section 3].

Definition 1.1. Let 𝛼(0,1), 𝑝(0,), 𝑞(1,), and 𝜏[0,]. The space ̇𝐹𝛼,𝜏𝑝,𝑞(𝑛) is defined to be the space of all measurable functions 𝑓 on 𝑛 such that 𝑓̇𝐹𝛼,𝜏𝑝,𝑞(𝑛)1=sup||𝐼||𝜏𝐼|𝑥𝑦|<2𝑙(𝐼)||𝑓||(𝑥)𝑓(𝑦)𝑞||||𝑥𝑦𝑛+𝑞𝛼𝑑𝑦𝑝/𝑞𝑑𝑥1/𝑝<,(1.5) where the supremum is taken over all cubes 𝐼 with the edges parallel to the coordinate axes in 𝑛.

Motivated by the above facts, a natural question is following: under what condition on 𝜓 the weighted Hardy operator and its adjoint are bounded on Triebel-Lizorkin-type spaces ̇𝐹𝛼,𝜏𝑝,𝑞(𝑛)? The main purpose of this paper is to give the sufficient and necessary condition on 𝜓 such that the operators 𝑈𝜓 and 𝑉𝜓 are bounded on Triebel-Lizorkin-type spaces ̇𝐹𝛼,𝜏𝑝,𝑞(𝑛). We have the following main results.

Theorem 1.2. Let 𝜓[0,1][0,) be a function, and let 𝑝[1,], 𝑞[1,), 𝜏(0,), and 𝛼(0,1) such that 𝜏>1/𝑝1/𝑞 and 𝛼𝑛(1/𝑝𝜏). Then 𝑈𝜓̇𝐹𝛼,𝜏𝑝,𝑞(𝑛̇𝐹)𝛼,𝜏𝑝,𝑞(𝑛) exists as a bounded operator if 10𝑡𝑛(1/𝑝𝜏)+𝛼𝜓(𝑡)𝑑𝑡<.(1.6) Moreover, when (1.6) holds, the norm of 𝑈𝜓 on ̇𝐹𝛼,𝜏𝑝,𝑞(𝑛) is given by 𝑈𝜓̇𝐹𝛼,𝜏𝑝,𝑞(𝑛̇𝐹)𝛼,𝜏𝑝,𝑞(𝑛)=10𝑡𝑛(1/𝑝𝜏)+𝛼𝜓(𝑡)𝑑𝑡.(1.7)

We give the proof of Theorem 1.2 in Section 2.

Finally, we make some conventions on notations. Throughout the paper, 𝑛 denotes the 𝑛-dimensional Euclidean space, with Euclidean norm |𝑥|, and Lebesgue measure 𝑑𝑥. A cube 𝐼 will always mean a cube in 𝑛 with the edges parallel to the coordinate axes with sidelength 𝑙(𝐼) and volume |𝐼|. The dilated cube 𝜆𝐼, 𝜆>0, is the cube with the same center as 𝐼 and sidelength 𝜆𝑙(𝐼). For all 𝑞(0,], 𝑞 denotes the adjoint index of 𝑞, namely, 1/𝑞+1/𝑞=1. 𝐶 will often be used to denote a positive constant, but it may vary from line to line.

2. Proof of Main Results

We begin with the proof of Theorem 1.2.

Proof of Theorem 1.2. Suppose (1.6) holds. If ̇𝐹𝑓𝛼,𝜏𝑝,𝑞(𝑛), then for any cube 𝐼𝑛, applying Minkowski's inequality, we have 𝐼|𝑥𝑦|<2𝑙(𝐼)||𝑈𝜓𝑓(𝑥)𝑈𝜓||𝑓(𝑦)𝑞||||𝑥𝑦𝑛+𝑞𝛼𝑑𝑦𝑝/𝑞𝑑𝑥1/𝑝=𝐼|𝑥𝑦|<2𝑙(𝐼)|||10[]|||𝑓(𝑡𝑥)𝑓(𝑡𝑦)𝜓(𝑡)𝑑𝑡𝑞||||𝑥𝑦𝑛+𝑞𝛼𝑑𝑦𝑝/𝑞𝑑𝑥1/𝑝𝐼10|𝑥𝑦|<2𝑙(𝐼)|𝑓(𝑡𝑥)𝑓(𝑡𝑦)|𝑞||||𝑥𝑦𝑛+𝑞𝛼𝑑𝑦1/𝑞𝜓(𝑡)𝑑𝑡𝑝𝑑𝑥1/𝑝10𝐼|𝑥𝑦|<2𝑙(𝐼)||||𝑓(𝑡𝑥)𝑓(𝑡𝑦)𝑞||||𝑥𝑦𝑛+𝑞𝛼𝑑𝑦𝑝/𝑞𝑑𝑥1/𝑝𝜓(𝑡)𝑑𝑡.(2.1) Notice that 𝐼|𝑥𝑦|<2𝑙(𝐼)||||𝑓(𝑡𝑥)𝑓(𝑡𝑦)𝑞||||𝑥𝑦𝑛+𝑞𝛼𝑑𝑦𝑝/𝑞=𝑑𝑥𝑡𝐼|𝑢𝑣|<2𝑙(𝑡𝐼)||||𝑓(𝑢)𝑓(𝑣)𝑞|𝑢𝑣|𝑛+𝑞𝛼𝑑𝑣𝑝/𝑞𝑑𝑢𝑡𝑝𝛼𝑛.(2.2) Then, by (1.6), we have 1||𝐼||𝜏𝐼|𝑥𝑦|<2𝑙(𝐼)||𝑈𝜓𝑓(𝑥)𝑈𝜓||𝑓(𝑦)𝑞||||𝑥𝑦𝑛+𝑞𝛼𝑑𝑦𝑝/𝑞𝑑𝑥1/𝑝1||||𝑡𝐼𝜏10𝑡𝐼|𝑢𝑣|<2𝑙(𝑡𝐼)||||𝑓(𝑢)𝑓(𝑣)𝑞|𝑢𝑣|𝑛+𝑞𝛼𝑑𝑣𝑝/𝑞𝑑𝑢1/𝑝𝑡𝛼𝑛/𝑝+𝑛𝜏𝜓(𝑡)𝑑𝑡𝑓̇𝐹𝛼,𝜏𝑝,𝑞(𝑛)10𝑡𝛼𝑛(1/𝑝𝜏)𝜓(𝑡)𝑑𝑡.(2.3) Thus, the operator 𝑈𝜓 is bounded on ̇𝐹𝛼,𝜏𝑝,𝑞(𝑛) with operator norm no more than 10𝑡𝛼𝑛(1/𝑝𝜏)𝜓(𝑡)𝑑𝑡.(2.4)
Conversely, if 𝑈𝜓 is bounded on ̇𝐹𝛼,𝜏𝑝,𝑞(𝑛), then we can choose the function 𝑓0(𝑥)=|𝑥|𝑛(1/𝑝𝜏)+𝛼,𝑥𝑛𝑙,|𝑥|𝑛(1/𝑝𝜏)+𝛼,𝑥𝑛𝑟,(2.5) where 𝑛𝑙 and 𝑛𝑟 denote, respectively, the left and right halves of 𝑛, separated by the hyperplane 𝑥1=0 (𝑥1 is the first coordinate of 𝑥𝑛).
We now show that 0<𝑓0̇𝐹𝛼,𝜏𝑝,𝑞(𝑛)<. Notice that the assumption on 𝛼 implies that 𝜏<1/𝑝. For any cube 𝐼 in 𝑛, we consider two cases.
Case 1. If 𝑝<𝑞, then 𝑝/𝑞<1. Since 𝜏>1/𝑝1/𝑞, then 𝛼>0>𝑛(1/𝑝1/𝑞)𝑛𝜏, and hence 𝐼+||𝑓0(||𝑧)𝑞𝑑𝑧𝐼{|𝑧|||}|𝑧|[𝑛(1/𝑝𝜏)+𝛼]𝑞𝑑𝑧+𝐼{|𝑧|>||}|𝑧|[𝑛(1/𝑝𝜏)+𝛼]𝑞||||𝑑𝑧[𝑛(1/𝑝𝜏)+𝛼]𝑞+𝑛+||||[𝑛(1/𝑝𝜏)+𝛼]𝑞||𝐼||,(2.6) which together with Hölder's inequality, 𝜏>1/𝑝1/𝑞, yields that 𝐼||<2𝑙(𝐼)||𝑓0(||𝑥+)𝑞||||𝑛+𝑞𝛼𝑑𝑝/𝑞𝑑𝑥𝐼||<2𝑙(𝐼)||𝑓0||(𝑥+)𝑞||||𝑛+𝑞𝛼𝑑𝑑𝑥𝑝/𝑞||𝐼||1𝑝/𝑞=||<2𝑙(𝐼)𝐼+||𝑓0||(𝑧)𝑞1𝑑𝑧||||𝑛+𝑞𝛼𝑑𝑝/𝑞||𝐼||1𝑝/𝑞||<2𝑙(𝐼)||||[𝑛(1/𝑝𝜏)+𝛼]𝑞+𝑛+||||[𝑛(1/𝑝𝜏)+𝛼]𝑞||𝐼||||||𝑛+𝑞𝛼𝑑𝑝/𝑞||𝐼||1𝑝/𝑞||<2𝑙(𝐼)||||[𝑛(1/𝑝𝜏)+𝛼]𝑞+𝑛+||||[𝑛(1/𝑝𝜏)+𝛼]𝑞||𝐼||||||𝑛+𝑞𝛼𝑑𝑝/𝑞||𝐼||1𝑝/𝑞𝐶𝑙(𝐼)𝑛𝜏𝑝.(2.7)Case 2. If 𝑝𝑞, then 𝑝/𝑞1. Using Minkowski's inequality, similar to the computation of Case 1, we also obtain 𝐼||<2𝑙(𝐼)||𝑓0||(𝑥+)𝑞||||𝑛+𝑞𝛼𝑑𝑝/𝑞𝑑𝑥1/𝑝||<2𝑙(𝐼)𝐼||𝑓0||(𝑥+)𝑞||||𝑛+𝑞𝛼𝑝/𝑞𝑑𝑥𝑞/𝑝𝑑1/𝑞𝐶𝑙(𝐼)𝑛𝜏.(2.8) Combining Cases 1 and 2 yields that 1||𝐼||𝜏𝐼||<2𝑙(𝐼)||𝑓0(𝑥+)𝑓0||(𝑥)𝑞||||𝑛+𝑞𝛼𝑑𝑝/𝑞𝑑𝑥1/𝑝1||𝐼||𝜏𝐼||<2𝑙(𝐼)||𝑓0||(𝑥+)𝑞+||𝑓0||(𝑥)𝑞||||𝑛+𝑞𝛼𝑑𝑝/𝑞𝑑𝑥1/𝑝𝐶,(2.9) namely, 𝑓0̇𝐹𝛼,𝜏𝑝,𝑞(𝑛) and 0𝑓0̇𝐹𝛼,𝜏𝑝,𝑞(𝑛)<.
Since 𝑈𝜓𝑓0(𝑥)=𝑓0(𝑥)10𝑡𝑛(1/𝑝𝜏)+𝛼𝜓(𝑡)𝑑𝑡,(2.10) we then have 𝑈𝜓̇𝐹𝛼,𝜏𝑝,𝑞(𝑛̇𝐹)𝛼,𝜏𝑝,𝑞(𝑛)=10𝑡(𝑛/𝑝𝜏)+𝛼𝜓(𝑡)𝑑𝑡,(2.11) which completes the proof of Theorem 1.2.

Remark 2.1. In fact, if we choose 𝑝=𝑞 and 𝜏=1/𝑝1/𝑟, then 10𝑡𝛼𝑛(1/𝑝𝜏)𝜓(𝑡)𝑑𝑡=10𝑡𝛼𝑛/𝑟𝜓(𝑡)𝑑𝑡.(2.12) Therefore, Theorem 1.2 is consistent with [4, Theorem 4.1] since 𝑄𝑟𝛼,𝑝(𝑛̇𝐹)=𝛼,1/𝑝1/𝑟𝑝,𝑝(𝑛).

The weighted Hardy operator 𝑈𝜓 and the weighted Cesàro average operator 𝑉𝜓 are adjoint mutually, namely, 𝑛𝑔(𝑥)𝑈𝜓𝑓(𝑥)𝑑𝑥=𝑛𝑓(𝑥)𝑉𝜓𝑔(𝑥)𝑑𝑥.(2.13) Recall that 𝑉𝜓𝑓(𝑥)=10𝑓𝑥𝑡𝑡𝑛𝜓(𝑡)𝑑𝑡,𝑥𝑛,(2.14) then we have 𝐼|𝑥𝑦|<2𝑙(𝐼)||𝑉𝜓𝑓(𝑥)𝑉𝜓||𝑓(𝑦)𝑞||||𝑥𝑦𝑛+𝑞𝛼𝑑𝑦𝑝/𝑞𝑑𝑥1/𝑝=𝐼|𝑥𝑦|<2𝑙(𝐼)|||10[]𝑡𝑓(𝑥/𝑡)𝑓(𝑦/𝑡)𝑛|||𝜓(𝑡)𝑑𝑡𝑞||||𝑥𝑦𝑛+𝑞𝛼𝑑𝑦𝑝/𝑞𝑑𝑥1/𝑝𝐼10|𝑥𝑦|<2𝑙(𝐼)||||𝑓(𝑥/𝑡)𝑓(𝑦/𝑡)𝑞||||𝑥𝑦𝑛+𝑞𝛼𝑑𝑦1/𝑞𝑡𝑛𝜓(𝑡)𝑑𝑡𝑝𝑑𝑥1/𝑝10𝐼|𝑥𝑦|<2𝑙(𝐼)||||𝑓(𝑥/𝑡)𝑓(𝑦/𝑡)𝑞||||𝑥𝑦𝑛+𝑞𝛼𝑑𝑦𝑝/𝑞𝑑𝑥1/𝑝𝑡𝑛𝜓(𝑡)𝑑𝑡,(2.15) where 𝐼|𝑥𝑦|<2𝑙(𝐼)||||𝑓(𝑥/𝑡)𝑓(𝑦/𝑡)𝑞||||𝑥𝑦𝑛+𝑞𝛼𝑑𝑦𝑝/𝑞=𝑑𝑥(1/𝑡)𝐼|𝑢𝑣|<2𝑙((1/𝑡)𝐼)||||𝑓(𝑢)𝑓(𝑣)𝑞|𝑢𝑣|𝑛+𝑞𝛼𝑑𝑣𝑝/𝑞𝑑𝑢𝑡𝑝𝛼+𝑛.(2.16)

Similar to the proof of Theorem 1.2, we have next theorem.

Theorem 2.2. Let 𝜓[0,1][0,) be a function, and let 𝑝[1,], 𝑞[1,), 𝜏(0,), and 𝛼(0,1) such that 𝜏>1/𝑝1/𝑞 and 𝛼𝑛(1/𝑝𝜏). Then 𝑉𝜓̇𝐹𝛼,𝜏𝑝,𝑞(𝑛̇𝐹)𝛼,𝜏𝑝,𝑞(𝑛) exists as a bounded operator if 10𝑡𝑛(11/𝑝+𝜏)𝛼𝜓(𝑡)𝑑𝑡<.(2.17)
Moreover, when (2.17) holds, the norm of 𝑉𝜓 on ̇𝐹𝛼,𝜏𝑝,𝑞(𝑛) is given by 𝑉𝜓̇𝐹𝛼,𝜏𝑝,𝑞(𝑛̇𝐹)𝛼,𝜏𝑝,𝑞(𝑛)=10𝑡𝑛(11/𝑝+𝜏)𝛼𝜓(𝑡)𝑑𝑡.(2.18)

Recall that the Triebel-Lizorkin-Hausdorff spaces 𝐹̇𝐻𝑝𝛼,𝜏,𝑞(𝑛) were originally introduced in [7, 12] and proved therein to be the predual spaces of Triebel-Lizorkin-type spaces ̇𝐹𝛼,𝜏𝑝,𝑞(𝑛). Triebel-Lizorkin-Hausdorff spaces unify and generalize Triebel-Lizorkin spaces (see, e.g., [13]) and Hardy-Hausdorff spaces 𝐻𝐻1𝛼 in [14]. These spaces were further studied in [15].

By dual argument, we have the following theorems.

Theorem 2.3. Let 𝜓[0,1][0,) be a function, and let 𝑝[1,], 𝑞(1,], 𝜏(0,1/(max{𝑝,𝑞})], and 𝛼(0,1) such that 𝜏>1/𝑝1/𝑞 and 𝛼𝑛(1/𝑝𝜏). Then 𝑈𝜓̇𝐻𝐹𝛼,𝜏𝑝,𝑞(𝑛̇𝐻)𝐹𝛼,𝜏𝑝,𝑞(𝑛) exists as a bounded operator if 10𝑡𝑛(1/𝑝𝜏)𝛼𝜓(𝑡)𝑑𝑡<.(2.19) Moreover, when (2.19) holds, the norm of 𝑈𝜓 on 𝐹̇𝐻𝛼,𝜏𝑝,𝑞(𝑛) is given by 𝑈𝜓𝐹̇𝐻𝛼,𝜏𝑝,𝑞(𝑛̇𝐻)𝐹𝛼,𝜏𝑝,𝑞(𝑛)=10𝑡𝑛(1/𝑝𝜏)𝛼𝜓(𝑡)𝑑𝑡.(2.20)

Theorem 2.4. Let 𝜓[0,1][0,) be a function, and let 𝑝[1,], 𝑞(1,], 𝜏(0,1/(max{𝑝,𝑞})], and 𝛼(0,1) such that 𝜏>1/𝑝1/𝑞 and 𝛼𝑛(1/𝑝𝜏). Then 𝑉𝜓̇𝐻𝐹𝛼,𝜏𝑝,𝑞(𝑛̇𝐻)𝐹𝛼,𝜏𝑝,𝑞(𝑛) exists as a bounded operator if 10𝑡𝑛(11/𝑝+𝜏)𝛼𝜓(𝑡)𝑑𝑡<.(2.21) Moreover, when (2.21) holds, the norm of 𝑉𝜓 on 𝐹̇𝐻𝛼,𝜏𝑝,𝑞(𝑛) is given by 𝑉𝜓𝐹̇𝐻𝛼,𝜏𝑝,𝑞(𝑛̇𝐻)𝐹𝛼,𝜏𝑝,𝑞(𝑛)=10𝑡𝑛(11/𝑝+𝜏)𝛼𝜓(𝑡)𝑑𝑡.(2.22)

Remark 2.5. When 𝑝=𝑞 and 𝜏=1/𝑞1/𝑟, then 10𝑡𝑛(11/𝑝+𝜏)𝛼𝜓(𝑡)𝑑𝑡=10𝑡𝑛(11/𝑟)𝛼𝜓(𝑡)𝑑𝑡,(2.23) and Theorem 2.4 is consistent with [4, Theorem 5.8] since 𝐹̇𝐻𝛼,1/𝑞1/𝑟𝑞,𝑞(𝑛) is predual space of 𝑄𝑟𝛼,𝑞(𝑛).

Our Theorems actually induce the following results: 𝑉𝜓𝐹̇𝐻𝛼,𝜏𝑝,𝑞(𝑛̇𝐻)𝐹𝛼,𝜏𝑝,𝑞(𝑛)=𝑈𝜓𝐹̇𝐻𝛼,𝜏𝑝,𝑞(𝑛̇𝐻)𝐹𝛼,𝜏𝑝,𝑞(𝑛),𝑈𝜓𝐹̇𝐻𝛼,𝜏𝑝,𝑞(𝑛̇𝐻)𝐹𝛼,𝜏𝑝,𝑞(𝑛)=𝑉𝜓𝐹̇𝐻𝛼,𝜏𝑝,𝑞(𝑛̇𝐻)𝐹𝛼,𝜏𝑝,𝑞(𝑛).(2.24)

In particular, 𝑈𝜓𝐹̇𝐻𝛼,1/𝑞1/𝑝𝑞,𝑞(𝑛̇𝐻)𝐹𝛼,1/𝑞1/𝑝𝑞,𝑞(𝑛)=𝑉𝜓𝑄𝑝𝛼,𝑞(𝑛)𝑄𝑝𝛼,𝑞(𝑛),𝑉𝜓𝐹̇𝐻𝛼,1/𝑞1/𝑝𝑞,𝑞(𝑛̇𝐻)𝐹𝛼,1/𝑞1/𝑝𝑞,𝑞(𝑛)=𝑈𝜓𝑄𝑝𝛼,𝑞(𝑛)𝑄𝑝𝛼,𝑞(𝑛).(2.25)

Acknowledgments

The authors cordially thank Professor D. C. Yang and Dr. W. Yuan for their valuable comments. This work was supported by the Fundamental Research Funds for the Central Universities (2011QN058) and (2011QN150).

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