A New Characterization of Generalized Weighted Composition Operators from the Bloch Space into the Zygmund Space

Abstract

A new criterion for the boundedness and the compactness of the generalized weighted composition operators from the Bloch space into the Zygmund space is given in this paper.

1. Introduction

Let 𝐻(𝔻) be the space of analytic functions on the unit disk 𝔻={𝑧∶|𝑧|<1}. An 𝑓∈𝐻(𝔻) is said to belong to the Bloch space, denoted by ℬ, if 𝑓_ℬ=𝑓(0)+sup_𝑧∈𝔻1−|𝑧|²𝑓^′(𝑧)<∞. (1) Let ℬ₀ be the little Bloch space, which consists of those 𝑓∈ℬ such that (1−|𝑧|²)|𝑓^′(𝑧)|→0, as |𝑧|→1. Denote by 𝒵 the set of all 𝑓∈𝐻(𝔻)⋂𝐶(𝔻) such that 𝑓𝑓𝑒=sup^{𝑖(𝜃+ℎ)}+𝑓𝑒^{𝑖(𝜃−ℎ)}−2𝑓𝑒^𝑖𝜃ℎ<∞, (2) where the supremum is taken over all 𝑒^𝑖𝜃∈𝜕𝔻 and ℎ>0. In fact, 𝑓∈𝒵 if and only if sup_𝑧∈𝔻(1−|𝑧|²)|𝑓^′′(𝑧)|<∞ by Theorem 5.3 of [1] and the Closed Graph Theorem. The class 𝒵 with the norm 𝑓_𝒵=𝑓(0)+𝑓^′(0)+sup_𝑧∈𝔻1−|𝑧|²𝑓^′′(𝑧) (3) is a Banach space. We call 𝒵 the Zygmund space. For more information on the Zygmund space, see, for example, [1, 2].

Let 𝑆(𝔻) denote the set of analytic self-map of 𝔻. Associated with 𝜑∈𝑆(𝔻) is the composition operator 𝐶_𝜑, which is defined by 𝐶_𝜑(𝑓)(𝑧)=𝑓(𝜑(𝑧)) for 𝑧∈𝔻 and 𝑓∈𝐻(𝔻). We refer the readers to the book [3] for the theory of the composition operator on various function spaces.

Let 𝑢∈𝐻(𝔻). The weighted composition operator, denoted by 𝑢𝐶_𝜑, is defined by 𝑢𝐶_𝜑𝑓(𝑧)=𝑢(𝑧)𝑓𝜑(𝑧),𝑧∈𝔻,𝑓∈𝐻(𝔻). (4)

Let 𝑛 be a nonnegative integer. The generalized weighted composition operator, denoted by 𝐷^𝑛_𝜑,𝑢, is defined as follows (see [4–7]): 𝐷^𝑛_𝜑,𝑢𝑓(𝑧)=𝑢(𝑧)⋅𝑓^(𝑛)𝜑(𝑧),𝑧∈𝔻,𝑓∈𝐻(𝔻). (5) When 𝑛=0, then 𝐷^𝑛_𝜑,𝑢 becomes the weighted composition operator. If 𝑛=0 and 𝑢(𝑧)=1, then 𝐷^𝑛_𝜑,𝑢=𝐶_𝜑. If 𝑛=1 and 𝑢(𝑧)=𝜑^′(𝑧), then 𝐷^𝑛_𝜑,𝑢=𝐷𝐶_𝜑. If 𝑛=1 and 𝑢(𝑧)=1, then 𝐷^𝑛_𝜑,𝑢=𝐶_𝜑𝐷. The operators 𝐷𝐶_𝜑 and 𝐶_𝜑𝐷 were studied, for example, in [8–15].

Composition operators, weighted composition operators, and related operators on the Bloch space were studied in [10, 12–22], while composition operators and weighted composition operators between the Zygmund space and some other spaces were studied in [2, 12, 23–28].

Recently, many researchers studied the generalized weighted composition operator 𝐷^𝑛_𝜑,𝑢 on various spaces; see, for example, in [4–7, 25, 26, 29–32]. In [26], Stević has studied the operator 𝐷^𝑛_𝜑,𝑢 from the Bloch space to the 𝑛th weighted-type space, which includes the Zygmund space. Among other things, he obtained the following result (see [26]).

Theorem A. Let 𝑢∈𝐻(𝔻), 𝜑, an analytic self-map of  𝔻 and 𝑛 a positive integer. Then the following assertions hold.(a)The operator 𝐷^𝑛_𝜑,𝑢∶ℬ(𝑜𝑟ℬ₀)→𝒵 is bounded if and only if 𝑀₁∶=sup_𝑧∈𝔻1−|𝑧|²𝑢^′′(𝑧)1−𝜑(𝑧)^2𝑛𝑀<∞,₂∶=sup_𝑧∈𝔻1−|𝑧|²2𝑢^′(𝑧)𝜑^′(𝑧)+𝑢(𝑧)𝜑^′′(𝑧)1−𝜑(𝑧)^2𝑛+1𝑀<∞,₃∶=sup_𝑧∈𝔻1−|𝑧|²|𝑢(𝑧)|𝜑^′(𝑧)²1−𝜑(𝑧)^2𝑛+2<∞. (6)(b)Suppose that the operator 𝐷^𝑛_𝜑,𝑢∶ℬ(𝑜𝑟ℬ₀)→𝒵 is bounded then 𝐷^𝑛_𝜑,𝑢∶ℬ(𝑜𝑟ℬ₀)→𝒵 is compact if and only if lim_{|𝜑(𝑧)|→1}1−|𝑧|²𝑢^′′(𝑧)1−𝜑(𝑧)^2𝑛=0,lim_{𝜑(𝑧)→1}1−|𝑧|²|𝑢(𝑧)|𝜑^′(𝑧)²1−𝜑(𝑧)^2𝑛+2=0,lim_{𝜑(𝑧)→1}1−|𝑧|²2𝑢^′(𝑧)𝜑^′(𝑧)+𝑢(𝑧)𝜑^′′(𝑧)1−𝜑(𝑧)^2𝑛+1=0. (7)

Here we give a new criterion for the boundedness or compactness of the operator 𝐷^𝑛_𝜑,𝑢; namely, we use three families of functions to characterize the operator 𝐷^𝑛_𝜑,𝑢∶ℬ→𝒵.

Throughout the paper, 𝐶 denotes a positive constant which may differ from one occurrence to the other. The notation 𝐴≍𝐵 means that there exists a positive constant 𝐶 such that 𝐵/𝐶≤𝐴≤𝐶𝐵.

2. Main Results

In this section we give our main results and proofs. For 𝑎∈𝔻, set 𝑓_𝑎,𝑗(𝑧)=1−|𝑎|²1−𝑎𝑧^𝑗,𝑗=1,2,3. (8) Next, we will use these three families of functions to characterize generalized weighted composition operators 𝐷^𝑛_𝜑,𝑢∶ℬ→𝒵.

Theorem 1. Let 𝑢∈𝐻(𝔻), 𝜑 be an analytic self-map of 𝔻, and 𝑛 a positive integer. Then the following conditions are equivalent:(a)the operator 𝐷^𝑛_𝜑,𝑢∶ℬ→𝒵 is bounded;(b)the operator 𝐷^𝑛_𝜑,𝑢∶ℬ₀→𝒵 is bounded;(c)𝑢∈𝒵, 𝑁₂∶=sup_𝑧∈𝔻(1−|𝑧|²)|𝑢(𝑧)||𝜑^′(𝑧)|²<∞, 𝑁₃∶=sup_𝑧∈𝔻(1−|𝑧|²)|2𝑢^′(𝑧)𝜑^′(𝑧)+𝑢(𝑧)𝜑^′′(𝑧)|<∞, 𝐴∶=maxsup_𝑤∈𝔻𝐷^𝑛_𝜑,𝑢𝑓_{𝜑(𝑤),1𝒵},sup_𝑤∈𝔻𝐷^𝑛_𝜑,𝑢𝑓_{𝜑(𝑤),2𝒵},sup_𝑤∈𝔻𝐷^𝑛_𝜑,𝑢𝑓_{𝜑(𝑤),3𝒵}<∞. (9)

Proof. (a)⇒(b). This implication is obvious.
(b)⇒(c). Assume that 𝐷^𝑛_𝜑,𝑢∶ℬ₀→𝒵 is bounded. Taking the functions 𝑧^𝑛, 𝑧^𝑛+1, and 𝑧^𝑛+2 and using the boundedness of 𝐷^𝑛_𝜑,𝑢, we see that 𝑢∈𝒵, and 𝑁₂ and 𝑁₃ are finite. For each 𝑎∈𝔻, it is easy to check that 𝑓_𝑎,𝑗∈ℬ₀,𝑗=1,2,3. Moreover ‖𝑓_𝑎,𝑗‖_ℬ,𝑗=1,2,3, are bounded by constants independent of 𝑎. By the boundedness of 𝐷^𝑛_𝜑,𝑢∶ℬ₀→𝒵, we get sup_𝑎∈𝔻𝐷^𝑛_𝜑,𝑢𝑓_{𝜑(𝑎),𝑗𝒵}≤𝐷^𝑛_𝜑,𝑢sup_𝑎∈𝔻𝑓_{𝜑(𝑎),𝑗ℬ}≤𝐶𝐷^𝑛_𝜑,𝑢<∞,𝑗=1,2,3, (10) as desired.
(c)⇒(a). Suppose that 𝑢∈𝒵, and 𝑁₂ and 𝑁₃ and 𝐴 are finite. To prove this implication, we only need to show that these conditions imply (6). A calculation shows that 𝑓^(𝑛)_𝑎,1(𝑎)=𝑛!𝑎^𝑛1−|𝑎|^2𝑛,𝑓^(𝑛)_𝑎,2(𝑎)=(𝑛+1)!𝑎^𝑛1−|𝑎|^2𝑛,𝑓^(𝑛)_𝑎,3(𝑎)=(𝑛+2)!𝑎^𝑛21−|𝑎|^2𝑛. (11) For the simplicity, we denote 2𝑢^′(𝑧)𝜑^′(𝑧)+𝑢(𝑧)𝜑^′′(𝑧) by 𝑣(𝑧). From (11), for 𝑤∈𝔻, we have 𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),1^′′(𝑤)=𝑛!𝑢^′′(𝑤)𝜑(𝑤)^𝑛1−𝜑(𝑤)^2𝑛+(𝑛+1)!𝑣(𝑤)𝜑(𝑤)^𝑛+11−𝜑(𝑤)^21+𝑛+(𝑛+2)!𝑢(𝑤)𝜑^′(𝑤)²𝜑(𝑤)^𝑛+21−𝜑(𝑤)^22+𝑛,𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),2^′′(𝑤)=(𝑛+1)!𝑢^′′(𝑤)𝜑(𝑤)^𝑛1−𝜑(𝑤)^2𝑛+(𝑛+2)!𝑣(𝑤)𝜑(𝑤)^𝑛+11−𝜑(𝑤)^21+𝑛+(𝑛+3)!𝑢(𝑤)𝜑^′(𝑤)²𝜑(𝑤)^𝑛+21−𝜑(𝑤)^22+𝑛,𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),3^′′(𝑤)=(𝑛+2)!𝑢^′′(𝑤)𝜑(𝑤)^𝑛21−𝜑(𝑤)^2𝑛+(𝑛+3)!𝑣(𝑤)𝜑(𝑤)^𝑛+121−𝜑(𝑤)^21+𝑛+(𝑛+4)!𝑢(𝑤)𝜑^′(𝑤)²𝜑(𝑤)^𝑛+221−𝜑(𝑤)^22+𝑛. (12)(13)(14) Multiplying (12) by −(𝑛+1) and then adding (13), we get −(𝑛+1)𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),1^′′(𝑤)+𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),2^′′(𝑤)=(𝑛+1)!𝑣(𝑤)𝜑(𝑤)^𝑛+11−𝜑(𝑤)^21+𝑛+2(𝑛+2)!𝑢(𝑤)𝜑^′(𝑤)²𝜑(𝑤)^𝑛+21−𝜑(𝑤)^22+𝑛. (15) Multiplying (12) by −(𝑛+1)(𝑛+2) and (14) by 2, respectively, we obtain −(𝑛+1)(𝑛+2)𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),1^′′(𝑤)+2𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),3^′′(𝑤)=2(𝑛+2)!𝑣(𝑤)𝜑(𝑤)^𝑛+11−𝜑(𝑤)^21+𝑛+(4𝑛+10)(𝑛+2)!𝑢(𝑤)𝜑^′(𝑤)²𝜑(𝑤)^𝑛+21−𝜑(𝑤)^22+𝑛. (16) Multiplying (15) by 2(𝑛+2), we get −2(𝑛+1)(𝑛+2)𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),1^′′(𝑤)+2(𝑛+2)𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),2^′′(𝑤)=2(𝑛+2)!𝑣(𝑤)𝜑(𝑤)^𝑛+11−𝜑(𝑤)^21+𝑛+(4𝑛+8)(𝑛+2)!𝑢(𝑤)𝜑^′(𝑤)²𝜑(𝑤)^𝑛+21−𝜑(𝑤)^22+𝑛. (17) Subtracting (17) from (16), we obtain 2(𝑛+2)!𝑢(𝑤)𝜑^′(𝑤)²𝜑(𝑤)^𝑛+21−𝜑(𝑤)^22+𝑛=(𝑛+1)(𝑛+2)𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),1^′′(𝑤)−2(𝑛+2)𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),2^′′(𝑤)+2𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),3^′′(𝑤), (18) which implies that 1−|𝑤|²𝑢(𝑤)𝜑^′(𝑤)²𝜑(𝑤)^𝑛+21−𝜑(𝑤)^22+𝑛≤1|𝑤|2𝑛!1−²𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),1^′′(𝑤+1)()!|𝑤|𝑛+11−²𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),2^′′(𝑤+1)()!|𝑤|𝑛+21−²𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),3^′′(𝑤≤1)𝐷2𝑛!^{𝑛𝜑,𝑢}𝑓_{𝜑(𝑤),1𝒵}+1()!𝐷𝑛+1^{𝑛𝜑,𝑢}𝑓_{𝜑(𝑤),2𝒵}+1()!𝐷𝑛+2^{𝑛𝜑,𝑢}𝑓_{𝜑(𝑤),3𝒵}≤𝐴+𝐴2𝑛!()!+𝐴𝑛+1()!.𝑛+2 (19)(20)(21) From (16) and (18), we obtain (𝑛+2)!𝑣(𝑤)𝜑(𝑤)^𝑛+11−𝜑(𝑤)^21+𝑛=−(𝑛+1)(𝑛+2)(𝑛+3)𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),1^′′(𝑤)+(𝑛+2)(2𝑛+5)𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),2^′′(𝑤)−2(𝑛+2)𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),3^′′(𝑤), (22) which implies that 1−|𝑤|²|𝑣(𝑤)|𝜑(𝑤)^𝑛+11−𝜑(𝑤)^21+𝑛≤(𝑛+3)|𝑤|𝑛!1−²𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),1^′′(𝑤+())2𝑛+5()!|𝑤|𝑛+11−²𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),2^′′(𝑤+2)()!|𝑤|𝑛+11−²𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),3^′′(𝑤≤())𝑛+3𝐷𝑛!^{𝑛𝜑,𝑢}𝑓_{𝜑(𝑤),1𝒵}+(2𝑛+5)()!𝐷𝑛+1^{𝑛𝜑,𝑢}𝑓_{𝜑(𝑤),2𝒵}+2()!𝐷𝑛+1^{𝑛𝜑,𝑢}𝑓_{𝜑(𝑤),3𝒵}≤(𝑛+3)()𝑛!𝐴+2𝑛+5()!2𝑛+1𝐴+(𝑛+1)!𝐴. (23)(24)(25) By (12), (18), and (22), we have 𝑛!𝑢^′′(𝑤)𝜑(𝑤)^𝑛1−𝜑(𝑤)^2𝑛=2+(𝑛+1)(𝑛+4)2𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),1^′′(𝑤)−(𝑛+3)𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),2^′′(𝑤)+𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),3^′′(𝑤), (26) which implies that 1−|𝑤|²𝑢^′′(𝑤)𝜑(𝑤)^𝑛1−𝜑(𝑤)^2𝑛≤2+(𝑛+1)(𝑛+4)|𝑤|2𝑛!1−²𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),1^′′(𝑤+())𝑛+3|𝑤|𝑛!1−²𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),2^′′(𝑤+1)|𝑤|𝑛!1−²𝐷^𝑛_𝜑,𝑢𝑓_𝜑(𝑤),3^′′(𝑤≤())2+𝑛+1)(𝑛+4𝐷2𝑛!^{𝑛𝜑,𝑢}𝑓_{𝜑(𝑤),1𝒵}+(𝑛+3)𝐷𝑛!^{𝑛𝜑,𝑢}𝑓_{𝜑(𝑤),2𝒵}+1𝐷𝑛!^{𝑛𝜑,𝑢}𝑓_{𝜑(𝑤),3𝒵}≤2+(𝑛+1)(𝑛+4)()2𝑛!𝐴+𝑛+31𝑛!𝐴+𝑛!𝐴. (27)(28)(29) Fix 𝑟∈(0,1). If |𝜑(𝑤)|>𝑟, then from (21) we obtain 1−|𝑤|²|𝑢(𝑤)|𝜑^′(𝑤)²1−𝜑(𝑤)^22+𝑛≤𝐴2𝑛!𝑟^𝑛+2+𝐴(𝑛+1)!𝑟^𝑛+2+𝐴(𝑛+2)!𝑟^𝑛+2<∞. (30) On the other hand, if |𝜑(𝑤)|≤𝑟, we get 1−|𝑤|²|𝑢(𝑤)|𝜑^′(𝑤)²1−𝜑(𝑤)^22+𝑛≤11−𝑟^22+𝑛sup_𝑧∈𝔻1−|𝑧|²|𝑢(𝑧)|𝜑^′(𝑧)²<∞. (31) From (30) and (31) we see that 𝑀₃ is finite. Similarly, from (25) and (29) we can obtain that 𝑀₁ and 𝑀₂ are finite as well. The proof of this theorem is finished.

To get the characterization of the compactness of 𝐷^𝑛_𝜑,𝑢∶ℬ(orℬ₀)→𝒵, we need the following criterion, which follows from standard arguments similar to those outlined in Proposition 3.11 of [3].

Lemma 2. Let 𝑢∈𝐻(𝔻), 𝜑, an analytic self-map of 𝔻 and 𝑛 be a positive integer. The operator 𝐷^𝑛_𝜑,𝑢∶ℬ(𝑜𝑟ℬ₀)→𝒵 is compact if and only if 𝐷^𝑛_𝜑,𝑢∶ℬ(𝑜𝑟ℬ₀)→𝒵 is bounded, and for any bounded sequence (𝑓_𝑘)_𝑘∈ℕ in ℬ(𝑜𝑟ℬ₀) which converges to zero uniformly on compact subsets of 𝔻, one has ‖𝐷^𝑛_𝜑,𝑢𝑓_𝑘‖_𝒵→0 as 𝑘→∞.

Theorem 3. Let 𝑢∈𝐻(𝔻), 𝜑 be an analytic self-map of 𝔻, and 𝑛 a positive integer. Suppose that the operator 𝐷^𝑛_𝜑,𝑢∶ℬ→𝒵 is bounded; then the following conditions are equivalent:(a)the operator 𝐷^𝑛_𝜑,𝑢∶ℬ→𝒵 is compact;(b)the operator 𝐷^𝑛_𝜑,𝑢∶ℬ₀→𝒵 is compact;(c)lim_{|𝜑(𝑤)|→1}‖𝐷^𝑛_𝜑,𝑢𝑓_{𝜑(𝑤),𝑗}‖_𝒵=0,𝑗=1,2,3.

Proof. (a)⇒(b). This implication is clear.
(b)⇒(c). Assume that 𝐷^𝑛_𝜑,𝑢∶ℬ₀→𝒵 is compact. Let {𝑤_𝑘}_𝑘∈ℕ be a sequence in 𝔻 such that lim_𝑘→∞|𝜑(𝑤_𝑘)|=1 (if such a sequence does not exist, then the limits in (c) automatically hold). Since the sequences {𝑓_{𝜑(𝑤𝑘),𝑗}}, 𝑗=1,2,3, are bounded in ℬ₀ and converge to 0 uniformly on compact subsets of 𝔻, by Lemma 2, we get ‖𝐷^𝑛_𝜑,𝑢𝑓_{𝜑(𝑤𝑘),𝑗}‖_𝒵→0, as 𝑘→∞, which means that (c) holds.
(c)⇒(a). Suppose that the limits in (c) are 0. To prove this implication, we only need to show that (7) hold. Using the inequality (28), we get 1−|𝑤|²𝑢^′′(𝑤)1−𝜑(𝑤)^2𝑛≤[2+(𝑛+1)(𝑛+4)]𝐷^𝑛_𝜑,𝑢𝑓_{𝜑(𝑤),1𝒵}2𝑛!𝜑(𝑤)^𝑛+(𝑛+3)𝐷^𝑛_𝜑,𝑢𝑓_{𝜑(𝑤),2𝒵}𝑛!𝜑(𝑤)^𝑛+𝐷^𝑛_𝜑,𝑢𝑓_{𝜑(𝑤),3𝒵}𝑛!𝜑(𝑤)^𝑛⟶0, (32) as |𝜑(𝑤)|→1.
Using inequality (24), we get 1−|𝑤|²2𝑢^′(𝑤)𝜑^′(𝑤)+𝑢(𝑤)𝜑^′′(𝑤)1−𝜑(𝑤)^21+𝑛≤(𝑛+3)𝐷^𝑛_𝜑,𝑢𝑓_{𝜑(𝑤),1𝒵}𝑛!𝜑(𝑤)^𝑛+1+(2𝑛+5)𝐷^𝑛_𝜑,𝑢𝑓_{𝜑(𝑤),2𝒵}(𝑛+1)!𝜑(𝑤)^𝑛+1+2𝐷^𝑛_𝜑,𝑢𝑓_{𝜑(𝑤),3𝒵}(𝑛+1)!𝜑(𝑤)^𝑛+1⟶0, (33) as |𝜑(𝑤)|→1.
Using inequality (20), we get 1−|𝑤|²𝑢(𝑤)𝜑^′(𝑤)²1−𝜑(𝑤)^22+𝑛≤𝐷^𝑛_𝜑,𝑢𝑓_{𝜑(𝑤),1𝒵}2𝑛!𝜑(𝑤)^𝑛+2+𝐷^𝑛_𝜑,𝑢𝑓_{𝜑(𝑤),2𝒵}(𝑛+1)!𝜑(𝑤)^𝑛+2+𝐷^𝑛_𝜑,𝑢𝑓_{𝜑(𝑤),3𝒵}(𝑛+2)!𝜑(𝑤)^𝑛+2⟶0, (34) as |𝜑(𝑤)|→1. The desired result follows. The proof of this theorem is complete.

Acknowledgment

The first author is supported by National Natural Science Foundation of China (no. 11126284).