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

Abstract

Let 𝑝∈[1,∞], 𝑞∈[1,∞), 𝜏∈(0,∞), and 𝛼∈(0,1) such that 𝜏>1/𝑝−1/𝑞 and 𝛼≤𝑛(1/𝑝−𝜏), let 𝑈_𝜓 be the weighted Hardy operator and 𝑉_𝜓 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 𝑈_𝜓 and 𝑉_𝜓 on the Triebel-Lizorkin-type spaces 𝐹̇^{𝛼,𝜏𝑝,𝑞}(ℝ^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 𝑈_𝜓 and its adjoint operator 𝑉_𝜓 on Triebel-Lizorkin-type spaces and their predual spaces. Recall that, for a fixed function 𝜓∶[0,1]→[0,∞), the weighted Hardy operator 𝑈_𝜓 is defined by 𝑈_𝜓𝑓(𝑥)=¹⁰𝑓(𝑡𝑥)𝜓(𝑡)𝑑𝑡,𝑥∈ℝ^𝑛,(1.1) see Carton-Lebrun and Fosset [1]. Accordingly, the adjoint operator 𝑉_𝜓 of 𝑈_𝜓, named by the weighted Cesàro average operator, is defined by 𝑉_𝜓𝑓(𝑥)=¹⁰𝑓𝑥𝑡𝑡^−𝑛𝜓(𝑡)𝑑𝑡,𝑥∈ℝ^𝑛.(1.2)

This paper is motivated by the following facts. On one hand, 𝑈_𝜓 is related to the Hardy-Littlewood maximal operators in harmonic analysis. If 𝜓≡1 and 𝑛=1, then 𝑈_𝜓 goes back to the classical Hardy-Littlewood average 𝑈: 𝑈𝑓(𝑥)=1𝑥^𝑥0𝑓(𝑥)𝑑𝑦,𝑥≠0.(1.3) Its adjoint operator 𝑉_𝜓 is the classical Cesàro average operator: 𝑉𝑓(𝑥)=⎧⎪⎪⎨⎪⎪⎩^∞𝑥𝑓(𝑦)−𝑦𝑑𝑦,𝑥>0,_𝑥−∞𝑓(𝑦)𝑦𝑑𝑦,𝑥<0.(1.4)

Xiao in [2] obtained the boundedness of 𝑈_𝜓 on BMO(ℝ^𝑛), the boundedness of 𝑉_𝜓 on 𝐻¹(ℝ^𝑛), and their corresponding operator norms. Recall that 𝑄_𝛼(ℝ^𝑛) can be viewed as a generalization of BMO(ℝ^𝑛) (cf. [3]). Recently Tang and Zhai in [4] obtained the boundedness and norm of 𝑈_𝜓 on 𝑄_𝛼(ℝ^𝑛). They even work to more general spaces 𝑄^{𝑝𝛼,𝑞}(ℝ^𝑛). The boundedness of 𝑈_𝜓 and 𝑉_𝜓 on 𝑄^{𝑝𝛼,𝑞}(ℝ^𝑛) and its dual space are worked out. Recall that 𝑄 spaces (𝑄_𝛼 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 ℝ^𝑛 by Essén et al. By [3, Theorem 2.3], 𝑄_𝛼(ℝ^𝑛) is always a subspace of BMO(ℝ^𝑛). As a generation of 𝑄 spaces, the spaces 𝑄^{𝑝𝛼,𝑞}(ℝ^𝑛) when 𝛼∈(0,1) and 2≤𝑞<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, 8–11] for more studies on these spaces.

On the other hand, it is well known that Besov spaces 𝐵̇^{𝛼𝑝,𝑞}(ℝ^𝑛) and Triebel-Lizorkin spaces 𝐹̇^{𝛼𝑝,𝑞}(ℝ^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 𝐵̇^{𝛼,𝜏𝑝,𝑞}(ℝ^𝑛) and 𝐹̇^{𝛼,𝜏𝑝,𝑞}(ℝ^𝑛) for all 𝛼∈ℝ, 𝜏∈[0,∞), and 𝑝,𝑞∈(0,∞]. These spaces unify and generalize the homogeneous Besov spaces 𝐵̇^{𝛼𝑝,𝑞}(ℝ^𝑛), Triebel-Lizorkin spaces 𝐹̇^{𝛼𝑝,𝑞}(ℝ^𝑛), and the spaces 𝑄_𝛼(ℝ^𝑛) simultaneously. The following equivalent definition of the spaces 𝐹̇^{𝛼,𝜏𝑝,𝑞}(ℝ^𝑛) were given in [7, Section 3].

Definition 1.1. Let 𝛼∈(0,1), 𝑝∈(0,∞), 𝑞∈(1,∞), and 𝜏∈[0,∞]. The space 𝐹̇^{𝛼,𝜏𝑝,𝑞}(ℝ^𝑛) is defined to be the space of all measurable functions 𝑓 on ℝ^𝑛 such that ‖𝑓‖_̇𝐹^{𝛼,𝜏𝑝,𝑞}_(ℝ^𝑛₎=sup1||𝐼||^𝜏⎧⎪⎨⎪⎩_𝐼_{|𝑥−𝑦|<2𝑙(𝐼)}||𝑓(𝑥)−𝑓(𝑦)||^𝑞||𝑥−𝑦||^{𝑛+𝑞𝛼}𝑑𝑦^𝑝/𝑞𝑑𝑥⎫⎪⎬⎪⎭^1/𝑝<∞,(1.5) where the supremum is taken over all cubes 𝐼 with the edges parallel to the coordinate axes in ℝ^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 𝐹̇^{𝛼,𝜏𝑝,𝑞}(ℝ^𝑛)? The main purpose of this paper is to give the sufficient and necessary condition on 𝜓 such that the operators 𝑈_𝜓 and 𝑉_𝜓 are bounded on Triebel-Lizorkin-type spaces 𝐹̇^{𝛼,𝜏𝑝,𝑞}(ℝ^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/𝑝−1/𝑞 and 𝛼≤𝑛(1/𝑝−𝜏). Then 𝑈_𝜓∶𝐹̇^{𝛼,𝜏𝑝,𝑞}(ℝ^𝑛)→𝐹̇^{𝛼,𝜏𝑝,𝑞}(ℝ^𝑛) exists as a bounded operator if ¹⁰𝑡^{−𝑛(1/𝑝−𝜏)+𝛼}𝜓(𝑡)𝑑𝑡<∞.(1.6) Moreover, when (1.6) holds, the norm of 𝑈_𝜓 on 𝐹̇^{𝛼,𝜏𝑝,𝑞}(ℝ^𝑛) is given by ‖‖𝑈_𝜓‖‖_{̇𝐹𝛼,𝜏𝑝,𝑞(ℝ𝑛̇𝐹)→𝛼,𝜏𝑝,𝑞(ℝ𝑛)}=¹⁰𝑡^{−𝑛(1/𝑝−𝜏)+𝛼}𝜓(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, ℝ^𝑛 denotes the 𝑛-dimensional Euclidean space, with Euclidean norm |𝑥|, and Lebesgue measure 𝑑𝑥. A cube 𝐼 will always mean a cube in ℝ^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 𝑓∈𝐹̇^{𝛼,𝜏𝑝,𝑞}(ℝ^𝑛), then for any cube 𝐼⊂ℝ^𝑛, applying Minkowski's inequality, we have ⎧⎪⎨⎪⎩_𝐼_{|𝑥−𝑦|<2𝑙(𝐼)}||𝑈_𝜓𝑓(𝑥)−𝑈_𝜓𝑓(𝑦)||^𝑞||𝑥−𝑦||^{𝑛+𝑞𝛼}𝑑𝑦^𝑝/𝑞𝑑𝑥⎫⎪⎬⎪⎭^1/𝑝=⎧⎪⎨⎪⎩_𝐼⎛⎜⎜⎜⎝_{|𝑥−𝑦|<2𝑙(𝐼)}|||∫¹⁰[𝑓(𝑡𝑥)−𝑓(𝑡𝑦)]𝜓(𝑡)𝑑𝑡|||^𝑞||𝑥−𝑦||^{𝑛+𝑞𝛼}𝑑𝑦⎞⎟⎟⎟⎠^𝑝/𝑞𝑑𝑥⎫⎪⎬⎪⎭^1/𝑝≤⎧⎪⎨⎪⎩_𝐼⎡⎢⎢⎣¹⁰_{|𝑥−𝑦|<2𝑙(𝐼)}|𝑓(𝑡𝑥)−𝑓(𝑡𝑦)|^𝑞||𝑥−𝑦||^{𝑛+𝑞𝛼}𝑑𝑦^1/𝑞𝜓(𝑡)𝑑𝑡⎤⎥⎥⎦^𝑝𝑑𝑥⎫⎪⎬⎪⎭^1/𝑝≤¹⁰⎧⎪⎨⎪⎩_𝐼_{|𝑥−𝑦|<2𝑙(𝐼)}||𝑓(𝑡𝑥)−𝑓(𝑡𝑦)||^𝑞||𝑥−𝑦||^{𝑛+𝑞𝛼}𝑑𝑦^𝑝/𝑞𝑑𝑥⎫⎪⎬⎪⎭^1/𝑝𝜓(𝑡)𝑑𝑡.(2.1) Notice that _𝐼_{|𝑥−𝑦|<2𝑙(𝐼)}||𝑓(𝑡𝑥)−𝑓(𝑡𝑦)||^𝑞||𝑥−𝑦||^{𝑛+𝑞𝛼}=𝑑𝑦^𝑝/𝑞𝑑𝑥_𝑡𝐼_{|𝑢−𝑣|<2𝑙(𝑡𝐼)}||𝑓(𝑢)−𝑓(𝑣)||^𝑞|𝑢−𝑣|^{𝑛+𝑞𝛼}𝑑𝑣^𝑝/𝑞𝑑𝑢⋅𝑡^{𝑝𝛼−𝑛}.(2.2) Then, by (1.6), we have 1||𝐼||^𝜏⎧⎪⎨⎪⎩_𝐼_{|𝑥−𝑦|<2𝑙(𝐼)}||𝑈_𝜓𝑓(𝑥)−𝑈_𝜓𝑓(𝑦)||^𝑞||𝑥−𝑦||^{𝑛+𝑞𝛼}𝑑𝑦^𝑝/𝑞𝑑𝑥⎫⎪⎬⎪⎭^1/𝑝≤1||𝑡𝐼||^𝜏¹⁰⎧⎪⎨⎪⎩_𝑡𝐼_{|𝑢−𝑣|<2𝑙(𝑡𝐼)}||𝑓(𝑢)−𝑓(𝑣)||^𝑞|𝑢−𝑣|^{𝑛+𝑞𝛼}≤‖𝑓‖_{̇𝐹𝛼,𝜏𝑝,𝑞(ℝ𝑛)}𝑑𝑣^𝑝/𝑞𝑑𝑢⎫⎪⎬⎪⎭^1/𝑝𝑡^{𝛼−𝑛/𝑝+𝑛𝜏}𝜓(𝑡)𝑑𝑡¹⁰𝑡^{𝛼−𝑛(1/𝑝−𝜏)}𝜓(𝑡)𝑑𝑡.(2.3) Thus, the operator 𝑈_𝜓 is bounded on 𝐹̇^{𝛼,𝜏𝑝,𝑞}(ℝ^𝑛) with operator norm no more than ¹⁰𝑡^{𝛼−𝑛(1/𝑝−𝜏)}𝜓(𝑡)𝑑𝑡.(2.4)
Conversely, if 𝑈_𝜓 is bounded on 𝐹̇^{𝛼,𝜏𝑝,𝑞}(ℝ^𝑛), then we can choose the function 𝑓₀(𝑥)=−|𝑥|^{−𝑛(1/𝑝−𝜏)+𝛼}|𝑥|_{−𝑛(1/𝑝−𝜏)+𝛼},𝑥∈ℝ^𝑛𝑙,,𝑥∈ℝ^𝑛𝑟,(2.5) where ℝ^𝑛𝑙 and ℝ^𝑛𝑟 denote, respectively, the left and right halves of ℝ^𝑛, separated by the hyperplane 𝑥₁=0 (𝑥₁ is the first coordinate of 𝑥∈ℝ^𝑛).
We now show that 0<‖𝑓₀‖_̇𝐹^{𝛼,𝜏𝑝,𝑞}_(ℝ𝑛)<∞. Notice that the assumption on 𝛼 implies that 𝜏<1/𝑝. For any cube 𝐼 in ℝ^𝑛, we consider two cases.
Case 1. If 𝑝<𝑞, then 𝑝/𝑞<1. Since 𝜏>1/𝑝−1/𝑞, then 𝛼>0>𝑛(1/𝑝−1/𝑞)−𝑛𝜏, and hence _𝐼+ℎ≤||𝑓₀(𝑧)||^𝑞𝑑𝑧_{𝐼∩{|𝑧|≤|ℎ|}}≤||ℎ|||𝑧|^{[−𝑛(1/𝑝−𝜏)+𝛼]𝑞}𝑑𝑧+_{𝐼∩{|𝑧|>|ℎ|}}|𝑧|^{[−𝑛(1/𝑝−𝜏)+𝛼]𝑞}𝑑𝑧_{[−𝑛(1/𝑝−𝜏)+𝛼]𝑞+𝑛}+||ℎ||^{[−𝑛(1/𝑝−𝜏)+𝛼]𝑞}||𝐼||,(2.6) which together with Hölder's inequality, 𝜏>1/𝑝−1/𝑞, yields that _𝐼_{|ℎ|<2𝑙(𝐼)}||𝑓₀(𝑥+ℎ)||^𝑞||ℎ||^{𝑛+𝑞𝛼}≤𝑑ℎ^𝑝/𝑞𝑑𝑥_𝐼_{|ℎ|<2𝑙(𝐼)}||𝑓₀(𝑥+ℎ)||^𝑞||ℎ||^{𝑛+𝑞𝛼}𝑑ℎ𝑑𝑥^𝑝/𝑞||𝐼||^{1−𝑝/𝑞}=_{|ℎ|<2𝑙(𝐼)}_𝐼+ℎ||𝑓₀(𝑧)||^𝑞𝑑𝑧1||ℎ||^{𝑛+𝑞𝛼}𝑑ℎ^𝑝/𝑞||𝐼||^{1−𝑝/𝑞}≤_{|ℎ|<2𝑙(𝐼)}||ℎ||^{[−𝑛(1/𝑝−𝜏)+𝛼]𝑞+𝑛}+||ℎ||^{[−𝑛(1/𝑝−𝜏)+𝛼]𝑞}||𝐼||||ℎ||^{𝑛+𝑞𝛼}𝑑ℎ^𝑝/𝑞||𝐼||^{1−𝑝/𝑞}≤_{|ℎ|<2𝑙(𝐼)}||ℎ||^{[−𝑛(1/𝑝−𝜏)+𝛼]𝑞+𝑛}+||ℎ||^{[−𝑛(1/𝑝−𝜏)+𝛼]𝑞}||𝐼||||ℎ||^{𝑛+𝑞𝛼}𝑑ℎ^𝑝/𝑞||𝐼||^{1−𝑝/𝑞}≤𝐶𝑙(𝐼)^𝑛𝜏𝑝.(2.7)Case 2. If 𝑝≥𝑞, then 𝑝/𝑞≥1. Using Minkowski's inequality, similar to the computation of Case 1, we also obtain ⎧⎪⎨⎪⎩_𝐼_{|ℎ|<2𝑙(𝐼)}||𝑓₀(𝑥+ℎ)||^𝑞||ℎ||^{𝑛+𝑞𝛼}𝑑ℎ^𝑝/𝑞𝑑𝑥⎫⎪⎬⎪⎭^1/𝑝≤⎧⎪⎨⎪⎩_{|ℎ|<2𝑙(𝐼)}⎡⎢⎢⎣_𝐼||𝑓⁰(𝑥+ℎ)||^𝑞||ℎ||^{𝑛+𝑞𝛼}^𝑝/𝑞𝑑𝑥⎤⎥⎥⎦^𝑞/𝑝𝑑ℎ⎫⎪⎬⎪⎭^1/𝑞≤𝐶𝑙(𝐼)^𝑛𝜏.(2.8) Combining Cases 1 and 2 yields that 1||𝐼||^𝜏⎧⎪⎨⎪⎩_𝐼_{|ℎ|<2𝑙(𝐼)}||𝑓₀(𝑥+ℎ)−𝑓₀(𝑥)||^𝑞||ℎ||^{𝑛+𝑞𝛼}𝑑ℎ^𝑝/𝑞𝑑𝑥⎫⎪⎬⎪⎭^1/𝑝≤1||𝐼||^𝜏⎧⎪⎨⎪⎩_𝐼_{|ℎ|<2𝑙(𝐼)}||𝑓₀(𝑥+ℎ)||^𝑞+||𝑓₀(𝑥)||^𝑞||ℎ||^{𝑛+𝑞𝛼}𝑑ℎ^𝑝/𝑞𝑑𝑥⎫⎪⎬⎪⎭^1/𝑝≤𝐶,(2.9) namely, 𝑓₀∈𝐹̇^{𝛼,𝜏𝑝,𝑞}(ℝ^𝑛) and 0≠‖𝑓₀‖_̇𝐹^{𝛼,𝜏𝑝,𝑞}_(ℝ𝑛)<∞.
Since 𝑈_𝜓𝑓₀(𝑥)=𝑓₀(𝑥)¹⁰𝑡^{𝑛(1/𝑝−𝜏)+𝛼}𝜓(𝑡)𝑑𝑡,(2.10) we then have ‖‖𝑈_𝜓‖‖_{̇𝐹𝛼,𝜏𝑝,𝑞(ℝ𝑛̇𝐹)→𝛼,𝜏𝑝,𝑞(ℝ𝑛)}=¹⁰𝑡^{−(𝑛/𝑝−𝜏)+𝛼}𝜓(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/𝑝−1/𝑟, then ¹⁰𝑡^{𝛼−𝑛(1/𝑝−𝜏)}𝜓(𝑡)𝑑𝑡=¹⁰𝑡^{𝛼−𝑛/𝑟}𝜓(𝑡)𝑑𝑡.(2.12) Therefore, Theorem 1.2 is consistent with [4, Theorem 4.1] since 𝑄^{𝑟𝛼,𝑝}(ℝ^𝑛)=𝐹̇^{𝛼,1/𝑝−1/𝑟𝑝,𝑝}(ℝ^𝑛).

The weighted Hardy operator 𝑈_𝜓 and the weighted Cesàro average operator 𝑉_𝜓 are adjoint mutually, namely, _ℝ𝑛𝑔(𝑥)𝑈_𝜓𝑓(𝑥)𝑑𝑥=_ℝ𝑛𝑓(𝑥)𝑉_𝜓𝑔(𝑥)𝑑𝑥.(2.13) Recall that 𝑉_𝜓𝑓(𝑥)=¹⁰𝑓𝑥𝑡𝑡^−𝑛𝜓(𝑡)𝑑𝑡,𝑥∈ℝ^𝑛,(2.14) then we have ⎧⎪⎨⎪⎩_𝐼_{|𝑥−𝑦|<2𝑙(𝐼)}||𝑉_𝜓𝑓(𝑥)−𝑉_𝜓𝑓(𝑦)||^𝑞||𝑥−𝑦||^{𝑛+𝑞𝛼}𝑑𝑦^𝑝/𝑞𝑑𝑥⎫⎪⎬⎪⎭^1/𝑝=⎧⎪⎨⎪⎩_𝐼⎛⎜⎜⎜⎝_{|𝑥−𝑦|<2𝑙(𝐼)}|||∫¹⁰[𝑓(𝑥/𝑡)−𝑓(𝑦/𝑡)]𝑡^−𝑛𝜓(𝑡)𝑑𝑡|||^𝑞||𝑥−𝑦||^{𝑛+𝑞𝛼}𝑑𝑦⎞⎟⎟⎟⎠^𝑝/𝑞𝑑𝑥⎫⎪⎬⎪⎭^1/𝑝≤⎧⎪⎨⎪⎩_𝐼⎡⎢⎢⎣¹⁰_{|𝑥−𝑦|<2𝑙(𝐼)}||𝑓(𝑥/𝑡)−𝑓(𝑦/𝑡)||^𝑞||𝑥−𝑦||^{𝑛+𝑞𝛼}𝑑𝑦^1/𝑞𝑡^−𝑛𝜓(𝑡)𝑑𝑡⎤⎥⎥⎦^𝑝𝑑𝑥⎫⎪⎬⎪⎭^1/𝑝≤¹⁰⎧⎪⎨⎪⎩_𝐼_{|𝑥−𝑦|<2𝑙(𝐼)}||𝑓(𝑥/𝑡)−𝑓(𝑦/𝑡)||^𝑞||𝑥−𝑦||^{𝑛+𝑞𝛼}𝑑𝑦^𝑝/𝑞𝑑𝑥⎫⎪⎬⎪⎭^1/𝑝𝑡^−𝑛𝜓(𝑡)𝑑𝑡,(2.15) where _𝐼_{|𝑥−𝑦|<2𝑙(𝐼)}||𝑓(𝑥/𝑡)−𝑓(𝑦/𝑡)||^𝑞||𝑥−𝑦||^{𝑛+𝑞𝛼}=𝑑𝑦^𝑝/𝑞𝑑𝑥_(1/𝑡)𝐼_{|𝑢−𝑣|<2𝑙((1/𝑡)𝐼)}||𝑓(𝑢)−𝑓(𝑣)||^𝑞|𝑢−𝑣|^{𝑛+𝑞𝛼}𝑑𝑣^𝑝/𝑞𝑑𝑢⋅𝑡^{−𝑝𝛼+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/𝑝−1/𝑞 and 𝛼≤𝑛(1/𝑝−𝜏). Then 𝑉_𝜓∶𝐹̇^{𝛼,𝜏𝑝,𝑞}(ℝ^𝑛)→𝐹̇^{𝛼,𝜏𝑝,𝑞}(ℝ^𝑛) exists as a bounded operator if ¹⁰𝑡^{−𝑛(1−1/𝑝+𝜏)−𝛼}𝜓(𝑡)𝑑𝑡<∞.(2.17)
Moreover, when (2.17) holds, the norm of 𝑉_𝜓 on 𝐹̇^{𝛼,𝜏𝑝,𝑞}(ℝ^𝑛) is given by ‖‖𝑉_𝜓‖‖_{̇𝐹𝛼,𝜏𝑝,𝑞(ℝ𝑛̇𝐹)→𝛼,𝜏𝑝,𝑞(ℝ𝑛)}=¹⁰𝑡^{−𝑛(1−1/𝑝+𝜏)−𝛼}𝜓(𝑡)𝑑𝑡.(2.18)

Recall that the Triebel-Lizorkin-Hausdorff spaces 𝐹𝐻̇^{𝑝−𝛼,𝜏}_′,𝑞′(ℝ^𝑛) were originally introduced in [7, 12] and proved therein to be the predual spaces of Triebel-Lizorkin-type spaces 𝐹̇^{𝛼,𝜏𝑝,𝑞}(ℝ^𝑛). Triebel-Lizorkin-Hausdorff spaces unify and generalize Triebel-Lizorkin spaces (see, e.g., [13]) and Hardy-Hausdorff spaces 𝐻𝐻^1−𝛼 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/𝑝^−1/𝑞^ and 𝛼≤𝑛(1/𝑝^−𝜏). Then 𝑈_𝜓∶𝐹𝐻̇^{−𝛼,𝜏𝑝,𝑞}(ℝ^𝑛)→𝐹𝐻̇^{−𝛼,𝜏𝑝,𝑞}(ℝ^𝑛) exists as a bounded operator if ¹⁰𝑡^{−𝑛(1/𝑝′−𝜏)−𝛼}𝜓(𝑡)𝑑𝑡<∞.(2.19) Moreover, when (2.19) holds, the norm of 𝑈_𝜓 on 𝐹𝐻̇^{−𝛼,𝜏𝑝,𝑞}(ℝ^𝑛) is given by ‖‖𝑈_𝜓‖‖_{𝐹̇𝐻−𝛼,𝜏𝑝,𝑞(ℝ𝑛̇𝐻)→𝐹−𝛼,𝜏𝑝,𝑞(ℝ𝑛)}=¹⁰𝑡^{−𝑛(1/𝑝′−𝜏)−𝛼}𝜓(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/𝑝^−1/𝑞^ and 𝛼≤𝑛(1/𝑝^−𝜏). Then 𝑉_𝜓∶𝐹𝐻̇^{−𝛼,𝜏𝑝,𝑞}(ℝ^𝑛)→𝐹𝐻̇^{−𝛼,𝜏𝑝,𝑞}(ℝ^𝑛) exists as a bounded operator if ¹⁰𝑡^{−𝑛(1−1/𝑝′+𝜏)−𝛼}𝜓(𝑡)𝑑𝑡<∞.(2.21) Moreover, when (2.21) holds, the norm of 𝑉_𝜓 on 𝐹𝐻̇^{−𝛼,𝜏𝑝,𝑞}(ℝ^𝑛) is given by ‖‖𝑉_𝜓‖‖_{𝐹̇𝐻−𝛼,𝜏𝑝,𝑞(ℝ𝑛̇𝐻)→𝐹−𝛼,𝜏𝑝,𝑞(ℝ𝑛)}=¹⁰𝑡^{−𝑛(1−1/𝑝′+𝜏)−𝛼}𝜓(𝑡)𝑑𝑡.(2.22)

Remark 2.5. When 𝑝^=𝑞^ and 𝜏=1/𝑞^−1/𝑟, then ¹⁰𝑡^{−𝑛(1−1/𝑝′+𝜏)−𝛼}𝜓(𝑡)𝑑𝑡=¹⁰𝑡^{−𝑛(1−1/𝑟)−𝛼}𝜓(𝑡)𝑑𝑡,(2.23) and Theorem 2.4 is consistent with [4, Theorem 5.8] since 𝐹𝐻̇^{−𝛼,1/𝑞−1/𝑟𝑞,𝑞}(ℝ^𝑛) is predual space of 𝑄^{𝑟𝛼,𝑞}(ℝ^𝑛).

Our Theorems actually induce the following results: ‖‖‖‖𝑈𝑉_𝜓𝜓‖‖_{𝐹̇𝐻−𝛼,𝜏𝑝′,𝑞′(ℝ𝑛̇𝐻)→𝐹−𝛼,𝜏𝑝′,𝑞′(ℝ𝑛)}=‖‖𝑈_𝜓‖‖_{𝐹̇𝐻𝛼,𝜏𝑝,𝑞(ℝ𝑛̇𝐻)→𝐹𝛼,𝜏𝑝,𝑞(ℝ𝑛)},‖‖_{𝐹̇𝐻−𝛼,𝜏𝑝′,𝑞′(ℝ𝑛̇𝐻)→𝐹−𝛼,𝜏𝑝′,𝑞′(ℝ𝑛)}=‖‖𝑉_𝜓‖‖_{𝐹̇𝐻𝛼,𝜏𝑝,𝑞(ℝ𝑛̇𝐻)→𝐹𝛼,𝜏𝑝,𝑞(ℝ𝑛)}.(2.24)

In particular, ‖‖‖‖𝑈𝑉_𝜓𝜓‖‖_{𝐹̇𝐻−𝛼,1/𝑞−1/𝑝𝑞′,𝑞′(ℝ𝑛̇𝐻)→𝐹−𝛼,1/𝑞−1/𝑝𝑞′,𝑞′(ℝ𝑛)}=‖‖𝑉_𝜓‖‖_{𝑄𝑝𝛼,𝑞(ℝ𝑛)→𝑄𝑝𝛼,𝑞(ℝ𝑛)},‖‖_{𝐹̇𝐻−𝛼,1/𝑞−1/𝑝𝑞′,𝑞′(ℝ𝑛̇𝐻)→𝐹−𝛼,1/𝑞−1/𝑝𝑞′,𝑞′(ℝ𝑛)}=‖‖𝑈_𝜓‖‖_{𝑄𝑝𝛼,𝑞(ℝ𝑛)→𝑄𝑝𝛼,𝑞(ℝ𝑛)}.(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).