Boundedness of Weighted Hardy Operator and Its Adjoint on Triebel-Lizorkin-Type Spaces
Let , , , and such that and , let be the weighted Hardy operator and its adjoint operator with respect to the weight function . In this paper, the authors establish a sufficient and necessary condition on weight function to ensure the boundedness of and on the Triebel-Lizorkin-type spaces and their predual spaces, Triebel-Lizorkin-Hausdorff spaces, which unify and generalize the known results on -type spaces.
This paper focuses on the boundedness of the weighted Hardy operator and its adjoint operator on Triebel-Lizorkin-type spaces and their predual spaces. Recall that, for a fixed function , the weighted Hardy operator is defined by see Carton-Lebrun and Fosset . Accordingly, the adjoint operator of , named by the weighted Cesàro average operator, is defined by
This paper is motivated by the following facts. On one hand, is related to the Hardy-Littlewood maximal operators in harmonic analysis. If and , then goes back to the classical Hardy-Littlewood average : Its adjoint operator is the classical Cesàro average operator:
Xiao in  obtained the boundedness of on , the boundedness of on , and their corresponding operator norms. Recall that can be viewed as a generalization of (cf. ). Recently Tang and Zhai in  obtained the boundedness and norm of on . They even work to more general spaces . The boundedness of and on and its dual space are worked out. Recall that spaces ( spaces) were originally defined by Aulaskari et al.  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 -dimensional Euclidean spaces by Essén et al. By [3, Theorem 2.3], is always a subspace of . As a generation of spaces, the spaces when and were first introduced by Cui and Yang  and later extended to , , and in . We also refer to [4, 8–11] for more studies on these spaces.
On the other hand, it is well known that Besov spaces and Triebel-Lizorkin spaces 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 and for all , , and . These spaces unify and generalize the homogeneous Besov spaces , Triebel-Lizorkin spaces , and the spaces simultaneously. The following equivalent definition of the spaces were given in [7, Section 3].
Definition 1.1. Let , , , and . The space is defined to be the space of all measurable functions on such that where the supremum is taken over all cubes with the edges parallel to the coordinate axes in .
Motivated by the above facts, a natural question is following: under what condition on the weighted Hardy operator and its adjoint are bounded on Triebel-Lizorkin-type spaces ? The main purpose of this paper is to give the sufficient and necessary condition on such that the operators and are bounded on Triebel-Lizorkin-type spaces . We have the following main results.
Theorem 1.2. Let be a function, and let , , , and such that and . Then exists as a bounded operator if Moreover, when (1.6) holds, the norm of on is given by
Finally, we make some conventions on notations. Throughout the paper, denotes the -dimensional Euclidean space, with Euclidean norm , and Lebesgue measure . A cube will always mean a cube in with the edges parallel to the coordinate axes with sidelength and volume . The dilated cube , , is the cube with the same center as and sidelength . For all , denotes the adjoint index of , namely, . 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 , then for any cube , applying Minkowski's inequality, we have
Then, by (1.6), we have
Thus, the operator is bounded on with operator norm no more than
Conversely, if is bounded on , then we can choose the function where and denote, respectively, the left and right halves of , separated by the hyperplane ( is the first coordinate of ).
We now show that . Notice that the assumption on implies that . For any cube in , we consider two cases.
Case 1. If , then . Since , then , and hence which together with Hölder's inequality, , yields that Case 2. If , then . Using Minkowski's inequality, similar to the computation of Case 1, we also obtain Combining Cases 1 and 2 yields that namely, and .
Since we then have which completes the proof of Theorem 1.2.
The weighted Hardy operator and the weighted Cesàro average operator are adjoint mutually, namely, Recall that then we have where
Similar to the proof of Theorem 1.2, we have next theorem.
Theorem 2.2. Let be a function, and let , , , and such that and . Then exists as a bounded operator if
Moreover, when (2.17) holds, the norm of on is given by
Recall that the Triebel-Lizorkin-Hausdorff spaces were originally introduced in [7, 12] and proved therein to be the predual spaces of Triebel-Lizorkin-type spaces . Triebel-Lizorkin-Hausdorff spaces unify and generalize Triebel-Lizorkin spaces (see, e.g., ) and Hardy-Hausdorff spaces in . These spaces were further studied in .
By dual argument, we have the following theorems.
Theorem 2.3. Let be a function, and let , , , and such that and . Then exists as a bounded operator if Moreover, when (2.19) holds, the norm of on is given by
Theorem 2.4. Let be a function, and let , , , and such that and . Then exists as a bounded operator if Moreover, when (2.21) holds, the norm of on is given by
Our Theorems actually induce the following results:
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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