`Journal of Function Spaces and ApplicationsVolume 2012, Article ID 431016, 37 pageshttp://dx.doi.org/10.1155/2012/431016`
Research Article

## Fourier Multipliers on Triebel-Lizorkin-Type Spaces

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

Received 20 November 2011; Accepted 10 January 2012

Copyright © 2012 Dachun Yang et al. 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

The authors study the mapping properties of Fourier multipliers, with symbols satisfying some generalized Hörmander's condition, on Triebel- Lizorkin-type spaces and Triebel-Lizorkin-Hausdorff spaces. To this end, the authors first establish a new characterization of these spaces via some generalized (weighted) functions, which essentially improves the known result for Triebel-Lizorkin spaces even when . Applying this new characterization, the authors then obtain the boundedness of Fourier multipliers on Triebel-Lizorkin-type spaces and Triebel-Lizorkin-Hausdorff spaces, which also give a new proof of the Sobolev embedding theorems for these spaces.

#### 1. Introduction

It is well known that many classical operators, including some convolution operators, fractional differential operators, and pseudodifferential operators with constant coefficients, fall into the framework of Fourier multipliers. The study of mapping properties of Fourier multipliers on Besov and Triebel-Lizorkin spaces has a long history; see, for example, [110]. Indeed, the best-known Fourier multiplier on for , which is nowadays called Hörmander’s multiplier theorem, was obtained by Hörmander [3, Theorem 2.5], preceded by Mihlin [1, 2]. Triebel [4, Theorem 3.5] gave a very useful generalization of Hörmander’s multiplier theorem [3, Theorem 2.5] from the scalar-valued case to the vector-valued case, which further induced the introduction of the nowadays called Triebel-Lizorkin spaces; see also [5, pages 161–168] for more details including some history of the study on Fourier multipliers. Later, Triebel [9, Theorem 2] established a Fourier multiplier theorem for inhomogeneous Triebel-Lizorkin spaces, which was even proved to be sharp in [9, Remark 12]; see also [10, pages 73–77] for a detailed discussion.

Recently, Cho and Kim [11] and Cho [12] introduced a new family of Fourier multipliers with symbols satisfying some generalized Hörmander’s condition and studied the mapping properties of these Fourier multipliers on the classical homogeneous Besov spaces and Triebel-Lizorkin spaces via first establishing some equivalent characterizations of these spaces. This family of Fourier multipliers contains the classical Riesz potential operator and the differential operator as special cases. As an application, Cho and Kim [11] and Cho [12] presented a new proof of the Sobolev embedding theorems for Besov and Triebel-Lizorkin spaces.

The main purpose of this paper is to clarify the behaviors of these Fourier multipliers in [11, 12] on four new classes of function spaces: the Besov-type space , the Triebel-Lizorkin-type space , and their preduals, the Besov-Hausdorff space and the Triebel-Lizorkin-Hausdorff space . These spaces were recently introduced and investigated in [1318] and proved therein to cover many classical function spaces such as Besov spaces and Triebel-Lizorkin spaces (see, e.g., [10, 19, 20]), spaces and Hardy-Hausdorff spaces (see, e.g., [2124]), Triebel-Lizorkin-Morrey spaces and Morrey spaces (see, e.g., [16, 2528]). To study the boundedness of Fourier multipliers on and , we first establish a new characterization of these spaces in terms of generalized (weighted) functions, which essentially improve the known results in [12] for Triebel-Lizorkin spaces even when . Applying this new characterization, we then obtain the Fourier multiplier results on and , which also essentially improve the known results for Triebel-Lizorkin spaces obtained by Cho in [12] and, moreover, give a new proof of the Sobolev embedding theorems, obtained in [14, 15], for these spaces. Besides, for the Besov-type space and the Besov-Hausdorff space , some of the corresponding results are also presented.

We begin with some notions and notation. In what follows, let and ; let be the space of all the Schwartz functions on with the classical topology and its topological dual space, namely, the set of all continuous linear functionals on endowed with the weak- topology.

Following Triebel [10], let and consider as a subspace of , including the topology. Use to denote the topological dual space of , namely, the set of all continuous linear functionals on . We also endow with the weak- topology. Let be the set of all polynomials on . It is well known that as topological spaces. Similarly, for any , the space is defined to be the set of all Schwartz functions satisfying that for all multi-indices with and its topological dual space. We also let . As usual, denotes the Fourier transform of an integrable function on , which is defined as for all .

The following notion of Fourier multipliers when was originally introduced by Cho and Kim in [11] and Cho in [12]. For and , assume that satisfies that for all , where for , . The Fourier multiplier is defined by setting, for all , .

We remark that the condition (1.2) when is just the classical Hörmander condition (see [3, Theorem 2.5]) and, moreover, the condition (1.2) when with maximum norms instead of norms is called the Mihlin condition (see [1, 2]). One typical example satisfying (1.2) with is the kernels of Riesz transforms given by for and . When , a typical example satisfying (1.2) for any is given by another example is the symbol of a differential operator of order with .

To recall the notions of and in [14] and, their predual spaces, and in [13, 14], we need the following notation.

For and , denote by the dyadic cube and its side length. Let , and for all .

Let and . The space with is defined to be the set of all sequences of measurable functions on such that Similarly, the space with is defined to be the space of all sequences of measurable functions on such that Throughout the whole paper, we denote by C a positive constant which is independent of the main parameters, but it may vary from line to line. Let be the space of all functions   such that Now we recall the notions of the Besov-type space and the Triebel-Lizorkin-type space from [14]. In what follows, for any and , let for all .

Definition 1.1. Let and .(i)The Besov-type space with is defined to be the space of all such that .(ii)The Triebel-Lizorkin-type space with is defined to be the space of all such that .
Obviously, and . We also remark that the spaces and are independent of the choice of ; see [14].

Remark 1.2. Let .(i)For , it was proved in [29, Theorem 1(i)] that when and , and when with equivalent quasinorms. In [30, Corollary 5.7], it was proved that with equivalent quasinorms for and .(ii)For , it was proved in [29, Theorem 1(ii)] that when and , and when with equivalent quasinorms.
Next we recall the Hausdorff-type counterparts of and . To this end, for and , let . For and , the d-dimensional Hausdorff capacity of is defined by where the infimum is taken over all countable open ball coverings of ; see, for example, [31, 32].
For any function , the Choquet integral of with respect to is then defined by
In what follows, we write . For any measurable function on and , its nontangential maximal function is defined by
For and , the space with is defined to be the space of all sequences of measurable functions on such that and the space with is defined to be the space of all sequences of measurable functions on such that where the infimums are taken over all nonnegative Borel measurable functions on satisfying and with the restriction that for any is allowed to vanish only where vanishes. Here and, in what follows, for all , the symbol denotes and, for , the symbol denotes its conjugate index, namely, .

Remark 1.3. By [15, Remark 2.1], we know that if , then for all nonnegative measurable functions on ,
We now recall the notion of the spaces and introduced in [17].

Definition 1.4. Let and .(i)The Besov-Hausdorff space with and is defined to be the space of all such that(ii)The Triebel-Lizorkin-Hausdorff space with and is defined to be the space of all such that
Recall that and . Moreover, the dual spaces of and are, respectively, and ; see [13, 14].
Now we present the main results of this paper as follows.

Theorem 1.5. Let , and . Suppose that satisfies (1.2) with .(i)If and , then there exists a positive constant such that for all , .(ii)If and , then there exists a positive constant such that for all , .

We remark that the Fourier multiplier is originally defined on . Although may not be dense in and , can still be defined on the whole spaces and in a suitable way; see (3.10) and Lemma 3.4 below.

We also remark Theorem 1.5 when completely covers the known results obtained in [12, Theorem 5.1]. The proof of Theorem 1.5 is given in Section 3.

From Theorem 1.5 and [14, Proposition 3.3], we immediately deduce the following conclusion. We omit the details.

Corollary 1.6. Let , , and . Assume that satisfies (1.2) with .(i)If and such that , then there exists a positive constant such that for all , (ii)If and such that , then there exists a positive constant such that for all ,

We point out that Corollary 1.6(ii) when completely covers Cho and Kim [11, Theorem 1.1] and Cho [12, Theorem 7.1].

Moreover, the range of in Corollary 1.6(i) can be essentially improved as indicated by the following theorem.

Theorem 1.7. Let and such that . Let such that . Assume that satisfies (1.2) with and . Then there exists a positive constant such that for all ,

As an immediate consequence of Theorem 1.7 and the lifting property of the space (see, [14, Proposition 3.5]), we have the following conclusion, which shows that Theorem 1.7 has variant for any instead of .

Corollary 1.8. Given and , let be real number with and such that . Assume that satisfies (1.2) with and . Then there exists a positive constant such that for all , .

Remark 1.9. (i) We remark that, by taking , , , , , and for all , then Theorem 1.7 (and also Corollary 1.8 with ) is just the well-known Hardy-Littlewood-Sobolev theorem for fractional integrals (see, e.g., [33, page 119, Theorem 1(b)]), namely, the Riesz potential maps boundedly from to , where . In this sense, Theorem 1.7 (and hence Corollary 1.8) is a generalization of the Hardy-Littlewood-Sobolev for fractional integrals.
(ii) Theorem 1.7 (resp., Corollary 1.8) is not true in the case that and hence (resp., and hence ). Indeed, the assumption (resp., ) and hence play a crucial role in the proof of Theorem 1.7 in Section 3, which is not valid for the case that (resp., ) and hence .
For , let be the well-known Sobolev-Slobodeckij space on . Recall that Triebel [9, Theorem 2] proved that for all , and , if and then the Fourier multiplier is bounded from the inhomogeneous Triebel-Lizorkin space to itself, where and are Schwartz functions satisfying that , , on , and on . From this, together with the embedding theorem [10, Theorem ], we further deduce that, under the above assumptions on , is also bounded from to with , , and .
Notice that, if is as in Theorem 1.7 or Corollary 1.8, then is not necessary to belong to . For example, if , then as in (1.4) satisfies all the assumptions of Theorem 1.7 and Corollary 1.8, but . Thus, the assumptions in both Theorem 1.7 (or Corollary 1.8) and Triebel [9, Theorem 2] are not comparable. This is quite natural, since we are considering the multiplier on homogeneous function spaces, while Triebel [9, Theorem 2] (see also [10, pages 73–77]) studied the multipliers on inhomogeneous function spaces. In some sense, Theorem 1.7 and Corollary 1.8 might be regarded as fractional variants of the homogeneous version of [9, Theorem 2] which corresponds to the case that of Theorem 1.7 and Corollary 1.8. This might also be the reason why the assumption on in [9, Theorem 2] is quite different from the requirement of in Theorem 1.7 and Corollary 1.8. Moreover, the restriction in [9, Theorem 2] is sharp; see Triebel [9, Remark 12] or [10, pages 73–77].
(iii) Recall that in [34], Marschall introduced a very general class of symbols with , , , , and . For a symbol , and any and , the nonregular pseudodifferential operator is defined as Then Marschall [34, Theorem 9(a)] proved that for all , , , , , , and if either ( in case that ) or , then the operator with is bounded from to , where denotes the inhomogeneous Triebel-Lizorkin space. This, together with the Sobolev embedding properties of Triebel-Lizorkin spaces, further implies that the operator is bounded from to with and .
Notice that, if satisfies the assumptions of Theorem 1.7 or Corollary 1.8, then is not necessary to belong to ; see, for example, as in (1.4) with . Thus, by the same reason as in (ii), the assumptions in both Theorem 1.7 (or Corollary 1.8) and Marschall [34, Theorem 9(a)] are not comparable.
(iv) Recall that it was proved by Cho in [12, Theorem 5.2] that when if , or if , the operator maps boundedly into . However, from Theorem 1.7, we deduce that this conclusion is also true when if . Therefore, even when , Theorem 1.7 also essentially improves [12, Theorem 5.2]. Moreover, there exists a gap in the proof of [12, Theorem 5.2] in the endpoint case when , namely, the formula [12, (5.6)] seems not enough for the first inequality in [12, page 853]. The proof of Theorem 1.7 seals this gap and is given in Section 3.
Theorems 1.5 and 1.7 have the following counterparts for Hausdorff-type spaces.

Theorem 1.10. Let , and satisfy (1.2) with .(i)If and , then there exists a positive constant such that for all ,(ii)If and , then there exists a positive constant such that for all ,

Differently from the spaces and , it is known that is dense in the spaces and ; see [13, Lemma 5.3] and [14, Lemma 6.3]. Thus, although is originally defined on , we can extend into the whole spaces and by a density argument.

We remark that Theorem 1.10(i) when coincides with [12, Theorem 5.1] in the case that . The proof of Theorem 1.10 is also given in Section 3.

From Theorem 1.10 and [15, Theorem 4.1], we immediately deduce the following conclusion and omit the details.

Corollary 1.11. Let , and such that . Assume that satisfies the condition (1.2) with .(i)Let and such that . If , then there exists a positive constant such that for all , (ii)Let and such that . If , then there exists a positive constant such that for all ,

Moreover, similar to Corollary 1.6(i), we can further improve the range of in Corollary 1.11(i) as follows.

Theorem 1.12. Let , with and such that . Let and such that . Assume that satisfies (1.2) with and Then there exists a positive constant such that for all ,

The proof of Theorem 1.12 is given in Section 3.

Similar to Corollary 1.8, we have the following conclusion, which is an immediate consequence of Theorem 1.12 and the lifting property of the space that can be deduced directly from [15, Theorem 4.1].

Corollary 1.13. Let and . Let be a real number with and such that . Assume that satisfies that and satisfies (1.2) with as in (1.27). Then there exists a positive constant such that for all , .

Corollary 1.13 implies that Theorem 1.12 has variant for any instead of .

Remark 1.14. Recall that when , the Triebel-Lizorkin-Hausdorff space is just the classical Triebel-Lizorkin space . Thus, when , Theorem 1.12 coincides with Theorem 1.7. In this sense, Theorem 1.12 when also essentially improves [12, Theorem 5.2]; see Remark 1.9(iv).
The proofs of Theorems 1.5, 1.7 and 1.10 strongly depend on the Peetre-type maximal function characterizations of , , , and obtained in [18]. Additionally, to prove Theorems 1.7 and 1.12, we need first establish the generalized (weighted) -function equivalent characterizations of and , respectively, in Theorems 2.7 and 2.9 below. We point out that Theorems 2.7 and 2.9 consist of two parts: sufficiency part and necessary part. The proofs of the sufficiency part are essentially deduced from the corresponding generalized Lusin-area function characterizations, obtained in [18], of these function spaces. The approach used in the proofs of the necessary part of Theorems 2.7 and 2.9 is totally different from that used in the proof of [12, Lemma 3.2(3)] for , which induces an essential improvement of [12, Lemma 3.2(3)] such that we can replace the restriction in [12, Lemma 3.2(3)] by . The proof of [12, Lemma 3.2(3)] strongly depends on the exact equivalent relations between the norms of the generalized Lusin-area functions with different apertures, which is not clear whether it is still true if norm is replaced by the Morrey norm. Instead of that, in the proofs of Theorems 2.7 and 2.9, we use the Lusin-area function characterization of these spaces and the homogeneity of the Euclidean space . This improvement further induces an improvement of Theorems 1.7 and 1.12 even when , compared to [12, Theorem 5.2].
To prove Theorems 1.7 and 1.12, we need two technical lemmas from [12, Lemmas 4.1 and 4.2] (see also Lemmas 3.2 and 3.5 below). However, [12, Lemma 4.1(2)] therein is not accurate; see Remark 3.3 below. We give a corrected version in Lemma 3.2(ii) of this paper. We also remark that there exists a gap in the proof of [12, Theorem 5.2] for Triebel-Lizorkin spaces in the endpoint case when ; see Remark 1.9(iv). In this paper, we seal this gap via a subtle application of the equivalence between the Triebel-Lizorkin space and the Triebel-Lizorkin-type space obtained by Frazier and Jawerth [30, Corollary 5.7] (see also [14, Proposition 3.1]).
The paper is organized as follows. In Section 2, we present Theorems 2.7 and 2.9 and their proofs by first recalling some known characterizations, obtained in [18], of , , , and in terms of the Peetre-type maximal function and the Lusin-area function of local means. Section 3 is devoted to the proofs of Theorems 1.5, 1.7, 1.10, and 1.12. Finally in Section 4, as an application, we give a new proof of the Sobolev-type embedding theorems for and .
We point out that so far, for the Besov-type space and the Besov-Hausdorff space , it is unclear whether the corresponding results of Theorems 1.7 and 1.12 are true or not. The proofs of Theorems 1.7 and 1.12 strongly depend on the generalized (weighted) -function equivalent characterizations of and , which are not available for and . Moreover, it is also interesting to establish the inhomogeneous variants of these results.
Finally, we make more conventions on the notation. Throughout the whole paper, the symbol means that , where is a positive constant independent of the main parameter. If and , then we write . If is a subset of , we denote by the characteristic function of .

#### 2. Some Equivalent Characterizations of ̇𝐹𝑠,𝜏𝑝,𝑞(ℝ𝑛), ̇𝐵𝑠,𝜏𝑝,𝑞(ℝ𝑛), 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(ℝ𝑛), and 𝐵̇𝐻𝑠,𝜏𝑝,𝑞(ℝ𝑛)

In this section, we first recall some equivalent characterizations, established in [18], of , , , and , in terms of the Peetre-type maximal function and the Lusin-area function of local means. Using these characterizations, we further establish some new characterizations of these spaces in terms of the generalized (weighted) -functions, which play a key role in the proofs of Theorems 1.7 and 1.12 in Section 3.

Let and satisfy that In what follows, for any function , and , . For all , , , and , let which is called the Peetre-type maximal function of local means; see, for example, [18].

The following characterization of was obtained in [18].

Theorem 2.1. Let and such that and be as in (2.1). Then the space is characterized by where with the usual modification made when .

Remark 2.2. Recall that when , the Triebel-Lizorkin-type spaces are just the Triebel-Lizorkin-Morrey spaces, that is, in the definition of Triebel-Lizorkin-type space, the sum can be replaced by ; see [16, Theorem 1.1]. By an argument similar to that used in [18, Theorem 3.1], we can prove that Theorem 2.1 is also true with replaced by in , and when . We omit the details.
The following Theorems 2.3 through 2.5 were established in [18].

Theorem 2.3. Let , and such that and be as in (2.1). Then the space is characterized by where with the usual modifications made when or .

Theorem 2.4. Let and such that and be as in (2.1). Then the space is characterized by where where the infimums are taken over all nonnegative Borel measurable functions on satisfying (1.13).

Theorem 2.5. Let and such that and be as in (2.1). Then the space is characterized by where where the infimums are taken over all nonnegative Borel measurable functions on satisfying (1.13).

Remark 2.6. (i) The space is a quasi-Banach space; see [13, 14, 17]. Indeed, by [17, Remarks 7.1 and 7.3], we know that for any ,
(ii)By the Aoki-Rolewicz theorem ([35, 36]), there exists such that for all . Indeed, does the job.
(iii)The conclusions in (i) and (ii) are also true for the space .
Next we establish a new characterization of the spaces and . Let , , and be a nonnegative Borel measurable function. In what follows, for , and in (2.1), set for all and . For all , and , recall that the generalized weighted Lusin-area function and the generalized weighted -function are defined, respectively, by If , then and are called, respectively, the generalized Lusin-area function, denoted by , and the generalized -function, denoted by .
In what follows, for and , let be the set of all functions   such that

Theorem 2.7. Let , , , , and such that . Then if and only if and , where is as in (2.13). Moreover, there exists a positive constant such that for all ,

Proof. Assume and . Notice that for any and , Then, from Remark 2.2, we deduce that and which completes the proof of the sufficiency of the theorem.
Conversely, suppose that . Then by Theorem 2.1, . Moreover, similar to the proof of [18, Theorem 3.1], for any , we see that where is a positive constant independent of and . Then by changing variables, we conclude that where the last inequality follows from the equivalence between Triebel-Lizorkin spaces and Triebel-Lizorkin-Morrey spaces when ; see [16] and also Remark 2.2. By changing variables, we know that for all , Thus, when , from the well-known inequality that for all and , it follows that where the last inequality follows from (2.20) and . Similarly, when , by Minkowski’s inequality and (2.20), we see that