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Journal of Function Spaces and Applications
Volume 2012, Article ID 431016, 37 pages
http://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

Academic Editor: Hans Triebel

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) 𝑔𝜆 functions, which essentially improves the known result for Triebel-Lizorkin spaces even when 𝜏=0. 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 𝐿𝑝(𝑛) for 𝑝(1,), 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 ̇𝐵𝑠𝑝,𝑞(𝑛) and Triebel-Lizorkin spaces ̇𝐹𝑠𝑝,𝑞(𝑛) via first establishing some equivalent characterizations of these spaces. This family of Fourier multipliers contains the classical Riesz potential operator 𝐼𝛼 and the differential operator 𝜕𝛼 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 ̇𝐵𝑠,𝜏𝑝,𝑞(𝑛), the Triebel-Lizorkin-type space ̇𝐹𝑠,𝜏𝑝,𝑞(𝑛), and their preduals, the Besov-Hausdorff space 𝐵̇𝐻𝑠,𝜏𝑝,𝑞(𝑛) and the Triebel-Lizorkin-Hausdorff space 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛). 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]), 𝑄 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 ̇𝐹𝑠,𝜏𝑝,𝑞(𝑛) and 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛), we first establish a new characterization of these spaces in terms of generalized (weighted) 𝑔𝜆 functions, which essentially improve the known results in [12] for Triebel-Lizorkin spaces even when 𝜏=0. Applying this new characterization, we then obtain the Fourier multiplier results on ̇𝐹𝑠,𝜏𝑝,𝑞(𝑛) and 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛), 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 ̇𝐵𝑠,𝜏𝑝,𝑞(𝑛) and the Besov-Hausdorff space 𝐵̇𝐻𝑠,𝜏𝑝,𝑞(𝑛), some of the corresponding results are also presented.

We begin with some notions and notation. In what follows, let ={1,2,} and +={0}; let 𝒮(𝑛) be the space of all the Schwartz functions on 𝑛 with the classical topology and 𝒮(𝑛) its topological dual space, namely, the set of all continuous linear functionals on 𝒮(𝑛) endowed with the weak- topology.

Following Triebel [10], let𝒮(𝑛)=𝜑𝒮(𝑛)𝑛𝜑(𝑥)𝑥𝛾𝑑𝑥=0multi-indices𝛾({0})𝑛(1.1) and consider 𝒮(𝑛) as a subspace of 𝒮(𝑛), including the topology. Use 𝒮(𝑛) to denote the topological dual space of 𝒮(𝑛), namely, the set of all continuous linear functionals on 𝒮(𝑛). We also endow 𝒮(𝑛) with the weak- topology. Let 𝒫(𝑛) be the set of all polynomials on 𝑛. It is well known that 𝒮(𝑛)=𝒮(𝑛)/𝒫(𝑛) as topological spaces. Similarly, for any 𝑁+, the space 𝒮𝑁(𝑛) is defined to be the set of all Schwartz functions satisfying that 𝑛𝜑(𝑥)𝑥𝛾𝑑𝑥=0 for all multi-indices 𝛾𝑛+ with |𝛾|𝑁 and 𝒮𝑁(𝑛) its topological dual space. We also let 𝒮1(𝑛)=𝒮(𝑛). As usual, 𝜙 denotes the Fourier transform of an integrable function 𝜙 on 𝑛, which is defined as 𝜙(𝜉)=𝑛𝑒𝑖𝜉𝑥𝜙(𝑥)𝑑𝑥 for all 𝜉𝑛.

The following notion of Fourier multipliers when 𝛼0 was originally introduced by Cho and Kim in [11] and Cho in [12]. For and 𝛼, assume that 𝑚𝐶(𝑛{0}) satisfies that for all |𝜎|,sup𝑅(0,)𝑅𝑛+2𝛼+2|𝜎|||𝜉||𝑅<2𝑅||𝜕𝜎𝜉||𝑚(𝜉)2𝑑𝜉𝐴𝜎<,(1.2) where for 𝜎=(𝜎1,,𝜎𝑛)𝑛+, 𝜕𝜎=(𝜕/𝜕𝑥1)𝜎1(𝜕/𝜕𝑥𝑛)𝜎𝑛. The Fourier multiplier 𝑇𝑚 is defined by setting, for all 𝑓𝒮(𝑛), (𝑇𝑚𝑓𝑓)=𝑚.

We remark that the condition (1.2) when 𝛼=0 is just the classical Hörmander condition (see [3, Theorem 2.5]) and, moreover, the condition (1.2) when 𝛼=0 with maximum norms instead of 𝐿2 norms is called the Mihlin condition (see [1, 2]). One typical example satisfying (1.2) with 𝛼=0 is the kernels of Riesz transforms 𝑅𝑗 given by 𝑅𝑗𝑓(𝜉𝜉)=𝑖𝑖/||𝜉||𝑓(𝜉)(1.3) for 𝜉𝑛{0} and 𝑗{1,,𝑛}. When 𝛼0, a typical example satisfying (1.2) for any is given by||𝜉||𝑚(𝜉)=𝛼for𝜉𝑛{0},(1.4) another example is the symbol of a differential operator 𝜕𝜎 of order 𝛼=𝜎1++𝜎𝑛 with 𝜎=(𝜎1,,𝜎𝑛)𝑛+.

To recall the notions of ̇𝐵𝑠,𝜏𝑝,𝑞(𝑛) and ̇𝐹𝑠,𝜏𝑝,𝑞(𝑛) in [14] and, their predual spaces, 𝐵̇𝐻𝑠,𝜏𝑝,𝑞(𝑛) and 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛) in [13, 14], we need the following notation.

For 𝑗 and 𝑘𝑛, denote by 𝑄𝑗𝑘 the dyadic cube 2𝑗([0,1)𝑛+𝑘) and (𝑄𝑗𝑘) its side length. Let 𝒬={𝑄𝑗𝑘𝑗,𝑘𝑛}, 𝒬𝑗={𝑄𝒬(𝑄)=2𝑗} and 𝑗𝑄=log2(𝑄) for all 𝑄𝒬.

Let 𝑞(0,] and 𝜏[0,). The space 𝐿𝑝𝜏(𝑞(𝑛,)) with 𝑝(0,) is defined to be the set of all sequences 𝐺={𝑔𝑗}𝑗 of measurable functions on 𝑛 such that𝐺𝐿𝑝𝜏(𝑞(𝑛,))=sup𝑃𝒬1||𝑃||𝜏𝑃𝑗=𝑗𝑃||𝑔𝑗||(𝑥)𝑞𝑝/𝑞𝑑𝑥1/𝑝<.(1.5) Similarly, the space 𝑞(𝐿𝑝𝜏(𝑛,)) with 𝑝(0,] is defined to be the space of all sequences 𝐺={𝑔𝑗}𝑗 of measurable functions on 𝑛 such that𝐺𝑞(𝐿𝑝𝜏(𝑛,))=sup𝑃𝒬1||𝑃||𝜏𝑗=𝑗𝑃𝑃||𝑔𝑗||(𝑥)𝑝𝑑𝑥𝑞/𝑝1/𝑞<.(1.6) 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 𝒜 be the space of all functions 𝜑𝒮(𝑛)  such that supp𝜑𝜉𝑛12||𝜉||,||||32𝜑(𝜉)𝐶>0if5||𝜉||53.(1.7) Now we recall the notions of the Besov-type space ̇𝐵𝑠,𝜏𝑝,𝑞(𝑛) and the Triebel-Lizorkin-type space ̇𝐹𝑠,𝜏𝑝,𝑞(𝑛) from [14]. In what follows, for any 𝑗 and 𝜑𝒜, let 𝜑𝑗(𝑥)=2𝑗𝑛𝜑(2𝑗𝑥) for all 𝑥𝑛.

Definition 1.1. Let 𝑠,𝜏[0,),𝑞(0,] and 𝜑𝒜.(i)The Besov-type space ̇𝐵𝑠,𝜏𝑝,𝑞(𝑛) with 𝑝(0,] is defined to be the space of all 𝑓𝒮(𝑛) such that 𝑓̇𝐵𝑠,𝜏𝑝,𝑞(𝑛)={2𝑗𝑠(𝜑𝑗𝑓)}𝑗𝑞(𝐿𝑝𝜏(𝑛,))<.(ii)The Triebel-Lizorkin-type space ̇𝐹𝑠,𝜏𝑝,𝑞(𝑛) with 𝑝(0,) is defined to be the space of all 𝑓𝒮(𝑛) such that 𝑓̇𝐹𝑠,𝜏𝑝,𝑞(𝑛)={2𝑗𝑠(𝜑𝑗𝑓)}𝑗𝐿𝑝𝜏(𝑞(𝑛,))<.
Obviously, ̇𝐵𝑠,0𝑝,𝑞(𝑛̇𝐵)=𝑠𝑝,𝑞(𝑛) and ̇𝐹𝑠,0𝑝,𝑞(𝑛̇𝐹)=𝑠𝑝,𝑞(𝑛). We also remark that the spaces ̇𝐵𝑠,𝜏𝑝,𝑞(𝑛) and ̇𝐹𝑠,𝜏𝑝,𝑞(𝑛) are independent of the choice of 𝜑𝒜; see [14].

Remark 1.2. Let 𝑠.(i)For 𝑝(0,), it was proved in [29, Theorem 1(i)] that ̇𝐹𝑠,𝜏𝑝,𝑟(𝑛̇𝐹)=𝑠+𝑛(𝜏1/𝑝),(𝑛) when 𝑟(0,) and 𝜏(1/𝑝,), and ̇𝐹𝑠,𝜏𝑝,(𝑛̇𝐹)=𝑠+𝑛(𝜏1/𝑝),(𝑛) when 𝜏[1/𝑝,) with equivalent quasinorms. In [30, Corollary 5.7], it was proved that ̇𝐹𝑠,1/𝑝𝑝,𝑞(𝑛̇𝐹)=𝑠,𝑞(𝑛) with equivalent quasinorms for 𝑝(0,) and 𝑞(0,].(ii)For p(0,], it was proved in [29, Theorem 1(ii)] that ̇𝐵𝑠,𝜏𝑝,𝑟(𝑛̇𝐵)=𝑠+𝑛(𝜏1/𝑝),(𝑛) when 𝑟(0,) and 𝜏(1/𝑝,), and ̇𝐵𝑠,𝜏𝑝,(𝑛̇𝐵)=𝑠+𝑛(𝜏1/𝑝),(𝑛) when 𝜏[1/𝑝,) with equivalent quasinorms.
Next we recall the Hausdorff-type counterparts of 𝐿𝑝𝜏(𝑞(𝑛,)) and 𝑞(𝐿𝑝𝜏(𝑛,)). To this end, for 𝑥𝑛 and 𝑟(0,), let 𝐵(𝑥,𝑟)={𝑦𝑛|𝑥𝑦|<𝑟}. For 𝐸𝑛 and 𝑑(0,𝑛], the d-dimensional Hausdorff capacity of 𝐸 is defined by𝐻𝑑(𝐸)=inf𝑗𝑟𝑑𝑗𝐸𝑗𝐵𝑥𝑗,𝑟𝑗,(1.8) where the infimum is taken over all countable open ball coverings {𝐵(𝑥𝑗,𝑟𝑗)}𝑗 of 𝐸; see, for example, [31, 32].
For any function 𝑓𝑛[0,], the Choquet integral of 𝑓 with respect to 𝐻𝑑 is then defined by𝑛𝑓(𝑥)𝑑𝐻𝑑(𝑥)=0𝐻𝑑({𝑥𝑛𝑓(𝑥)>𝜆})𝑑𝜆.(1.9)
In what follows, we write +𝑛+1=𝑛×(0,). For any measurable function 𝜔 on +𝑛+1 and 𝑥𝑛, its nontangential maximal function 𝑁𝜔 is defined by𝑁𝜔(𝑥)=sup|𝑦𝑥|<𝑡||||𝜔(𝑦,𝑡),𝑥𝑛.(1.10)
For 𝑝(1,) and 𝜏[0,), the space 𝐿𝑝𝜏(𝑞(𝑛,)) with 𝑞(1,) is defined to be the space of all sequences 𝐺={𝑔𝑗}𝑗 of measurable functions on 𝑛 such that𝐺𝐿𝑝𝜏(𝑞(𝑛,))=inf𝜔𝑛𝑗||𝑔𝑗||(𝑥)𝑞𝜔𝑥,2𝑗𝑞𝑝/𝑞𝑑𝑥1/𝑞<,(1.11) and the space 𝑞(𝐿𝑝𝜏(𝑛,)) with 𝑞[1,) is defined to be the space of all sequences 𝐺={𝑔𝑗}𝑗 of measurable functions on 𝑛 such that 𝐺𝑞(𝐿𝑝𝜏(𝑛,))=inf𝜔𝑗𝑛||𝑔𝑗||(𝑥)𝑝𝜔𝑥,2𝑗𝑝𝑑𝑥𝑞/𝑝1/𝑝<,(1.12) where the infimums are taken over all nonnegative Borel measurable functions 𝜔 on +𝑛+1 satisfying 𝑛[]𝑁𝜔(𝑥)(𝑝𝑞)𝑑𝐻𝑛𝜏(𝑝𝑞)(𝑥)1,(1.13) and with the restriction that for any 𝑗,𝜔(,2𝑗) is allowed to vanish only where 𝑔𝑗 vanishes. Here and, in what follows, for all 𝑎,𝑏, the symbol 𝑎𝑏 denotes max{𝑎,𝑏} and, for 𝑡[1,], the symbol 𝑡 denotes its conjugate index, namely, 1/𝑡+1/𝑡=1.

Remark 1.3. By [15, Remark 2.1], we know that if 0<𝑎𝑏1/𝜏, then for all nonnegative measurable functions 𝜔 on +𝑛+1, 𝑛[]𝑁𝜔(𝑥)𝑎𝑑H𝑛𝜏𝑎(𝑥)1impliesthat𝑛[]𝑁𝜔(𝑥)𝑏𝑑𝐻𝑛𝜏𝑏(𝑥)1.(1.14)
We now recall the notion of the spaces 𝐵̇𝐻𝑠,𝜏𝑝,𝑞(𝑛) and 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛) introduced in [17].

Definition 1.4. Let 𝑠,𝑝(1,) and 𝜑𝒜.(i)The Besov-Hausdorff space 𝐵̇𝐻𝑠,𝜏𝑝,𝑞(𝑛) with 𝑞[1,) and 𝜏[0,1/(𝑝𝑞)] is defined to be the space of all 𝑓𝒮(𝑛) such that𝑓𝐵̇𝐻𝑠,𝜏𝑝,𝑞(𝑛)2=𝑗𝑠𝜑𝑗𝑓𝑗𝑞(𝐿𝑝𝜏(𝑛,))<.(1.15)(ii)The Triebel-Lizorkin-Hausdorff space 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛) with 𝑞(1,) and 𝜏[0,1/(𝑝𝑞)] is defined to be the space of all 𝑓𝒮(𝑛) such that𝑓𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛)2=𝑗𝑠𝜑𝑗𝑓𝑗𝐿𝑝𝜏(𝑞(𝑛,))<.(1.16)
Recall that 𝐵̇𝐻𝑠,0𝑝,𝑞(𝑛̇𝐵)=𝑠𝑝,𝑞(𝑛) and 𝐹̇𝐻𝑠,0𝑝,𝑞(𝑛̇𝐹)=𝑠𝑝,𝑞(𝑛). Moreover, the dual spaces of 𝐵̇𝐻𝑠,𝜏𝑝,𝑞(𝑛) and 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛) are, respectively, ̇𝐵𝑠,𝜏𝑝,𝑞(𝑛) and ̇𝐹𝑠,𝜏𝑝,𝑞(𝑛); see [13, 14].
Now we present the main results of this paper as follows.

Theorem 1.5. Let 𝛼,𝛾,𝜏[0,), and 𝑟(0,]. Suppose that 𝑚 satisfies (1.2) with .(i)If >𝑛[max(1/𝑝,1/𝑟)+1/2] and 𝑝(0,), then there exists a positive constant 𝐶 such that for all ̇𝐹𝑓𝛾,𝜏𝑝,𝑟(𝑛), 𝑇𝑚𝑓̇𝐹𝛼+𝛾,𝜏𝑝,𝑟(𝑛)𝐶𝑓̇𝐹𝛾,𝜏𝑝,𝑟(𝑛).(ii)If >𝑛(1/p+1/2) and p(0,], then there exists a positive constant C such that for all ̇Bf𝛾,𝜏p,r(n), TmḟB𝛼+𝛾,𝜏p,r(n)CḟB𝛾,𝜏p,r(n).

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

We also remark Theorem 1.5 when 𝜏=0 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 𝛼,𝛽,𝛽<𝛼, 𝑝(0,),𝑞,𝑟(0,], and 𝜏[0,). Assume that 𝑚 satisfies (1.2) with .(i)If >𝑛[max(1/𝑝,1/𝑟)+1/2] and 𝑝(0,) such that 𝛽𝑛/𝑝=𝛼𝑛/𝑝, then there exists a positive constant 𝐶 such that for all ̇𝐹𝑓0,𝜏𝑝,𝑟(𝑛), 𝑇𝑚𝑓̇𝐹𝑝𝛽,𝜏,𝑞(𝑛)𝐶𝑓̇𝐹0,𝜏𝑝,𝑟(𝑛).(1.17)(ii)If >𝑛(1/𝑝+1/2) and 𝑝(0,] such that 𝛽𝑛/𝑝=𝛼𝑛/𝑝, then there exists a positive constant 𝐶 such that for all ̇𝐵𝑓0,𝜏𝑝,𝑟(𝑛), 𝑇𝑚𝑓̇𝐵𝑝𝛽,𝜏,𝑟(𝑛)𝐶𝑓̇𝐵0,𝜏𝑝,𝑟(𝑛).(1.18)

We point out that Corollary 1.6(ii) when 𝜏=0 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 𝛼,𝛽,𝑝(0,),𝜏[0,) and 𝑟,𝑞(0,] such that 𝛽<𝛼. Let 𝑝(0,) such that 𝛽𝑛/𝑝=𝛼𝑛/𝑝. Assume that 𝑚 satisfies (1.2) with and >𝑛/2. Then there exists a positive constant 𝐶 such that for all ̇𝐹𝑓0,𝜏𝑝,𝑟(𝑛), 𝑇𝑚𝑓̇𝐹𝑝𝛽,𝜏,𝑞(𝑛)𝐶𝑓̇𝐹0,𝜏𝑝,𝑟(𝑛).(1.19)

As an immediate consequence of Theorem 1.7 and the lifting property of the space ̇𝐹𝑠,𝜏𝑝,𝑞(𝑛) (see, [14, Proposition 3.5]), we have the following conclusion, which shows that Theorem 1.7 has variant for any 𝑠 instead of 𝑠=0.

Corollary 1.8. Given 𝛼,𝛾,𝑝(0,) and 𝑟,𝑞(0,], let 𝛽 be real number with 𝛽<𝛼+𝛾 and 𝑝(0,) such that 𝛽𝑛/𝑝=𝛼+𝛾𝑛/𝑝. Assume that 𝑚 satisfies (1.2) with and >𝑛/2. Then there exists a positive constant 𝐶 such that for all ̇𝐹𝑓𝛾,𝜏𝑝,𝑟(𝑛), 𝑇𝑚𝑓̇𝐹𝑝𝛽,𝜏,𝑞(𝑛)𝐶𝑓̇𝐹𝛾,𝜏𝑝,𝑟(𝑛).

Remark 1.9. (i) We remark that, by taking 𝛽=0, 𝛼(0,𝑛), 𝑝(1,𝑛/𝛼), 𝑞=𝑟=2, 𝜏=0, and 𝑚(𝜉)=|𝜉|𝛼 for all 𝜉𝑛{0}, then Theorem 1.7 (and also Corollary 1.8 with 𝛾=0) 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 𝐼𝛼 maps boundedly from 𝐿𝑝(𝑛) to 𝐿𝑝(𝑛), where 1/𝑝=1/𝑝𝛼/𝑛. 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 𝛽=𝛼 and hence 𝑝=𝑝 (resp., 𝛽=𝛾+𝛼 and hence 𝑝=𝑝). Indeed, the assumption 𝛽<𝛼 (resp., 𝛽<𝛾+𝛼) and hence 𝑝>𝑝 play a crucial role in the proof of Theorem 1.7 in Section 3, which is not valid for the case that 𝛽=𝛼 (resp., 𝛽=𝛾+𝛼) and hence 𝑝=𝑝.
For (0,), let 𝑊2(𝑛) be the well-known Sobolev-Slobodeckij space on 𝑛. Recall that Triebel [9, Theorem 2] proved that for all 𝑠, 𝑝,𝑞(0,) and >𝑛(1/min{𝑝,𝑞,1}1/2), if 𝑚𝐿(𝑛) and𝜓𝑚𝑊2(𝑛)+sup𝑗𝜑2()𝑚𝑗𝑊2(𝑛)<,(1.20) then the Fourier multiplier 𝑇𝑚 is bounded from the inhomogeneous Triebel-Lizorkin space 𝐹𝑠𝑝,𝑞(𝑛) to itself, where 𝜓 and 𝜑 are Schwartz functions satisfying that 0𝜓,𝜑1, supp𝜓𝐵(0,2), 𝜓1 on 𝐵(0,1), supp𝜑𝐵(0,4)𝐵(0,1/2) and 𝜑1 on 𝐵(0,2)𝐵(0,1). From this, together with the embedding theorem [10, Theorem 2.7.1], we further deduce that, under the above assumptions on 𝑚, 𝑇𝑚 is also bounded from 𝐹𝑠𝑝,𝑞(𝑛) to 𝐹𝑠+𝑛/𝑝𝑝𝑛/𝑝,𝑟(𝑛) with 𝑠, 𝑝,𝑞(0,), 𝑝(𝑝,) and 𝑟(0,].
Notice that, if 𝑚 is as in Theorem 1.7 or Corollary 1.8, then 𝑚 is not necessary to belong to 𝐿(𝑛). For example, if 𝛼0, then 𝑚 as in (1.4) satisfies all the assumptions of Theorem 1.7 and Corollary 1.8, but 𝑚𝐿(𝑛). 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 𝛼=0 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 >𝑛(1/min{𝑝,𝑞,1}1/2) 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 𝑆𝐵𝑘𝛿(𝑟,𝜇,𝜈;𝑁,𝜆) of symbols 𝑎𝒮(𝑛×𝑛) with 𝑘, 𝛿[0,1], 𝜇,𝜈(0,], 𝑟[𝑛/𝜇,)(0,), 𝜆[1,] and 𝑁(𝑛/𝜆,). For a symbol 𝑎𝑆𝐵𝑘𝛿(𝑟,𝜇,𝜈;𝑁,𝜆), and any 𝑓𝒮(𝑛) and 𝑥𝑛, the nonregular pseudodifferential operator 𝑎(𝑥,𝐷) is defined as1𝑎(𝑥,𝐷)𝑓(𝑥)=(2𝜋)𝑛𝑛𝑒𝑖𝑥𝜉𝑎(𝑥,𝜉)𝑓(𝜉)𝑑𝜉.(1.21) Then Marschall [34, Theorem 9(a)] proved that for all 𝑘, 𝑝,𝑞(0,], 𝛿[0,1], 𝜇,𝜈(0,], 𝑟(𝑛/(1𝛿)𝜇,), 𝜆[1,], 𝑁(𝑛max{1/𝜆,1/2,1/𝑝,1/𝑞},) and 𝑛1max1,𝑝+1𝜇11(1𝛿)𝑟=𝑠<𝑟𝑛max𝜇1𝑝,0,(1.22) if either 𝑝(0,1] (𝑝(0,1) in case that 𝜇=) or 𝑝(𝜇,][𝜈,], then the operator 𝑎(𝑥,𝐷) with 𝑎𝑆𝐵𝑘𝛿(𝑟,𝜇,𝜈;𝑁,𝜆) is bounded from 𝐹𝑠+𝑘𝑝,𝑞(𝑛) to 𝐹𝑠𝑝,𝑞(𝑛), where 𝐹𝑠𝑝,𝑞(𝑛) denotes the inhomogeneous Triebel-Lizorkin space. This, together with the Sobolev embedding properties of Triebel-Lizorkin spaces, further implies that the operator 𝑎(𝑥,𝐷) is bounded from 𝐹𝑠+𝑘𝑝,𝑞(𝑛) to 𝐹𝑠+𝑛/𝑝𝑝𝑛/𝑝,𝑡(𝑛) with 𝑝(𝑝,) and 𝑡(0,].
Notice that, if 𝑚 satisfies the assumptions of Theorem 1.7 or Corollary 1.8, then 𝑚 is not necessary to belong to 𝒮(𝑛); see, for example, 𝑚 as in (1.4) with 𝛼(0,). 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 >𝑛[max(1/𝑝,1/2)] if 𝑟(0,2), or >𝑛[max(1/𝑝,1/𝑟)+1/21/𝑟] if 𝑟[2,], the operator 𝑇𝑚 maps ̇𝐹0𝑝,𝑟(𝑛) boundedly into ̇𝐹𝛽𝑝,𝑞(𝑛). However, from Theorem 1.7, we deduce that this conclusion is also true when >𝑛/2 if 𝑟(0,]. Therefore, even when 𝜏=0, 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 𝑝=, 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 𝛼,𝛾, 𝑝(1,) and 𝑚 satisfy (1.2) with .(i)If 𝑟(1,),𝜏[0,1/(𝑝𝑟)] and >𝑛[max(1/p,1/𝑟)+𝜏+1/2], then there exists a positive constant 𝐶 such that for all ̇𝐻𝑓𝐹𝛾,𝜏𝑝,𝑟(𝑛),𝑇𝑚𝑓𝐹̇𝐻𝛼+𝛾,𝜏𝑝,𝑟(𝑛)𝐶𝑓𝐹̇𝐻𝛾,𝜏𝑝,𝑟(𝑛).(1.23)(ii)If 𝑟[1,),𝜏[0,1/(𝑝𝑟)] and >𝑛(1/𝑝+𝜏+1/2), then there exists a positive constant 𝐶 such that for all ̇𝐻𝑓𝐵𝛾,𝜏𝑝,𝑟(𝑛),𝑇𝑚𝑓𝐵̇𝐻𝛼+𝛾,𝜏𝑝,𝑟(𝑛)𝐶𝑓𝐵̇𝐻𝛾,𝜏𝑝,𝑟(𝑛).(1.24)

Differently from the spaces ̇𝐹𝑠,𝜏𝑝,𝑞(𝑛) and ̇𝐵𝑠,𝜏𝑝,𝑞(𝑛), it is known that 𝑆(𝑛) is dense in the spaces 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛) and 𝐵̇𝐻𝑠,𝜏𝑝,𝑞(𝑛); see [13, Lemma 5.3] and [14, Lemma 6.3]. Thus, although 𝑇𝑚𝑓 is originally defined on 𝒮(𝑛), we can extend 𝑇𝑚 into the whole spaces 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛) and 𝐵̇𝐻𝑠,𝜏𝑝,𝑞(𝑛) by a density argument.

We remark that Theorem 1.10(i) when 𝜏=0 coincides with [12, Theorem 5.1] in the case that 𝑝(0,). 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 𝛼,𝛽,𝛽<𝛼, 𝑝(1,) and 𝑝(1,) such that 𝛽𝑛/𝑝=𝛼𝑛/𝑝. Assume that 𝑚 satisfies the condition (1.2) with .(i)Let 𝑟,𝑞(1,) and 𝜏[0,min{1/(𝑝𝑟),1/(𝑝𝑞)}] such that 𝜏(𝑝𝑟)𝜏(𝑝𝑞). If >𝑛[max(1/𝑝,1/𝑟)+𝜏+1/2], then there exists a positive constant 𝐶 such that for all ̇𝐻𝑓𝐹0,𝜏𝑝,𝑟(𝑛), 𝑇𝑚𝑓𝐹̇𝐻𝑝𝛽,𝜏,𝑞(𝑛)𝐶𝑓𝐹̇𝐻0,𝜏𝑝,𝑟(𝑛).(1.25)(ii)Let 𝑟[1,) and 𝜏[0,min{1/(𝑝𝑟),1/(𝑝𝑟)}] such that 𝜏(𝑝𝑟)=𝜏(𝑝𝑟). If >𝑛(1/𝑝+𝜏+1/2), then there exists a positive constant 𝐶 such that for all ̇𝐻𝑓𝐵0,𝜏𝑝,𝑟(𝑛), 𝑇𝑚𝑓𝐵̇𝐻𝑝𝛽,𝜏,𝑟(𝑛)𝐶𝑓𝐵̇𝐻0,𝜏𝑝,𝑟(𝑛).(1.26)

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 𝛼,𝑝(1,), 𝛽 with 𝛽<𝛼 and 𝑝(1,) such that 𝛽𝑛/𝑝=𝛼𝑛/𝑝. Let 𝑟,𝑞(1,) and 𝜏[0,min{1/(𝑝𝑟),1/(𝑝𝑞)}] such that 𝜏(𝑝𝑟)𝜏(𝑝𝑞). Assume that 𝑚 satisfies (1.2) with and 𝑛1>𝜏+2,if𝜏(𝑝𝑟)𝑛12𝜏,𝑟+12,if𝑝,𝑟(1,2),𝜏0.(1.27) Then there exists a positive constant 𝐶 such that for all ̇𝐻𝑓𝐹0,𝜏𝑝,𝑟(𝑛), 𝑇𝑚𝑓𝐹̇𝐻𝑝𝛽,𝜏,𝑞(𝑛)𝐶𝑓𝐹̇𝐻0,𝜏𝑝,𝑟(𝑛).(1.28)

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 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛) that can be deduced directly from [15, Theorem 4.1].

Corollary 1.13. Let 𝛼,𝛾 and 𝑝,𝑟,𝑞(1,). Let 𝛽 be a real number with 𝛽<𝛼+𝛾 and 𝑝(1,) such that 𝛽𝑛/𝑝=𝛼+𝛾𝑛/𝑝. Assume that 𝜏[0,min{1/(𝑝𝑞),1/(𝑝𝑟)}] satisfies that 𝜏(𝑝𝑟)𝜏(𝑝𝑞) and 𝑚 satisfies (1.2) with as in (1.27). Then there exists a positive constant 𝐶 such that for all ̇𝐻𝑓𝐹𝛾,𝜏𝑝,𝑟(𝑛), 𝑇𝑚𝑓𝐹̇𝐻𝑝𝛽,𝜏,𝑞(𝑛)𝐶𝑓𝐹̇𝐻𝛾,𝜏𝑝,𝑟(𝑛).

Corollary 1.13 implies that Theorem 1.12 has variant for any 𝑠 instead of 𝑠=0.

Remark 1.14. Recall that when 𝜏=0, the Triebel-Lizorkin-Hausdorff space 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛) is just the classical Triebel-Lizorkin space ̇𝐹𝑠𝑝,𝑞(𝑛). Thus, when 𝜏=0, Theorem 1.12 coincides with Theorem 1.7. In this sense, Theorem 1.12 when 𝜏=0 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 ̇𝐵𝑠,𝜏𝑝,𝑞(𝑛), ̇𝐹𝑠,𝜏𝑝,𝑞(𝑛), 𝐵̇𝐻𝑠,𝜏𝑝,𝑞(𝑛), and 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛) obtained in [18]. Additionally, to prove Theorems 1.7 and 1.12, we need first establish the generalized (weighted) 𝑔𝜆-function equivalent characterizations of ̇𝐹𝑠,𝜏𝑝,𝑞(𝑛) and 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛), 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 ̇𝐹𝑠𝑝,𝑞(𝑛), which induces an essential improvement of [12, Lemma 3.2(3)] such that we can replace the restriction 𝜆>𝑛[max(1/𝑝,1/𝑟)] in [12, Lemma 3.2(3)] by 𝜆>𝑛/𝑟. The proof of [12, Lemma 3.2(3)] strongly depends on the exact equivalent relations between the 𝐿𝑝(𝑛) norms of the generalized Lusin-area functions with different apertures, which is not clear whether it is still true if 𝐿𝑝(𝑛) 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 𝑛. This improvement further induces an improvement of Theorems 1.7 and 1.12 even when 𝜏=0, 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 𝑝=; see Remark 1.9(iv). In this paper, we seal this gap via a subtle application of the equivalence between the Triebel-Lizorkin space ̇𝐹𝑠,𝑞(𝑛) and the Triebel-Lizorkin-type space ̇𝐹𝑠,1/𝑝𝑝,𝑞(𝑛) 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 ̇𝐵𝑠,𝜏𝑝,𝑞(𝑛), ̇𝐹𝑠,𝜏𝑝,𝑞(𝑛), 𝐵̇𝐻𝑠,𝜏𝑝,𝑞(𝑛), and 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛) 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 ̇𝐹𝑠,𝜏𝑝,𝑞(𝑛) and 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛).
We point out that so far, for the Besov-type space ̇𝐵𝑠,𝜏𝑝,𝑞(𝑛) and the Besov-Hausdorff space 𝐵̇𝐻𝑠,𝜏𝑝,𝑞(𝑛), 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) 𝑔𝜆-function equivalent characterizations of ̇𝐹𝑠,𝜏𝑝,𝑞(𝑛) and 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛), which are not available for ̇𝐵𝑠,𝜏𝑝,𝑞(𝑛) and 𝐵̇𝐻𝑠,𝜏𝑝,𝑞(𝑛). 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 𝐴𝐵 means that 𝐴𝐶𝐵, where 𝐶 is a positive constant independent of the main parameter. If 𝐴𝐵 and 𝐵𝐴, then we write 𝐴𝐵. If 𝐸 is a subset of , we denote by 𝜒𝐸 the characteristic function of 𝐸.

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

In this section, we first recall some equivalent characterizations, established in [18], of ̇𝐹𝑠,𝜏𝑝,𝑞(𝑛), ̇𝐵𝑠,𝜏𝑝,𝑞(𝑛), 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛), and 𝐵̇𝐻𝑠,𝜏𝑝,𝑞(𝑛), 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) 𝑔𝜆-functions, which play a key role in the proofs of Theorems 1.7 and 1.12 in Section 3.

Let 𝜀(0,),𝑅+{1} and Φ𝒮(𝑛) satisfy that||||Φ(𝜉)>0on𝜉𝑛𝜀2<||𝜉||<2𝜀,𝐷𝛼Φ(0)=0|𝛼|𝑅.(2.1) In what follows, for any function 𝜑,𝑡(0,) and 𝑥𝑛, 𝜑𝑡(𝑥)=𝑡𝑛𝜑(𝑥/𝑡). For all 𝜑𝒮𝑁(𝑛), 𝑓𝒮𝑁(𝑛), 𝑡(0,),𝜆(0,), and 𝑥𝑛, let𝜑𝑡𝑓𝜆(𝑥)=sup𝑦𝑛||𝜑𝑡||𝑓(𝑥+𝑦)||𝑦||1+/𝑡𝜆,(2.2) which is called the Peetre-type maximal function of local means; see, for example, [18].

The following characterization of ̇𝐹𝑠,𝜏𝑝,𝑞(𝑛) was obtained in [18].

Theorem 2.1. Let 𝑠,𝜏[0,),𝑝(0,),𝑞(0,],𝜆(𝑛(1/𝑝1/𝑞),) and 𝑅+{1} such that 𝑠+𝑛𝜏<𝑅+1 and Φ be as in (2.1). Then the space ̇𝐹𝑠,𝜏𝑝,𝑞(𝑛) is characterized by ̇𝐹𝑠,𝜏𝑝,𝑞(𝑛)=𝑓𝒮𝑅(𝑛̇𝐹)𝑓𝑠,𝜏𝑝,𝑞(𝑛)𝑖<,𝑖{1,2,3},(2.3) where ̇𝐹𝑓𝑠,𝜏𝑝,𝑞(𝑛)1=sup𝑃𝒬1||𝑃||𝜏𝑃0(𝑃)𝑡𝑠𝑞||Φ𝑡||𝑓(𝑥)𝑞𝑑𝑡𝑡𝑝/𝑞𝑑𝑥1/𝑝,̇𝐹𝑓𝑠,𝜏𝑝,𝑞(𝑛)2=sup𝑃𝒬1||𝑃||𝜏𝑃0(𝑃)𝑡𝑠𝑞||Φ𝑡𝑓𝜆||(𝑥)𝑞𝑑𝑡𝑡𝑝/𝑞𝑑𝑥1/𝑝,̇𝐹𝑓𝑠,𝜏𝑝,𝑞(𝑛)3=sup𝑃𝒬1||𝑃||𝜏𝑃0(𝑃)𝑡𝑠𝑞|𝑧|<𝑡||Φ𝑡||𝑓(𝑥+𝑧)𝑞𝑑𝑧𝑑𝑡𝑡𝑛+1𝑝/𝑞𝑑𝑥1/𝑝,(2.4) with the usual modification made when 𝑞=.

Remark 2.2. Recall that when 𝜏[0,1/𝑝), the Triebel-Lizorkin-type spaces are just the Triebel-Lizorkin-Morrey spaces, that is, in the definition of Triebel-Lizorkin-type space, the sum 𝑗=𝑗𝑃 can be replaced by 𝑗; 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 (𝑃) replaced by in ̇𝐹𝑓𝑠,𝜏𝑝,𝑞(𝑛)1, ̇𝐹𝑓𝑠,𝜏𝑝,𝑞(𝑛)2 and ̇𝐹𝑓𝑠,𝜏𝑝,𝑞(𝑛)3 when 𝜏[0,1/𝑝). We omit the details.
The following Theorems 2.3 through 2.5 were established in [18].

Theorem 2.3. Let 𝑠,𝜏[0,),𝑝,𝑞(0,],𝜆(𝑛/𝑝,), and 𝑅+{1} such that 𝑠+𝑛𝜏<𝑅+1 and Φ be as in (2.1). Then the space ̇𝐵𝑠,𝜏𝑝,𝑞(𝑛) is characterized by ̇𝐵𝑠,𝜏𝑝,𝑞(𝑛)=𝑓𝒮𝑅(𝑛̇𝐵)𝑓𝑠,𝜏𝑝,𝑞(𝑛)𝑖<,𝑖{1,2},(2.5) where ̇𝐵𝑓𝑠,𝜏𝑝,𝑞(𝑛)1=sup𝑃𝒬1||𝑃||𝜏0(𝑃)𝑡𝑠𝑞𝑃||Φ𝑡||𝑓(𝑥)𝑝𝑑𝑥𝑞/𝑝𝑑𝑡𝑡1/𝑞,̇𝐵𝑓𝑠,𝜏𝑝,𝑞(𝑛)2=sup𝑃𝒬1||𝑃||𝜏0(𝑃)𝑡𝑠𝑞𝑃||Φ𝑡𝑓𝜆||(𝑥)𝑝𝑑𝑥𝑞/𝑝𝑑𝑡𝑡1/𝑞,(2.6) with the usual modifications made when 𝑞= or 𝑝=.

Theorem 2.4. Let 𝑠,𝑝,q(1,),𝜏[0,1/(𝑝𝑞)],𝜆(𝑛[max{1/𝑝,1/𝑞}+𝜏],) and 𝑅+{1} such that 𝑠+𝑛𝜏<𝑅+1 and Φ be as in (2.1). Then the space 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛) is characterized by 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛)=𝑓𝒮𝑅(𝑛̇𝐻)𝑓𝐹𝑠,𝜏𝑝,𝑞(𝑛)𝑖<,𝑖{1,2,3},(2.7) where ̇𝐻𝑓𝐹𝑠,𝜏𝑝,𝑞(𝑛)1=inf𝜔0𝑡𝑠𝑞||Φ𝑡||𝑓𝑞[]𝜔(,𝑡)𝑞𝑑𝑡𝑡1/𝑞𝐿𝑝(𝑛),̇𝐻𝑓𝐹𝑠,𝜏𝑝,𝑞(𝑛)2=inf𝜔0𝑡𝑠𝑞Φ𝑡𝑓𝜆𝑞[]𝜔(,𝑡)𝑞𝑑𝑡𝑡1/𝑞𝐿𝑝(𝑛),̇𝐻𝑓𝐹𝑠,𝜏𝑝,𝑞(𝑛)3=inf𝜔0𝑡𝑠𝑞|𝑧|<𝑡||Φ𝑡||𝑓(+𝑧)𝑞[]𝜔(+𝑧,𝑡)𝑞𝑑𝑧𝑑𝑡𝑡𝑛+11/𝑞𝐿𝑝(𝑛),(2.8) where the infimums are taken over all nonnegative Borel measurable functions 𝜔 on +𝑛+1 satisfying (1.13).

Theorem 2.5. Let 𝑠,𝑝(1,),𝑞[1,),𝜏[0,1/(𝑝𝑞)],𝜆(𝑛(1/𝑝+𝜏),) and 𝑅+{1} such that 𝑠+𝑛𝜏<𝑅+1 and Φ be as in (2.1). Then the space 𝐵̇𝐻𝑠,𝜏𝑝,𝑞(𝑛) is characterized by 𝐵̇𝐻𝑠,𝜏𝑝,𝑞(𝑛)=𝑓𝒮𝑅(𝑛̇𝐻)𝑓𝐵𝑠,𝜏𝑝,𝑞(𝑛)𝑖<,𝑖{1,2},(2.9) where ̇𝐻𝑓𝐵𝑠,𝜏𝑝,𝑞(𝑛)1=inf𝜔0𝑡𝑠𝑞Φ𝑡[]𝑓𝜔(,𝑡)1𝑞𝐿𝑝(𝑛)𝑑𝑡𝑡1/𝑞,̇𝐻𝑓𝐵𝑠,𝜏𝑝,𝑞(𝑛)2=inf𝜔0𝑡𝑠𝑞Φ𝑡𝑓𝜆[]𝜔(,𝑡)1𝑞𝐿𝑝(𝑛)𝑑𝑡𝑡1/𝑞,(2.10) where the infimums are taken over all nonnegative Borel measurable functions 𝜔 on +𝑛+1 satisfying (1.13).

Remark 2.6. (i) The space 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛) is a quasi-Banach space; see [13, 14, 17]. Indeed, by [17, Remarks 7.1 and 7.3], we know that for any 𝑓1,𝑓2̇𝐻𝐹𝑠,𝜏𝑝,𝑞(𝑛), 𝑓1+𝑓2𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛)21/(𝑝𝑞)𝑓1𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛)+𝑓2𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛).(2.11)
(ii)By the Aoki-Rolewicz theorem ([35, 36]), there exists 𝑣(0,1] such that𝑗𝑓𝑗𝑣𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛)𝑗𝑓𝑗𝑣𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛)(2.12) for all {𝑓𝑗}𝑗̇𝐻𝐹𝑠,𝜏𝑝,𝑞(𝑛). Indeed, 𝑣=(𝑝𝑞)/(1+(𝑝𝑞)) does the job.
(iii)The conclusions in (i) and (ii) are also true for the space 𝐵̇H𝑠,𝜏𝑝,𝑞(𝑛).
Next we establish a new characterization of the spaces ̇𝐹𝑠,𝜏𝑝,𝑞(𝑛) and 𝐹̇𝐻𝑠,𝜏𝑝,𝑞(𝑛). Let 𝑞(0,], 𝜆(0,), and 𝜔 be a nonnegative Borel measurable function. In what follows, for 𝑅+{1}, 𝑓𝒮𝑅(𝑛) and 𝜑 in (2.1), set𝑢(𝑥,𝑡)=𝑓𝜑𝑡(𝑥),𝑢𝜆(𝑥,𝑡)=sup𝑦𝑛||||||||𝑢(𝑦,𝑡)1+𝑥𝑦𝑡𝜆,(2.13) for all 𝑥𝑛 and 𝑡(0,). For all 𝑏(0,),𝑠, and 𝑥𝑛, recall that the generalized weighted Lusin-area function 𝑆𝑠𝑏,𝑞(𝜔,𝑢)(𝑥) and the generalized weighted 𝑔𝜆-function 𝐺𝑠𝜆,𝑞(𝜔,𝑢)(𝑥) are defined, respectively, by 𝑆𝑠𝑏,𝑞(𝜔,𝑢)(𝑥)=0𝑡𝑠𝑞|𝑦𝑥|<𝑏𝑡||||𝑢(𝑦,𝑡)𝑞[]𝜔(𝑦,𝑡)𝑞(𝑏𝑡)𝑛𝑑𝑦𝑑𝑡𝑡1/𝑞,𝐺𝑠𝜆,𝑞(𝜔,𝑢)(𝑥)=0𝑡𝑠𝑞𝑛||||𝑢(𝑦,𝑡)𝑞||||1+𝑥𝑦𝑡𝜆𝑞[]𝜔(𝑦,𝑡)𝑞𝑑𝑦𝑑𝑡𝑡𝑛+11/𝑞.(2.14) If 𝜔(𝑥,𝑡)1, then 𝑆𝑠𝑏,𝑞(𝜔,𝑢) and 𝐺𝑠𝜆,𝑞(𝜔,𝑢) are called, respectively, the generalized Lusin-area function, denoted by 𝑆𝑠𝑏,𝑞(𝑢), and the generalized 𝑔𝜆-function, denoted by 𝐺𝑠𝜆,𝑞(𝑢).
In what follows, for 𝜏[0,) and 𝑝(0,), let 𝐿𝑝𝜏(𝑛) be the set of all functions 𝑓𝐿𝑝loc(𝑛)  such that 𝑓𝐿𝑝𝜏(𝑛)=sup𝑝𝒬1||𝑃||𝜏𝑃||||𝑓(𝑥)𝑝𝑑𝑥1/𝑝<.(2.15)

Theorem 2.7. Let 𝑠, 𝑝(0,), 𝜏[0,1/𝑝), 𝑞(0,], 𝜆(𝑛/𝑞,) and 𝑅+{1} such that 𝑠+𝑛𝜏<𝑅+1. Then ̇𝐹𝑓𝑠,𝜏𝑝,𝑞(𝑛) if and only if 𝑓𝒮𝑅(𝑛) and 𝐺𝑠𝜆,𝑞(𝑢)𝐿𝑝𝜏(𝑛), where 𝑢 is as in (2.13). Moreover, there exists a positive constant 𝐶 such that for all ̇𝐹𝑓𝑠,𝜏𝑝,𝑞(𝑛), 𝐶1𝑓̇𝐹𝑠,𝜏𝑝,𝑞(𝑛)𝐺𝑠𝜆,𝑞(𝑢)𝐿𝑝𝜏(𝑛)𝐶𝑓̇𝐹𝑠,𝜏𝑝,𝑞(𝑛).(2.16)

Proof. Assume 𝑓𝒮𝑅(𝑛) and 𝐺𝑠𝜆,𝑞(𝑢)𝐿𝑝𝜏(𝑛). Notice that for any 𝜆(0,) and 𝑥𝑛, 𝑆𝑠1,𝑞(𝑢)(𝑥)2𝜆0|𝑦𝑥|<𝑡𝑡𝑠||||𝑢(𝑦,𝑡)𝑞||||1+𝑥𝑦𝑡𝜆𝑞𝑡𝑛𝑑𝑦𝑑𝑡𝑡1/𝑞=2𝜆𝐺𝑠𝜆,𝑞(𝑢)(𝑥).(2.17) Then, from Remark 2.2, we deduce that ̇𝐹𝑓𝑠,𝜏𝑝,𝑞(𝑛) and 𝑓̇𝐹𝑠,𝜏𝑝,𝑞(𝑛)𝑆𝑠1,𝑞(𝑢)𝐿𝑝𝜏(𝑛)𝐺𝑠𝜆,𝑞(𝑢)𝐿𝑝𝜏(𝑛),(2.18) which completes the proof of the sufficiency of the theorem.
Conversely, suppose that ̇𝐹𝑓𝑠,𝜏𝑝,𝑞(𝑛). Then by Theorem 2.1, 𝑓𝒮𝑅(𝑛). Moreover, similar to the proof of [18, Theorem 3.1], for any 𝑘, we see thatsup𝑃𝒬1||𝑃||𝜏𝑃0|𝑦𝑥|<𝑡𝑡𝑠||𝜑2𝑘𝑡||𝑓(𝑦)𝑞𝑡𝑛𝑑𝑦𝑑𝑡𝑡𝑝/𝑞𝑑𝑥1/𝑝𝐶sup𝑃𝒬1||𝑃||𝜏𝑃𝑗=2𝑗𝑠𝑞||𝜑𝑘+𝑗||𝑓(𝑥)𝑞𝑝/𝑞𝑑𝑥1/𝑝,(2.19) where 𝐶 is a positive constant independent of 𝑘 and 𝑓. Then by changing variables, we conclude that sup𝑃𝒬1||𝑃||𝜏𝑃0|𝑦𝑥|<𝑡𝑡𝑠||𝜑2𝑘𝑡||𝑓(𝑦)𝑞𝑡𝑛𝑑𝑦𝑑𝑡𝑡𝑝/𝑞𝑑𝑥1/𝑝2𝑘𝑠sup𝑃𝒬1||𝑃||𝜏𝑃𝑗=2𝑗𝑠𝑞||𝜑𝑗||𝑓(𝑥)𝑞𝑝/𝑞𝑑𝑥1/𝑝2𝑘𝑠𝑓̇𝐹𝑠,𝜏𝑝,𝑞(𝑛),(2.20) where the last inequality follows from the equivalence between Triebel-Lizorkin spaces and Triebel-Lizorkin-Morrey spaces when 𝜏[0,1/𝑝); see [16] and also Remark 2.2. By changing variables, we know that for all 𝑥𝑛, 𝐺𝑠𝜆,𝑞(𝑢)(𝑥)=0|𝑦𝑥|<𝑡𝑡𝑠||𝜑𝑡||𝑓(𝑦)𝑞||||1+𝑥𝑦𝑡𝜆𝑞𝑡𝑛+𝑑𝑦𝑘=12𝑘1𝑡|𝑦𝑥|<2𝑘𝑡𝑑𝑦𝑑𝑡𝑡1/𝑞𝑘=02𝑘𝜆𝑞0|𝑦𝑥|<2𝑘𝑡𝑡𝑠||𝜑𝑡||𝑓(𝑦)𝑞𝑡𝑛𝑑𝑦𝑑𝑡𝑡1/𝑞𝑘=02𝑘(𝜆𝑞𝑠𝑞𝑛)0|𝑦𝑥|<𝑡𝑡𝑠||𝜑2𝑘𝑡||𝑓(𝑦)𝑞𝑡𝑛𝑑𝑦𝑑𝑡𝑡1/𝑞.(2.21) Thus, when 𝑝𝑞, from the well-known inequality that for all 𝑑(0,1] and {𝛼𝑗}𝑗, 𝑗||𝛼𝑗||𝑑𝑗||𝛼𝑗||𝑑(2.22) it follows that 𝐺𝑠𝜆,𝑞(𝑢)𝐿𝑝𝜏(𝑛)sup𝑃𝒬1||𝑃||𝜏𝑃𝑘=02𝑘(𝜆𝑞𝑠𝑞𝑛)0|𝑦𝑥|<𝑡𝑡𝑠||𝜑2𝑘𝑡||𝑓(𝑦)𝑞𝑡𝑛𝑑𝑦𝑑𝑡𝑡𝑝/𝑞𝑑𝑥1/𝑝sup𝑃𝒬1||𝑃||𝜏𝑃𝑘=02𝑘(𝜆𝑠𝑛/𝑞)𝑝0|𝑦𝑥|<𝑡𝑡𝑠||𝜑2𝑘𝑡||𝑓(𝑦)𝑞𝑡𝑛𝑑𝑦𝑑𝑡𝑡𝑝/𝑞𝑑𝑥1/𝑝𝑘=02𝑘(𝜆𝑠𝑛/𝑞)𝑝×sup𝑃𝒬1||𝑃||𝜏𝑝𝑃0|𝑦𝑥|<𝑡𝑡𝑠||𝜑2𝑘𝑡||𝑓(𝑦)𝑞𝑡𝑛𝑑𝑦𝑑𝑡𝑡𝑝/𝑞𝑑𝑥1/𝑝𝑘=02𝑘(𝜆𝑛/𝑞)𝑝𝑓𝑝̇𝐹𝑠,𝜏𝑝,𝑞(𝑛)1/𝑝𝑓̇𝐹𝑠,𝜏𝑝,𝑞(𝑛),(2.23) where the last inequality follows from (2.20) and 𝜆>𝑛/𝑞. Similarly, when 𝑝>𝑞, by Minkowski’s inequality and (2.20), we see that 𝐺𝑠𝜆,𝑞(𝑢)𝐿𝑝𝜏(𝑛)sup𝑃𝒬1||𝑃||𝜏𝑘=02𝑘(𝜆𝑞𝑠𝑞𝑛)×𝑃0|𝑦𝑥|<𝑡𝑡𝑠||𝜑2𝑘𝑡||𝑓(𝑦)𝑞𝑡𝑛𝑑𝑦𝑑𝑡𝑡𝑝/𝑞𝑑𝑥𝑞/𝑝1/𝑞𝑘=02𝑘(𝜆𝑞𝑠𝑞𝑛)sup𝑃𝒬1||𝑃||𝜏𝑞×𝑃0|𝑦𝑥|<𝑡𝑡𝑠||𝜑2𝑘𝑡||𝑓(𝑦)𝑞𝑡𝑛𝑑𝑦𝑑𝑡𝑡𝑝/𝑞𝑑𝑥𝑞/𝑝1/𝑞𝑘=02𝑘(𝜆𝑞𝑛)𝑓𝑞̇𝐹𝑠,𝜏𝑝,𝑞(𝑛)1/𝑞𝑓̇