# Fourier Multipliers on Triebel-Lizorkin-Type Spaces

Authors: Dachun Yang; Wen Yuan; Ciqiang Zhuo
Journal: Journal of Function Spaces and Applications
 (2012)

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## 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.


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## Body

## 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, [1–10]. Indeed, the best-known Fourier multiplier on Lp(ℝn) 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 Ḃp,qs(ℝn) and Triebel-Lizorkin spaces Ḟp,qs(ℝn) 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 Ḃp,qs,τ(ℝn), the Triebel-Lizorkin-type space Ḟp,qs,τ(ℝn), and their preduals, the Besov-Hausdorff space BḢp,qs,τ(ℝn) and the Triebel-Lizorkin-Hausdorff space FḢp,qs,τ(ℝn). These spaces were recently introduced and investigated in [13–18] 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., [21–24]), Triebel-Lizorkin-Morrey spaces and Morrey spaces (see, e.g., [16, 25–28]). To study the boundedness of Fourier multipliers on Ḟp,qs,τ(ℝn) and FḢp,qs,τ(ℝn), 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 Ḟp,qs,τ(ℝn) and FḢp,qs,τ(ℝn), 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 Ḃp,qs,τ(ℝn) and the Besov-Hausdorff space BḢp,qs,τ(ℝn), some of the corresponding results are also presented.We begin with some notions and notation. In what follows, letℕ:={1,2,…} and ℤ+:=ℕ∪{0}; let 𝒮(ℝn) be the space of all the Schwartz functions on ℝn with the classical topology and 𝒮′(ℝn) its topological dual space, namely, the set of all continuous linear functionals on 𝒮(ℝn) endowed with the weak-* topology.Following Triebel [10], let(1.1)S∞(Rn):={φ∈S(Rn):∫Rnφ(x)xγdx=0∀multi-indicesγ∈(N∪{0})n}
and consider 𝒮∞(ℝn) as a subspace of 𝒮(ℝn), including the topology. Use 𝒮∞′(ℝn) to denote the topological dual space of 𝒮∞(ℝn), namely, the set of all continuous linear functionals on 𝒮∞(ℝn). We also endow 𝒮∞′(ℝn) with the weak-* topology. Let 𝒫(ℝn) be the set of all polynomials on ℝn. It is well known that 𝒮∞′(ℝn)=𝒮′(ℝn)/𝒫(ℝn) as topological spaces. Similarly, for any N∈ℤ+, the space𝒮N(ℝn) is defined to be the set of all Schwartz functions satisfying that ∫ℝnφ(x)xγdx=0 for all multi-indices γ∈ℤ+n with |γ|≤N and 𝒮N′(ℝn) its topological dual space. We also let 𝒮-1(ℝn):=𝒮(ℝn). As usual, ϕ̂ denotes the Fourier transform of an integrable function ϕ on ℝn, which is defined as ϕ̂(ξ):=∫ℝne-iξ·xϕ(x)dx for all ξ∈ℝn.The following notion of Fourier multipliers whenα≠0 was originally introduced by Cho and Kim in [11] and Cho in [12]. For ℓ∈ℕ and α∈ℝ, assume that m∈Cℓ(ℝn∖{0}) satisfies that for all |σ|≤ℓ,(1.2)sup⁡R∈(0,∞)[R-n+2α+2|σ|∫R≤|ξ|<2R|∂ξσm(ξ)|2dξ]≤Aσ<∞,
where for σ:=(σ1,…,σn)∈ℤ+n, ∂σ:=(∂/∂x1)σ1⋯(∂/∂xn)σn. The Fourier multiplierTm is defined by setting, for all f∈𝒮∞(ℝn), (Tmf)̂:=mf̂.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
(1.3)(Rjf)̂(ξ):=-i(ξi/|ξ|)f̂(ξ)
for ξ∈ℝn∖{0} and j∈{1,…,n}. When α≠0, a typical example satisfying (1.2) for any ℓ∈ℕ is given by(1.4)m(ξ):=|ξ|-αforξ∈Rn∖{0},
another example is the symbol of a differential operator ∂σ of order α:=σ1+⋯+σn with σ:=(σ1,…,σn)∈ℤ+n.To recall the notions ofḂp,qs,τ(ℝn) and Ḟp,qs,τ(ℝn) in [14] and, their predual spaces, BḢp,qs,τ(ℝn) and FḢp,qs,τ(ℝn) in [13, 14], we need the following notation.Forj∈ℤ and k∈ℤn, denote by Qjk the dyadic cube2-j([0,1)n+k) and ℓ(Qjk) its side length. Let 𝒬:={Qjk:j∈ℤ,k∈ℤn}, 𝒬j:={Q∈𝒬:ℓ(Q)=2-j} and jQ:=-log2ℓ(Q) for all Q∈𝒬.Letq∈(0,∞] and τ∈[0,∞). The spaceLτp(ℓq(ℝn,ℤ)) with p∈(0,∞) is defined to be the set of all sequences G:={gj}j∈ℤ of measurable functions on ℝn such that(1.5)‖G‖Lτp(lq(Rn,Z)):=sup⁡P∈Q1|P|τ{∫P[∑j=jP∞|gj(x)|q]p/qdx}1/p<∞.
Similarly, the spaceℓq(Lτp(ℝn,ℤ)) with p∈(0,∞] is defined to be the space of all sequences G:={gj}j∈ℤ of measurable functions on ℝn such that(1.6)‖G‖lq(Lτp(Rn,Z)):=sup⁡P∈Q1|P|τ{∑j=jP∞[∫P|gj(x)|pdx]q/p}1/q<∞.
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 functionsφ∈𝒮(ℝn)  such that(1.7)supp⁡φ̂⊂{ξ∈Rn:12≤|ξ|≤2},|φ̂(ξ)|≥C>0if35≤|ξ|≤53.
Now we recall the notions of the Besov-type space Ḃp,qs,τ(ℝn) and the Triebel-Lizorkin-type space Ḟp,qs,τ(ℝn) from [14]. In what follows, for any j∈ℤ and φ∈𝒜, let φj(x):=2jnφ(2jx) for all x∈ℝn.Definition 1.1.
Lets∈ℝ,τ∈[0,∞),q∈(0,∞] and φ∈𝒜.(i)
TheBesov-type spaceḂp,qs,τ(ℝn) with p∈(0,∞] is defined to be the space of all f∈𝒮∞′(ℝn) such that ∥f∥Ḃp,qs,τ(ℝn):=∥{2js(φj*f)}j∈ℤ∥ℓq(Lτp(ℝn,ℤ))<∞.(ii)
TheTriebel-Lizorkin-type spaceḞp,qs,τ(ℝn) with p∈(0,∞) is defined to be the space of all f∈𝒮∞′(ℝn) such that ∥f∥Ḟp,qs,τ(ℝn):=∥{2js(φj*f)}j∈ℤ∥Lτp(ℓq(ℝn,ℤ))<∞.
Obviously,Ḃp,qs,0(ℝn)=Ḃp,qs(ℝn) and Ḟp,qs,0(ℝn)=Ḟp,qs(ℝn). We also remark that the spaces Ḃp,qs,τ(ℝn) and Ḟp,qs,τ(ℝn) are independent of the choice of φ∈𝒜; see [14].Remark 1.2.
Lets∈ℝ.(i)
Forp∈(0,∞), it was proved in [29, Theorem 1(i)] that Ḟp,rs,τ(ℝn)=Ḟ∞,∞s+n(τ-1/p)(ℝn) when r∈(0,∞) and τ∈(1/p,∞), and Ḟp,∞s,τ(ℝn)=Ḟ∞,∞s+n(τ-1/p)(ℝn) when τ∈[1/p,∞) with equivalent quasinorms. In [30, Corollary 5.7], it was proved that Ḟp,qs,1/p(ℝn)=Ḟ∞,qs(ℝn) with equivalent quasinorms for p∈(0,∞) and q∈(0,∞].(ii)
Forp∈(0,∞], it was proved in [29, Theorem 1(ii)] that Ḃp,rs,τ(ℝn)=Ḃ∞,∞s+n(τ-1/p)(ℝn) when r∈(0,∞) and τ∈(1/p,∞), and Ḃp,∞s,τ(ℝn)=Ḃ∞,∞s+n(τ-1/p)(ℝn) when τ∈[1/p,∞) with equivalent quasinorms.
Next we recall the Hausdorff-type counterparts ofLτp(ℓq(ℝn,ℤ)) and ℓq(Lτp(ℝn,ℤ)). To this end, for x∈ℝn and r∈(0,∞), let B(x,r):={y∈ℝn:|x-y|<r}. For E⊂ℝn and d∈(0,n], the d-dimensional Hausdorff capacity of E is defined by(1.8)Hd(E):=inf⁡{∑jrjd:E⊂⋃jB(xj,rj)},
where the infimum is taken over all countable open ball coverings {B(xj,rj)}j of E; see, for example, [31, 32].
For any functionf:ℝn→[0,∞], the Choquet integral of f with respect to Hd is then defined by(1.9)∫Rnf(x)dHd(x):=∫0∞Hd({x∈Rn:f(x)>λ})dλ.
In what follows, we writeℝ+n+1:=ℝn×(0,∞). For any measurable function ω on ℝ+n+1 and x∈ℝn, its nontangential maximal functionNω is defined by(1.10)Nω(x):=sup⁡|y-x|<t|ω(y,t)|,x∈Rn.
Forp∈(1,∞) and τ∈[0,∞), the spaceLτp(ℓq(ℝñ,ℤ)) with q∈(1,∞) is defined to be the space of all sequences G:={gj}j∈ℤ of measurable functions on ℝn such that(1.11)‖G‖Lτp(lq(Rñ,Z)):=inf⁡ω{∫Rn(∑j∈Z|gj(x)|q[ω(x,2-j)]-q)p/qdx}1/q<∞,
and the spaceℓq(Lτp(ℝñ,ℤ)) with q∈[1,∞) is defined to be the space of all sequences G:={gj}j∈ℤ of measurable functions on ℝn such that
(1.12)‖G‖lq(Lτp(Rñ,Z)):=inf⁡ω{∑j∈Z(∫Rn|gj(x)|p[ω(x,2-j)]-pdx)q/p}1/p<∞,
where the infimums are taken over all nonnegative Borel measurable functions ω on ℝ+n+1 satisfying
(1.13)∫Rn[Nω(x)](p∨q)′dHnτ(p∨q)′(x)≤1,
and with the restriction that for any j∈ℤ,ω(·,2-j) is allowed to vanish only where gj vanishes. Here and, in what follows, for all a,b∈ℝ, the symbola∨b denotes max{a,b} and, for t∈[1,∞], the symbolt′ denotes its conjugate index, namely, 1/t+1/t′=1.Remark 1.3.
By [15, Remark 2.1], we know that if 0<a≤b≤1/τ, then for all nonnegative measurable functions ω on ℝ+n+1,
(1.14)∫Rn[Nω(x)]adHnτa(x)≤1implies that∫Rn[Nω(x)]bdHnτb(x)≲1.
We now recall the notion of the spacesBḢp,qs,τ(ℝn) and FḢp,qs,τ(ℝn) introduced in [17].Definition 1.4.
Lets∈ℝ,p∈(1,∞) and φ∈𝒜.(i)
TheBesov-Hausdorff spaceBḢp,qs,τ(ℝn) with q∈[1,∞) and τ∈[0,1/(p∨q)′] is defined to be the space of all f∈𝒮∞′(ℝn) such that(1.15)‖f‖BḢp,qs,τ(Rn):=‖{2js(φj*f)}j∈Z‖lq(Lτp(Rñ,Z))<∞.(ii)
TheTriebel-Lizorkin-Hausdorff spaceFḢp,qs,τ(ℝn) with q∈(1,∞) and τ∈[0,1/(p∨q)′] is defined to be the space of all f∈𝒮∞′(ℝn) such that(1.16)‖f‖FḢp,qs,τ(Rn):=‖{2js(φj*f)}j∈Z‖Lτp(lq(Rñ,Z))<∞.
Recall thatBḢp,qs,0(ℝn)=Ḃp,qs(ℝn) and FḢp,qs,0(ℝn)=Ḟp,qs(ℝn). Moreover, the dual spaces of BḢp,qs,τ(ℝn) and FḢp,qs,τ(ℝn) are, respectively, Ḃp′,q′-s,τ(ℝn) and Ḟp′,q′-s,τ(ℝn); see [13, 14].
Now we present the main results of this paper as follows.Theorem 1.5.
Letα,γ∈ℝ,τ∈[0,∞), and r∈(0,∞]. Suppose that m satisfies (1.2) with ℓ∈ℕ.(i)
Ifℓ>n[max(1/p,1/r)+1/2] and p∈(0,∞), then there exists a positive constant C such that for all f∈Ḟp,rγ,τ(ℝn), ∥Tmf∥Ḟp,rα+γ,τ(ℝn)≤C∥f∥Ḟp,rγ,τ(ℝn).(ii)
Ifℓ>n(1/p+1/2) and p∈(0,∞], then there exists a positive constant C such that for all f∈Ḃp,rγ,τ(ℝn), ∥Tmf∥Ḃp,rα+γ,τ(ℝn)≤C∥f∥Ḃp,rγ,τ(ℝn).We remark that the Fourier multiplierTm is originally defined on 𝒮∞(ℝn). Although 𝒮∞(ℝn) may not be dense in Ḟp,qs,τ(ℝn) and Ḃp,qs,τ(ℝn), Tm can still be defined on the whole spaces Ḟp,qs,τ(ℝn) and Ḟp,qs,τ(ℝn) in a suitable way; see (3.10) and Lemma 3.4 below.We also remark Theorem1.5 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 Theorem1.5 and [14, Proposition 3.3], we immediately deduce the following conclusion. We omit the details.Corollary 1.6.
Letα,β∈ℝ,β<α, p∈(0,∞),q,r∈(0,∞], and τ∈[0,∞). Assume that m satisfies (1.2) with ℓ∈ℕ.(i)
Ifℓ>n[max(1/p,1/r)+1/2] and p*∈(0,∞) such that β-n/p*=α-n/p, then there exists a positive constant C such that for all f∈Ḟp,r0,τ(ℝn),
(1.17)‖Tmf‖Ḟp*,qβ,τ(Rn)≤C‖f‖Ḟp,r0,τ(Rn).(ii)
Ifℓ>n(1/p+1/2) and p*∈(0,∞] such that β-n/p*=α-n/p, then there exists a positive constant C such that for all f∈Ḃp,r0,τ(ℝn),
(1.18)‖Tmf‖Ḃp*,rβ,τ(Rn)≤C‖f‖Ḃp,r0,τ(Rn).We point out that Corollary1.6(ii) when τ=0 completely covers Cho and Kim [11, Theorem 1.1] and Cho [12, Theorem 7.1].Moreover, the range ofℓ in Corollary 1.6(i) can be essentially improved as indicated by the following theorem.Theorem 1.7.
Letα,β∈ℝ,p∈(0,∞),τ∈[0,∞) and r,q∈(0,∞] such that β<α. Let p*∈(0,∞) such that β-n/p*=α-n/p. Assume that m satisfies (1.2) with ℓ∈ℕ and ℓ>n/2. Then there exists a positive constant C such that for all f∈Ḟp,r0,τ(ℝn),
(1.19)‖Tmf‖Ḟp*,qβ,τ(Rn)≤C‖f‖Ḟp,r0,τ(Rn).As an immediate consequence of Theorem1.7 and the lifting property of the space Ḟp,qs,τ(ℝn) (see, [14, Proposition 3.5]), we have the following conclusion, which shows that Theorem 1.7 has variant for any s∈ℝ instead of s=0.Corollary 1.8.
Givenα,γ∈ℝ,p∈(0,∞) and r,q∈(0,∞], let β be real number with β<α+γ and p*∈(0,∞) such that β-n/p*=α+γ-n/p. Assume that m satisfies (1.2) with ℓ∈ℕ and ℓ>n/2. Then there exists a positive constant C such that for all f∈Ḟp,rγ,τ(ℝn), ∥Tmf∥Ḟp*,qβ,τ(ℝn)≤C∥f∥Ḟp,rγ,τ(ℝn).Remark 1.9.
(i) We remark that, by takingβ=0, α∈(0,n), p∈(1,n/α), q=r=2, τ=0, and m(ξ):=|ξ|-α for all ξ∈ℝn∖{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 potentialIα maps boundedly from Lp(ℝn) to Lp*(ℝn), where 1/p*=1/p-α/n. In this sense, Theorem 1.7 (and hence Corollary 1.8) is a generalization of the Hardy-Littlewood-Sobolev for fractional integrals.
(ii) Theorem1.7 (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.
Forℓ∈(0,∞), let W2ℓ(ℝn) be the well-known Sobolev-Slobodeckij space on ℝn. Recall that Triebel [9, Theorem 2] proved that for all s∈ℝ, p,q∈(0,∞) and ℓ>n(1/min{p,q,1}-1/2), if m∈L∞(ℝn) and(1.20)‖ψm‖W2l(Rn)+sup⁡j∈N‖φ(⋅)m(2j⋅)‖W2l(Rn)<∞,
then the Fourier multiplier Tm is bounded from the inhomogeneous Triebel-Lizorkin spaceFp,qs(ℝn)to itself, where ψ and φ are Schwartz functions satisfying that 0≤ψ,φ≤1, suppψ⊂B(0,2)¯, ψ≡1 on B(0,1)¯, suppφ⊂B(0,4)¯∖B(0,1/2) and φ≡1 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 fromFp,qs(ℝn)toFp*,rs+n/p*-n/p(ℝn) with s∈ℝ, p,q∈(0,∞), p*∈(p,∞) and r∈(0,∞].
Notice that, ifm is as in Theorem 1.7 or Corollary 1.8, then m is not necessary to belong to L∞(ℝn). For example, if α≠0, then m as in (1.4) satisfies all the assumptions of Theorem 1.7 and Corollary 1.8, but m∉L∞(ℝn). 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 ℓ in [9, Theorem 2] is quite different from the requirement of ℓ in Theorem 1.7 and Corollary 1.8. Moreover, the restriction ℓ>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 SBδk(r,μ,ν;N,λ) of symbols a∈𝒮′(ℝn×ℝn) with k∈ℝ, δ∈[0,1], μ,ν∈(0,∞], r∈[n/μ,∞)∩(0,∞), λ∈[1,∞] and N∈(n/λ,∞). For a symbol a∈SBδk(r,μ,ν;N,λ), and any f∈𝒮′(ℝn) and x∈ℝn, the nonregular pseudodifferential operatora(x,D) is defined as(1.21)a(x,D)f(x):=1(2π)n∫Rneix⋅ξa(x,ξ)f(ξ)dξ.
Then Marschall [34, Theorem 9(a)] proved that for all k∈ℝ, p,q∈(0,∞], δ∈[0,1], μ,ν∈(0,∞], r∈(n/(1-δ)μ,∞), λ∈[1,∞], N∈(nmax{1/λ,1/2,1/p,1/q},∞) and
(1.22)n(max⁡{1,1p+1μ}-1)-(1-δ)r=s<r-nmax⁡{1μ-1p,0},
if either p∈(0,1] (p∈(0,1) in case that μ=∞) or p∈(μ,∞]∩[ν,∞], then the operator a(x,D) with a∈SBδk(r,μ,ν;N,λ) is bounded from Fp,qs+k(ℝn) to Fp,qs(ℝn), where Fp,qs(ℝn) 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 Fp,qs+k(ℝn) to Fp*,ts+n/p*-n/p(ℝn) with p*∈(p,∞) and t∈(0,∞].
Notice that, ifm satisfies the assumptions of Theorem 1.7 or Corollary 1.8, then m is not necessary to belong to 𝒮′(ℝn); 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 ℓ>n[max(1/p,1/2)] if r∈(0,2), or ℓ>n[max(1/p,1/r)+1/2-1/r] if r∈[2,∞], the operator Tm maps Ḟp,r0(ℝn) boundedly into Ḟp*,qβ(ℝn). However, from Theorem 1.7, we deduce that this conclusion is also true when ℓ>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.
Theorems1.5 and 1.7 have the following counterparts for Hausdorff-type spaces.Theorem 1.10.
Letα,γ∈ℝ, p∈(1,∞) and m satisfy (1.2) with ℓ∈ℕ.(i)
Ifr∈(1,∞),τ∈[0,1/(p∨r)′] and ℓ>n[max(1/p,1/r)+τ+1/2], then there exists a positive constant C such that for all f∈FḢp,rγ,τ(ℝn),(1.23)‖Tmf‖FḢp,rα+γ,τ(Rn)≤C‖f‖FḢp,rγ,τ(Rn).(ii)
Ifr∈[1,∞),τ∈[0,1/(p∨r)′] and ℓ>n(1/p+τ+1/2), then there exists a positive constant C such that for all f∈BḢp,rγ,τ(ℝn),(1.24)‖Tmf‖BḢp,rα+γ,τ(Rn)≤C‖f‖BḢp,rγ,τ(Rn).Differently from the spacesḞp,qs,τ(ℝn) and Ḃp,qs,τ(ℝn), it is known that S∞(ℝn) is dense in the spaces FḢp,qs,τ(ℝn) and BḢp,qs,τ(ℝn); see [13, Lemma 5.3] and [14, Lemma 6.3]. Thus, although Tmf is originally defined on 𝒮∞(ℝn), we can extend Tm into the whole spaces FḢp,qs,τ(ℝn) and BḢp,qs,τ(ℝn) by a density argument.We remark that Theorem1.10(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 Theorem1.10 and [15, Theorem 4.1], we immediately deduce the following conclusion and omit the details.Corollary 1.11.
Letα,β∈ℝ,β<α, p∈(1,∞) and p*∈(1,∞) such that β-n/p*=α-n/p. Assume that m satisfies the condition (1.2) with ℓ∈ℕ.(i)
Letr,q∈(1,∞) and τ∈[0,min{1/(p∨r)′,1/(p*∨q)′}] such that τ(p∨r)′≤τ(p*∨q)′. If ℓ>n[max(1/p,1/r)+τ+1/2], then there exists a positive constant C such that for all f∈FḢp,r0,τ(ℝn),
(1.25)‖Tmf‖FḢp*,qβ,τ(Rn)≤C‖f‖FḢp,r0,τ(Rn).(ii)
Letr∈[1,∞) and τ∈[0,min{1/(p∨r)′,1/(p*∨r)′}] such that τ(p∨r)′=τ(p*∨r)′. If ℓ>n(1/p+τ+1/2), then there exists a positive constant C such that for all f∈BḢp,r0,τ(ℝn),
(1.26)‖Tmf‖BḢp*,rβ,τ(Rn)≤C‖f‖BḢp,r0,τ(Rn).Moreover, similar to Corollary1.6(i), we can further improve the range of ℓ in Corollary 1.11(i) as follows.Theorem 1.12.
Letα∈ℝ,p∈(1,∞), β∈ℝ with β<α and p*∈(1,∞) such that β-n/p*=α-n/p. Let r,q∈(1,∞) and τ∈[0,min{1/(p∨r)′,1/(p*∨q)′}] such that τ(p∨r)′≤τ(p*∨q)′. Assume that m satisfies (1.2) with ℓ∈ℕ and
(1.27)l>{n(τ+12),ifτ(p∨r)′≤2τ,n(1r+12),ifp,r∈(1,2),τ≠0.
Then there exists a positive constant C such that for all f∈FḢp,r0,τ(ℝn),
(1.28)‖Tmf‖FḢp*,qβ,τ(Rn)≤C‖f‖FḢp,r0,τ(Rn).The proof of Theorem1.12 is given in Section 3.Similar to Corollary1.8, we have the following conclusion, which is an immediate consequence of Theorem 1.12 and the lifting property of the space FḢp,qs,τ(ℝn) that can be deduced directly from [15, Theorem 4.1].Corollary 1.13.
Letα,γ∈ℝ and p,r,q∈(1,∞). Let β be a real number with β<α+γ and p*∈(1,∞) such that β-n/p*=α+γ-n/p. Assume that τ∈[0,min{1/(p*∨q)′,1/(p∨r)′}] satisfies that τ(p∨r)′≤τ(p*∨q)′ and m satisfies (1.2) with ℓ as in (1.27). Then there exists a positive constant C such that for all f∈FḢp,rγ,τ(ℝn), ∥Tmf∥FḢp*,qβ,τ(ℝn)≤C∥f∥FḢp,rγ,τ(ℝn).Corollary1.13 implies that Theorem 1.12 has variant for any s∈ℝ instead of s=0.Remark 1.14.
Recall that whenτ=0, the Triebel-Lizorkin-Hausdorff space FḢp,qs,τ(ℝn) is just the classical Triebel-Lizorkin space Ḟp,qs(ℝn). 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 Theorems1.5, 1.7 and 1.10 strongly depend on the Peetre-type maximal function characterizations of Ḃp,qs,τ(ℝn), Ḟp,qs,τ(ℝn), BḢp,qs,τ(ℝn), and FḢp,qs,τ(ℝn) obtained in [18]. Additionally, to prove Theorems 1.7 and 1.12, we need first establish the generalized (weighted) gλ*-function equivalent characterizations of Ḟp,qs,τ(ℝn) and FḢp,qs,τ(ℝn), 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 Ḟp,qs(ℝn), 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(ℝn) norms of the generalized Lusin-area functions with different apertures, which is not clear whether it is still true if Lp(ℝn) 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 ℝn. 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 Theorems1.7 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 Ḟ∞,qs(ℝn) and the Triebel-Lizorkin-type space Ḟp,qs,1/p(ℝn) obtained by Frazier and Jawerth [30, Corollary 5.7] (see also [14, Proposition 3.1]).
The paper is organized as follows. In Section2, we present Theorems 2.7 and 2.9 and their proofs by first recalling some known characterizations, obtained in [18], of Ḃp,qs,τ(ℝn), Ḟp,qs,τ(ℝn), BḢp,qs,τ(ℝn), and FḢp,qs,τ(ℝn) 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 Ḟp,qs,τ(ℝn) and FḢp,qs,τ(ℝn).
We point out that so far, for the Besov-type spaceḂp,qs,τ(ℝn) and the Besov-Hausdorff space BḢp,qs,τ(ℝn), 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 Ḟp,qs,τ(ℝn) and FḢp,qs,τ(ℝn), which are not available for Ḃp,qs,τ(ℝn) and BḢp,qs,τ(ℝn). 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, thesymbolA≲B means that A≤CB, where C is a positive constant independent of the main parameter. If A≲B and B≲A, then we write A~B. If E is a subset of ℝ, we denote by χE the characteristic function of E.


## 2. Some Equivalent Characterizations ofḞp,qs,τ(ℝn), Ḃp,qs,τ(ℝn), FḢp,qs,τ(ℝn), and BḢp,qs,τ(ℝn)

In this section, we first recall some equivalent characterizations, established in [18], of Ḟp,qs,τ(ℝn), Ḃp,qs,τ(ℝn), FḢp,qs,τ(ℝn), and BḢp,qs,τ(ℝn), 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.Letɛ∈(0,∞),R∈ℤ+∪{-1} and Φ∈𝒮(ℝn) satisfy that(2.1)|Φ̂(ξ)|>0on⁡{ξ∈Rn:ɛ2<|ξ|<2ɛ},Dα(Φ̂)(0)=0∀|α|≤R.
In what follows, for any function φ,t∈(0,∞) and x∈ℝn, φt(x):=t-nφ(x/t). For all φ∈𝒮N(ℝn), f∈𝒮N′(ℝn), t∈(0,∞),λ∈(0,∞), and x∈ℝn, let(2.2)(φt*f)λ(x):=sup⁡y∈Rn|(φt*f)(x+y)|(1+|y|/t)λ,
which is called the Peetre-type maximal function of local means; see, for example, [18].The following characterization ofḞp,qs,τ(ℝn) was obtained in [18].Theorem 2.1.
Lets∈ℝ,τ∈[0,∞),p∈(0,∞),q∈(0,∞],λ∈(n(1/p∨1/q),∞) and R∈ℤ+∪{-1} such that s+nτ<R+1 and Φ be as in (2.1). Then the space Ḟp,qs,τ(ℝn) is characterized by
(2.3)Ḟp,qs,τ(Rn)={f∈SR′(Rn):‖f∣Ḟp,qs,τ(Rn)‖i<∞},i∈{1,2,3},
where
(2.4)‖f∣Ḟp,qs,τ(Rn)‖1:=sup⁡P∈Q1|P|τ{∫P[∫0l(P)t-sq|(Φt*f)(x)|qdtt]p/qdx}1/p,‖f∣Ḟp,qs,τ(Rn)‖2:=sup⁡P∈Q1|P|τ{∫P[∫0l(P)t-sq|(Φt*f)λ(x)|qdtt]p/qdx}1/p,‖f∣Ḟp,qs,τ(Rn)‖3:=sup⁡P∈Q1|P|τ{∫P[∫0l(P)t-sq∫|z|<t|(Φt*f)(x+z)|qdzdttn+1]p/qdx}1/p,
with the usual modification made when q=∞.Remark 2.2.
Recall that whenτ∈[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∈ℤ; 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 ℓ(P) replaced by ∞ in ∥f∣Ḟp,qs,τ(ℝn)∥1, ∥f∣Ḟp,qs,τ(ℝn)∥2 and ∥f∣Ḟp,qs,τ(ℝn)∥3 when τ∈[0,1/p). We omit the details.
The following Theorems2.3 through 2.5 were established in [18].Theorem 2.3.
Lets∈ℝ,τ∈[0,∞),p,q∈(0,∞],λ∈(n/p,∞), and R∈ℤ+∪{-1} such that s+nτ<R+1 and Φ be as in (2.1). Then the space Ḃp,qs,τ(ℝn) is characterized by
(2.5)Ḃp,qs,τ(Rn)={f∈SR′(Rn):‖f∣Ḃp,qs,τ(Rn)‖i<∞},i∈{1,2},
where
(2.6)‖f∣Ḃp,qs,τ(Rn)‖1:=sup⁡P∈Q1|P|τ{∫0l(P)t-sq[∫P|(Φt*f)(x)|pdx]q/pdtt}1/q,‖f∣Ḃp,qs,τ(Rn)‖2:=sup⁡P∈Q1|P|τ{∫0l(P)t-sq[∫P|(Φt*f)λ(x)|pdx]q/pdtt}1/q,
with the usual modifications made when q=∞ or p=∞.Theorem 2.4.
Lets∈ℝ,p,q∈(1,∞),τ∈[0,1/(p∨q)′],λ∈(n[max{1/p,1/q}+τ],∞) and R∈ℤ+∪{-1} such that s+nτ<R+1 and Φ be as in (2.1). Then the space FḢp,qs,τ(ℝn) is characterized by
(2.7)FḢp,qs,τ(Rn)={f∈SR′(Rn):‖f∣FḢp,qs,τ(Rn)‖i<∞},i∈{1,2,3},
where
(2.8)‖f∣FḢp,qs,τ(Rn)‖1:=inf⁡ω‖{∫0∞t-sq|Φt*f|q[ω(⋅,t)]-qdtt}1/q‖Lp(Rn),‖f∣FḢp,qs,τ(Rn)‖2:=inf⁡ω‖{∫0∞t-sq[(Φt*f)λ]q[ω(⋅,t)]-qdtt}1/q‖Lp(Rn),‖f∣FḢp,qs,τ(Rn)‖3:=inf⁡ω‖{∫0∞t-sq∫|z|<t|Φt*f(⋅+z)|q[ω(⋅+z,t)]-qdzdttn+1}1/q‖Lp(Rn),
where the infimums are taken over all nonnegative Borel measurable functions ω on ℝ+n+1 satisfying (1.13).Theorem 2.5.
Lets∈ℝ,p∈(1,∞),q∈[1,∞),τ∈[0,1/(p∨q)′],λ∈(n(1/p+τ),∞) and R∈ℤ+∪{-1} such that s+nτ<R+1 and Φ be as in (2.1). Then the space BḢp,qs,τ(ℝn) is characterized by
(2.9)BḢp,qs,τ(Rn)={f∈SR′(Rn):‖f∣BḢp,qs,τ(Rn)‖i<∞},i∈{1,2},
where
(2.10)‖f∣BḢp,qs,τ(Rn)‖1:=inf⁡ω{∫0∞t-sq‖Φt*f[ω(⋅,t)]-1‖Lp(Rn)qdtt}1/q,‖f∣BḢp,qs,τ(Rn)‖2:=inf⁡ω{∫0∞t-sq‖(Φt*f)λ[ω(⋅,t)]-1‖Lp(Rn)qdtt}1/q,
where the infimums are taken over all nonnegative Borel measurable functions ω on ℝ+n+1 satisfying (1.13).Remark 2.6.
(i) The spaceFḢp,qs,τ(ℝn) is a quasi-Banach space; see [13, 14, 17]. Indeed, by [17, Remarks 7.1 and 7.3], we know that for any f1,f2∈FḢp,qs,τ(ℝn),
(2.11)‖f1+f2‖FḢp,qs,τ(Rn)≤21/(p∨q)′[‖f1‖FḢp,qs,τ(Rn)+‖f2‖FḢp,qs,τ(Rn)].
(ii)
By the Aoki-Rolewicz theorem ([35, 36]), there exists v∈(0,1] such that(2.12)‖∑j∈Zfj‖FḢp,qs,τ(Rn)v≲∑j∈Z‖fj‖FḢp,qs,τ(Rn)v
for all {fj}j∈ℤ⊂FḢp,qs,τ(ℝn). Indeed, v:=(p∨q)′/(1+(p∨q)′) does the job.
(iii)
The conclusions in (i) and (ii) are also true for the spaceBḢp,qs,τ(ℝn).
Next we establish a new characterization of the spacesḞp,qs,τ(ℝn) and FḢp,qs,τ(ℝn). Let q∈(0,∞], λ∈(0,∞), and ω be a nonnegative Borel measurable function. In what follows, for R∈ℤ+∪{-1}, f∈𝒮R′(ℝn) and φ in (2.1), set(2.13)u(x,t):=(f*φt)(x),uλ*(x,t):=sup⁡y∈Rn{|u(y,t)|(1+|x-y|t)-λ},
for all x∈ℝn and t∈(0,∞). For all b∈(0,∞),s∈ℝ, and x∈ℝn, recall that the generalized weighted Lusin-area functionSb,qs(ω,u)(x) and the generalized weightedgλ*-functionGλ,qs(ω,u)(x) are defined, respectively, by
(2.14)Sb,qs(ω,u)(x):={∫0∞t-sq∫|y-x|<bt|u(y,t)|q[ω(y,t)]-q(bt)-ndydtt}1/q,Gλ,qs(ω,u)(x):={∫0∞t-sq∫Rn|u(y,t)|q(1+|x-y|t)-λq[ω(y,t)]-qdydttn+1}1/q.
If ω(x,t)≡1, then Sb,qs(ω,u) and Gλ,qs(ω,u) are called, respectively, the generalized Lusin-area function, denoted by Sb,qs(u), and the generalizedgλ*-function, denoted by Gλ,qs(u).
In what follows, forτ∈[0,∞) and p∈(0,∞), let Lτp(ℝn) be the set of all functionsf∈Llocp(ℝn)  such that(2.15)‖f‖Lτp(Rn):=sup⁡p∈Q1|P|τ[∫P|f(x)|pdx]1/p<∞.Theorem 2.7.
Lets∈ℝ, p∈(0,∞), τ∈[0,1/p), q∈(0,∞], λ∈(n/q,∞) and R∈ℤ+∪{-1} such that s+nτ<R+1. Then f∈Ḟp,qs,τ(ℝn) if and only if f∈𝒮R′(ℝn) and Gλ,qs(u)∈Lτp(ℝn), where u is as in (2.13). Moreover, there exists a positive constant C such that for all f∈Ḟp,qs,τ(ℝn),
(2.16)C-1‖f‖Ḟp,qs,τ(Rn)≤‖Gλ,qs(u)‖Lτp(Rn)≤C‖f‖Ḟp,qs,τ(Rn).Proof.
Assumef∈𝒮R′(ℝn) and Gλ,qs(u)∈Lτp(ℝn). Notice that for any λ∈(0,∞) and x∈ℝn,
(2.17)S1,qs(u)(x)≤2λ{∫0∞∫|y-x|<t[t-s|u(y,t)|]q(1+|x-y|t)-λqt-ndydtt}1/q=2λGλ,qs(u)(x).
Then, from Remark 2.2, we deduce that f∈Ḟp,qs,τ(ℝn) and
(2.18)‖f‖Ḟp,qs,τ(Rn)~‖S1,qs(u)‖Lτp(Rn)≲‖Gλ,qs(u)‖Lτp(Rn),
which completes the proof of the sufficiency of the theorem.
Conversely, suppose thatf∈Ḟp,qs,τ(ℝn). Then by Theorem 2.1, f∈𝒮R′(ℝn). Moreover, similar to the proof of [18, Theorem 3.1], for any k∈ℕ, we see that(2.19)sup⁡P∈Q1|P|τ{∫P[∫0∞∫|y-x|<t(t-s|φ2-kt*f(y)|)qt-ndydtt]p/qdx}1/p≤Csup⁡P∈Q1|P|τ{∫P[∑j=-∞∞2jsq|φk+j*f(x)|q]p/qdx}1/p,
where C is a positive constant independent of k and f. Then by changing variables, we conclude that
(2.20)sup⁡P∈Q1|P|τ{∫P[∫0∞∫|y-x|<t(t-s|φ2-kt*f(y)|)qt-ndydtt]p/qdx}1/p≲2-kssup⁡P∈Q1|P|τ{∫P[∑j=-∞∞2jsq|φj*f(x)|q]p/qdx}1/p≲2-ks‖f‖Ḟp,qs,τ(Rn),
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 x∈ℝn,
(2.21)Gλ,qs(u)(x)={∫0∞[∫|y-x|<t(t-s|φt*f(y)|)q(1+|x-y|t)-λqt-ndy+∑k=1∞∫2k-1t≤|y-x|<2kt⋯dy]dtt∫0∞∫0∞}1/q≲{∑k=0∞2-kλq∫0∞∫|y-x|<2kt(t-s|φt*f(y)|)qt-ndydtt}1/q≲{∑k=0∞2-k(λq-sq-n)∫0∞∫|y-x|<t(t-s|φ2-kt*f(y)|)qt-ndydtt}1/q.
Thus, when p≤q, from the well-known inequality that for all d∈(0,1] and {αj}j⊂ℂ,
(2.22)(∑j|αj|)d≤∑j|αj|d
it follows that
(2.23)‖Gλ,qs(u)‖Lτp(Rn)≲sup⁡P∈Q1|P|τ{∫P[∑k=0∞2-k(λq-sq-n)∫0∞∫|y-x|<t(t-s|φ2-kt*f(y)|)qt-ndydtt]p/qdx}1/p≲sup⁡P∈Q1|P|τ{∫P∑k=0∞2-k(λ-s-n/q)p[∫0∞∫|y-x|<t(t-s|φ2-kt*f(y)|)qt-ndydtt]p/qdx}1/p≲{[∫0∞∫|y-x|<t(t-s|φ2-kt*f(y)|)qt-ndydtt]p/q∑k=0∞2-k(λ-s-n/q)p×sup⁡P∈Q1|P|τp∫P[∫0∞∫|y-x|<t(t-s|φ2-kt*f(y)|)qt-ndydtt]p/qdx}1/p≲{∑k=0∞2-k(λ-n/q)p‖f‖Ḟp,qs,τ(Rn)p}1/p≲‖f‖Ḟp,qs,τ(Rn),
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
(2.24)‖Gλ,qs(u)‖Lτp(Rn)≲sup⁡P∈Q1|P|τ({∫P[∫0∞∫|y-x|<t(t-s|φ2-kt*f(y)|)qt-ndydtt]p/qdx}q/p∑k=0∞2-k(λq-sq-n)×{∫P[∫0∞∫|y-x|<t(t-s|φ2-kt*f(y)|)qt-ndydtt]p/qdx}q/p)1/q≲({∫P[∫0∞∫|y-x|<t(t-s|φ2-kt*f(y)|)qt-ndydtt]p/qdx}q/p∑k=0∞2-k(λq-sq-n)sup⁡P∈Q1|P|τq×{∫P[∫0∞∫|y-x|<t(t-s|φ2-kt*f(y)|)qt-ndydtt]p/qdx}q/p)1/q≲{∑k=0∞2-k(λq-n)‖f‖Ḟp,qs,τ(Rn)q}1/q≲‖f‖Ḟp,qs,τ(Rn).
These estimates, together with Remark 2.2, imply the necessity of the theorem and hence complete the proof of Theorem 2.7.Remark 2.8.
We point that, by an argument similar to the proof of Theorem2.7, one can characterize Ḟp,qs,τ(ℝn) via a discrete version of the generalized weighted gλ*-function. More precisely, for all s∈ℝ, p∈(0,∞), τ∈[0,1/p), q∈(0,∞], λ∈(n/q,∞), and R∈ℤ+∪{-1} such that s+nτ<R+1, then f∈Ḟp,qs,τ(ℝn) if and only if f∈𝒮R′(ℝn) and [∑i∈ℤ∫ℝn2i(sq+n)|f*φi(z)|q(1+2i|z-·|)-λqdz]1/q∈Lτp(ℝn). Moreover,
(2.25)‖f‖Ḟp,qs,τ(Rn)~‖[∑i∈Z∫Rn2i(sq+n)|f*φi(z)|q(1+2i|z-⋅|)-λqdz]1/q‖Lτp(Rn).
We omit the details.
We also obtain the following analogy of Theorem2.7 for the space FḢp,qs,τ(ℝn).Theorem 2.9.
Lets∈ℝ,p∈(1,∞),q∈(1,∞],τ∈[0,1/(p∨q)′], λ∈(n/q,∞) and R∈ℤ+∪{-1} such that s+nτ<R+1. Then f∈FḢp,qs,τ(ℝn) if and only if f∈𝒮R′(ℝn) and Gλ,qs(ω,u)∈Lp(ℝn), where u is as in (2.13). Moreover, there exists a positive constant C such that for all f∈FḢp,qs,τ(ℝn),
(2.26)C-1‖f‖FḢp,qs,τ(Rn)≤inf⁡ω‖Gλ,qs(ω,u)‖Lp(Rn)≤C‖f‖FḢp,qs,τ(Rn),
where the infimum is taken over all nonnegative Borel measurable functions ω on ℝ+n+1 satisfying (1.13).Proof.
Assumef∈𝒮R′(ℝn) and Gλ,qs(ω,u)∈Lp(ℝn). For any λ∈(0,∞) and x∈ℝn, similar to the proof of Theorem 2.7, we know that
(2.27)S1,qs(ω,u)(x)≤2λqGλ,qs(ω,u)(x).
Then by Theorem 2.4, we see that
(2.28)‖f‖FḢp,qs,τ(Rn)~inf⁡ω‖S1,qs(ω,u)‖Lp(Rn)≲inf⁡ω‖Gλ,qs(ω,u)‖Lp(Rn),
which completes the proof of the sufficiency of the theorem.
Conversely, suppose thatf∈FḢp,qs,τ(ℝn). Then by Theorem 1.12, f∈𝒮R′(ℝn). By an argument similar to the proof of [18, Theorem 3.3], we see that for any k∈ℕ,(2.29)inf⁡ω‖{∫0∞∫|z-⋅|<tt-sq|((φk)t*f)(z)|q[ω(z,2-kt)]-qdzdttn+1}1/q‖Lp(Rn)≲inf⁡ω‖{∑j∈Z2jsq|φk+j*f|q[ω(⋅,2-k-j)]-q}1/q‖Lp(Rn)~2-ksinf⁡ω‖{∑j∈Z2jsq|φj*f|q[ω(⋅,2-j)]-q}1/q‖Lp(Rn)~2-ks‖f‖FḢp,qs,τ(Rn).
Letℝ+:=(0,∞). For all measurable functions F on ℝn×ℝ+×ℝn, let(2.30)‖F‖F:=inf⁡ω‖{∫0∞t-sq∫Rn|F(y,t,⋅)|[ω(y,t)]-qdydttn+1}1/q‖Lp(Rn),
where the infimum is taken over the same set as in (1.13). We claim that ∥·∥ℱ is a quasinorm with respect to F, precisely, for any measurable functions F1,F2 on ℝn×ℝ+×ℝn,
(2.31)‖F1+F2‖F≤21/(p∨q)′(‖F1‖F+‖F2‖F).
To see this, without loss of generality, we may assume that ∥F1∥ℱ+∥F2∥ℱ<∞. Then, for any ɛ∈(0,∞), choose nonnegative Borel measurable functions ω1,ω2 on ℝ+n+1 satisfying (1.13) such that
(2.32)‖{∫0∞t-sq∫Rn|Fi(y,t,⋅)|[ωi(y,t)]-qdydttn+1}1/q‖Lp(Rn)≤(1+ɛ)‖Fi‖F,
for i∈{1,2}. Notice that ω:=2-1/(p∨q)′max{ω1,ω2} still satisfies (1.13). Then by (2.22) and Minkowski’s inequality, we see that
(2.33)‖F1+F2‖F≤‖{∫0∞t-sq∫Rn|F1(y,t,⋅)+F2(y,t,⋅)|[ω(y,t)]-qdydttn+1}1/q‖Lp(Rn)≤21/(p∨q)′[‖{∫0∞t-sq∫Rn|F1(y,t,⋅)|[ω1(y,t)]-qdydttn+1}1/q‖Lp(Rn)+‖{∫0∞t-sq∫Rn|F2(y,t,⋅)|[ω2(y,t)]-qdydttn+1}1/q‖Lp(Rn)]≤21/(p∨q)′(1+ɛ)(‖F1‖F+‖F2‖F).
Letting ɛ→0 then concludes the above claim.
Thus, by the Aoki-Rolewica theorem [35, 36], we know that(2.34)‖∑j∈ZFj‖Fv≲∑j∈Z‖Fj‖Fv,
for all measurable functions {Fj}j∈ℤ on ℝn×ℝ+×ℝn, where v=(p∨q)′/(1+(p∨q)′).
Choosingλ>n/q, by (2.34), (2.29), and an estimate similar to (2.21), we conclude that(2.35)inf⁡ω‖Gλ,qs(ω,u)‖Lp(Rn)v=inf⁡ω‖{∫0∞t-sq∫Rn|φt*f(y)|q(1+|⋅-y|t)-λq[ω(y,t)]-qdydttn+1}1/q‖Lp(Rn)v≲inf⁡ω‖{∫0∞t-sq∑k=0∞∫B(⋅,2kt)2-λkq|φt*f(y)|q[ω(y,t)]-qdydttn+1}1/q‖Lp(Rn)v≲∑k=0∞2-λkvinf⁡ω‖{∫0∞t-sq∫Rn|φt*f(y)|qχB(⋅,2kt)(y)[ω(y,t)]-qdydttn+1}1/q‖Lp(Rn)v≲∑k=0∞2-k(λ-s-n/q)v×inf⁡ω‖[∫0∞∫|y-⋅|<tt-sq|(φk)t*f(y)|q[ω(y,2-kt)]-qdydttn+1]1/q‖Lp(Rn)v≲∑k=0∞2-k(λ-n/q)v‖f‖FḢp,qs,τ(Rn)v~‖f‖FḢp,qs,τ(Rn)v,
which implies the necessity of the theorem and hence completes the proof of Theorem 2.9.


## 3. Proofs of Theorems1.5, 1.7, 1.10, and 1.12

In this section, we give the proofs for Theorems1.5, 1.7, 1.10, and 1.12.In what follows,K always denotes the distribution whose Fourier transform is the functionmin (1.2). Then we have the following observation.Lemma 3.1.
Letm be as in (1.2) and K its inverse Fourier transform. Then K∈𝒮∞′(ℝn).Proof.
Letφ∈𝒮∞(ℝn). Then ∂γφ̂(0)=0 for all γ∈ℤ+n and hence for any L∈ℕ,
(3.1)〈K,φ〉=∫Rnm(ξ)φ̂(ξ)dξ=∫|ξ|≥1m(ξ)φ̂(ξ)dξ+∫|ξ|<1m(ξ)[φ̂(ξ)-∑|γ|≤L∂γφ̂(0)γ!ξγ]dξ=:I1+I2.
For I1, by Hölder’s inequality and (1.2), we see that(3.2)|I1|≲∑k=0∞∫2k≤|ξ|<2k+1|m(ξ)||φ̂(ξ)|dξ≲∑k=0∞1(1+2k)M∫2k≤|ξ|<2k+1|m(ξ)|dξ≲∑k=0∞2nk/2(1+2k)M[∫2k≤|ξ|<2k+1|m(ξ)|2dξ]1/2≲∑k=0∞2k(n-α)(1+2k)M≲1,
where M∈[0,∞) is chosen large enough such that M>n-α.
For I2, by the mean value theorem, there exists θ∈[0,1] such that(3.3)|I2|≲∑k=-∞-1∫2k≤|ξ|<2k+1|m(ξ)|sup⁡|γ|=L+1|∂γφ̂(θξ)||ξ||γ|dξ≲∑k=-∞-12k(L+1)∫2k≤|ξ|<2k+1|m(ξ)|dξ≲∑k=-∞-12k(L+1+n-α)≲1,
where L is chosen large enough such that L>α-n-1. This finishes the proof of Lemma 3.1.The following estimates play an important role in the proofs of Theorems1.5, 1.7, 1.10, and 1.12.Lemma 3.2.
Letψ,ζ be Schwartz functions on ℝn such that ψ̂,ζ̂ are supported in the annulus {ξ∈ℝn:1/2≤|ξ|≤2}. Assume that m satisfies (1.2).(i)
Ifλ∈(0,∞) and ℓ>λ+n/2, then there exists a positive constant C such that for all t∈(0,∞),
(3.4)∫Rn(1+|z|t)λ|(K*ψt)(z)|dz≤Ctα.(ii)
Letk,N be any two positive integers. If ℓ≥λ>0, then there exists a positive constant C such that for all s,t∈(0,∞),
(3.5)∫Rn(1+|z|s)2λ|(K*ζs*ψt)(z)|2dz≤C(min⁡{t,s})-n+2α{(1+ts)-2N,ifs≤t,(ts)2k,ift<s.Remark 3.3.
We remark that Lemma3.2(i) is just [12, Lemma 4.1(1)]. It was also claimed in [12, Lemma 4.1(2)] that the inequality in Lemma 3.2(ii) is valid with (min{t,s})-n+2α replaced by s-n+2α. However, the proof of [12, Lemma 4.1(2)] is problematic. Indeed, the last inequality in [12, page 849] seems to be true only when s≤t. We give a correct version in Lemma 3.2(ii).Proof of Lemma3.2(ii).
Sinceλ≤ℓ, by the Plancherel theorem, we see that
(3.6)∫Rn(1+|z|s)2λ|(K*ζs*ψt)(z)|2dz≲∑|σ|≤ls-2|σ|∫Rn|∂ξσ(m(ξ)ζŝ(ξ)ψt̂(ξ))|2dξ≲∑|σ|≤ls-2|σ|∑σ1+σ2+σ3=σs2|σ2|t2|σ3|∫Rn|(∂ξσ1m)(ξ)(∂ξσ2ζ̂)(sξ)(∂ξσ3ψ̂)(tξ)|2dξ.
When s≤t, by the support of ζ̂ and (1.2), we see that
(3.7)∫Rn(1+|z|s)2λ|(K*ζs*ψt)(z)|2dz≲∑|σ|≤ls-2|σ|∑σ1+σ2+σ3=σs2|σ2|t2|σ3|∫1/(2s)≤|ξ|≤2/s|(∂ξσ1m)(ξ)(∂ξσ3ψ̂)(tξ)|2dξ≲∑|σ|≤l∑σ1+σ2+σ3=σ(ts)2|σ3|1(1+t/s)2N+2ls-2|σ1|∫1/(2s)≤|ξ|≤2/s|(∂ξσ1m)(ξ)|2dξ≲s-n+2α1(1+t/s)2N.
When t≤s, by the support of ψ̂ and (1.2), we conclude that
(3.8)∫Rn(1+|z|s)2λ|(K*ζs*ψt)(z)|2dz≲∑|σ|≤l∑σ1+σ2+σ3=σs-2|σ|s2|σ2|t2|σ3|∫1/(2t)≤|ξ|≤2/t|(∂ξσ1m)(ξ)(∂ξσ2ζ̂)(sξ)|2dξ≲∑|σ|≤l∑σ1+σ2+σ3=σ(ts)2|σ1|+2|σ3|1(1+(s/t))2kt-2|σ1|∫1/(2t)≤|ξ|≤2/t|(∂ξσ1m)(ξ)|2dξ≲t-n+2α(ts)2k,
which completes the proof of Lemma 3.5(ii).Recall that𝒮∞(ℝn) is dense in the spaces FḢp,qs,τ(ℝn) and BḢp,qs,τ(ℝn) (see [13, Lemma 5.3] and [14, Lemma 6.3]). Then the definition of Fourier multiplier Tm can be extended to the whole spaces FḢp,qs,τ(ℝn) and BḢp,qs,τ(ℝn) via a dense argument. Next we show that, via a suitable way, Tm can also be defined on the whole spaces Ḟp,qs,τ(ℝn) and Ḃp,qs,τ(ℝn). To this end, let φ∈𝒜. Then by [37, Lemma (6.9)], there exists ψ∈𝒜 such that(3.9)∑i∈Zφ̂(2-iξ)¯ψ̂(2-iξ)=1,∀ξ∈Rn∖{0}.
For any f∈Ḟp,qs,τ(ℝn) or Ḃp,qs,τ(ℝn), we define Tmf by setting, for all ϕ∈𝒮∞(ℝn),(3.10)〈Tmf,ϕ〉:=∑i∈Zf*φi*ψi*ϕ*K(0).
In this sense, we say Tmf∈𝒮∞′(ℝn). The following result shows that Tmf in (3.10) is well defined.Lemma 3.4.
Letℓ∈(n/2,∞), s∈ℝ, τ∈[0,∞) and q∈(0,∞], f∈Ḟp,qs,τ(ℝn) with p∈(0,∞) or f∈Ḃp,qs,τ(ℝn) with p∈(0,∞]. Then Tmf in (3.10) is independent of the choices of the pair (φ,ψ) of functions in 𝒜 satisfying (3.9). Moreover, Tmf∈𝒮∞′(ℝn).Proof.
Assume first thatf∈Ḟp,∞s,τ(ℝn). Let φ̃ and ψ̃ be another pair of functions in 𝒜 satisfying (3.9). Since ϕ∈𝒮∞(ℝn), by the Calderón reproducing formula (see [13, Lemma 2.1]), we know that
(3.11)ϕ=∑j∈Zφ̃j*ψ̃*ϕ
in 𝒮∞(ℝn). Thus,
(3.12)∑i∈Zf*φi*ψi*ϕ*K(0)=∑i∈Zf*φi*ψi*(∑j∈Zφ̃j*ψ̃j*ϕ)*K(0)=∑i∈Z∑j=i-1i+1f*(φi*φ̃j)*(ψi*ψ̃j)*ϕ*K(0),
where the last equality follows from the fact that φi*φ̃j=0 if |i-j|≥2.
Letζ:=φ*φ̃ and η:=ψ*ψ̃. Then ζ, η∈𝒜. If τ∈[0,1/p) and q=∞, we see that for all x∈ℝn,(3.13)∑i∈Z|f*(φi*φ̃i)*(ψi*ψ̃i)*ϕ*K(0)|=∑i∈Z|∫Rnf*ζi(-z)ηi*ϕ*K(z)dz|≲[sup⁡i∈Zsup⁡z∈Rn2is|f*ζi(-z)|(1+2i|z+x|)-λ]×∑i∈Z∫Rn2-is(1+2i|z+x|)λ|K*ηi*ϕ(z)|dz=:I1(x)I2(x),
where λ is an arbitrary positive number.
For I1, by Remark 2.8, we know that ∥I1∥Lτp(ℝn)≲∥f∥Ḟp,∞s,τ(ℝn)<∞, which implies that there exists x∈B(0,1) such that |I1(x)|<∞.
For I2, choosing λ∈(0,∞) and μ∈(n/2,∞) such that λ+μ<ℓ, by Hölder’s inequality and Lemma 3.2(ii), we see that for all x∈B(0,1),(3.14)∫Rn(1+2i|z+x|)λ|K*ηi*ϕ(z)|dz≲(1+2i)λ∫Rn(1+2i|z|)λ|K*ηi*ϕ(z)|dz≲(1+2i)λ2-in/2[∫Rn(1+2i|z|)2(λ+μ)|K*ηi*ϕ(z)|2dz]1/2≲2-in/2(min⁡{1,2-i})-n/2+α(min⁡{1,2i})k(1+2i)λ-N.
Thus, by choosing k and N large enough such that k>n/2+|s| and N>λ+|α|+|s|, we know that
(3.15)I2(x)≲∑i=0∞2i(λ-α-s-N)+∑i=-∞-12i(k-n/2-s)≲1.
Therefore,
(3.16)∑i∈Z|f*(φi*φ̃i)*(ψi*ψ̃i)*ϕ*K(0)|<∞.
By an argument similar to the above, we see that
(3.17)∑i∈Z∑j=i-1i+1|f*(φi*φ̃j)*(ψi*ψ̃j)*ϕ*K(0)|<∞,
which, together with the Calderón reproducing formula, further induces that
(3.18)∑i∈Zf*φi*ψi*φ*K(0)=∑j∈Z∑i=j-1j+1f*(φi*φ̃j)*(ψi*ψ̃j)*ϕ*K(0)=∑j∈Zf*φ̃j*ψ̃j*(∑i∈Zφi*ψi*ϕ)*K(0)=∑j∈Zf*φ̃j*ψ̃j*φ*K(0).
Thus, Tmf in (3.10) is independent of the choices of the pair (φ,ψ). Moreover, the previous argument also implies that Tmf∈𝒮∞′(ℝn).
Ifτ∈[0,1/p) and q∈(0,∞), from the embedding Ḟp,qs,τ(ℝn)⊂Ḟp,∞s,τ(ℝn), we deduce that Tmf is also well defined in Ḟp,qs,τ(ℝn) and Tmf∈𝒮∞′(ℝn).
Ifτ∈(1/p,∞) and q∈(0,∞), by Remark 1.2 and [29, Corollary 1], we know that(3.19)Ḟp,qs,τ(Rn)=Ḟ∞,∞s+n(τ-1/p)(Rn)=Ḃ∞,∞s+n(τ-1/p)(Rn).
Then, by Theorem 2.3, we know that
(3.20)∑i∈Z|∫Rnf*ζi(-z)ηi*ϕ*K(z)dz|≤[sup⁡i∈Zsup⁡z∈Rn2i[s+n(τ-1/p)]|f*ζi(z)|]∑i∈Z2-i[s+n(τ-1/p)]∫Rn|ηi*ϕ*K(z)|dz~‖f‖Ḟ∞,∞s+n(τ-1/p)(Rn)∑i∈Z2-i[s+n(τ-1/p)]∫Rn|ηi*ϕ*K(z)|dz.
Choosing λ>n/2 such that ℓ>λ, by Hölder’s inequality and Lemma 3.2(ii), we conclude that
(3.21)∑i∈Z2-i[s+n(τ-1/p)]∫Rn|ηi*ϕ*K(z)|dz≲∑i∈Z2-i[s+n(τ-1/p)][∫Rn(1+2i|z|)-2λdz]1/2[∫Rn(1+2i|z|)2λ|ηi*ϕ*K(z)|2dz]1/2≲∑i∈Z2-i[s+n(τ-1/p+1/2)](min⁡{1,2-i})-n/2+α(min⁡{1,2i})k(1+2i)-N≲∑i=0∞2-i[s+nτ-n/p+α+N]+∑i=-∞-12-i(s+nτ-n/p+n/2-k)≲1,
where k and N are chosen large enough such that k>|s|+n(τ-n/p+n/2) and N>|s|+|α|+n/p. By an argument similar to the above, we see that
(3.22)∑i∈Z∑j=i-1i+1|f*(φi*φ̃j)*(ψi*ψ̃j)*ϕ*K(0)|≲‖f‖Ḟ∞,∞s+n(τ-1/p)(Rn)~‖f‖Ḟp,qs,τ(Rn),
which, together with the Calderón reproducing formula, further induces that
(3.23)∑i∈Zf*φi*ψi*ϕ*K(0)=∑j∈Zf*φ̃j*ψ̃j*ϕ*K(0).
Thus, in the case that τ∈(1/p,∞), Tmf in (3.10) is also independent of the choices of the pair (φ,ψ). Moreover, Tmf∈𝒮∞′(ℝn).
Finally, ifτ=1/p and q∈(0,∞), since Ḟp,qs,1/p(ℝn)=Ḟ∞,qs(ℝn)⊂Ḟ∞,∞s(ℝn) (see [30, Corollary 5.7]), from the previous argument, we deduce that Tm is also well defined in Ḟp,qs,1/p(ℝn). Therefore, we obtain the desired conclusion for the space Ḟp,qs,τ(ℝn) for all admissible indices.
Assume now thatf∈Ḃp,qs,τ(ℝn). If p∈(0,∞), by the obtained conclusion for Ḟp,qs,τ(ℝn), the embedding Ḃp,qs,τ(ℝn)⊂Ḟp,qs,τ(ℝn) when q≤p (see [14, Proposition 3.1(vii)]) and
(3.24)Ḃp,qs,τ(Rn)⊂Ḃp,ps-ϵ,τ+ϵ/n(Rn)⊂Ḟp,qs-ϵ,τ+ϵ/n(Rn)
for some ϵ when q>p (see (iii) and (vii) of [14, Proposition 3.1]), we know that Tm is well defined in Ḃp,qs,τ(ℝn). This, together with the embedding Ḃ∞,qs,τ(ℝn)⊂Ḃp0,qs,τ+1/p0(ℝn) for some p0∈(0,∞) (see [14, Proposition 3.1(ii)]), further induces the corresponding result for Ḃ∞,qs,τ(ℝn), and hence completes the proof of Lemma 3.4.Now we have the following technical lemma.Lemma 3.5.
Letα∈ℝ, λ∈(0,∞), r∈[2,∞], ℓ∈ℕ,φ,ψ∈𝒜, and u, uλ* be as in (2.13). Assume that m satisfies (1.2) and f∈𝒮∞′(ℝn) such that Tmf∈𝒮∞′(ℝn).(i)
Ifℓ>λ+n/2 and Φ=φ*ψ, then for all x,y∈ℝn and t∈(0,∞),(3.25)|(Tmf*Φt)(y)|≤Ctα(1+|x-y|t)λuλ*(x,t).(ii)
Ifℓ>λ+n(1/2-1/r), then for all x,y∈ℝn and t∈(0,∞) satisfying that |x-y|<t,(3.26)|(Tmf*ψt)(y)|≤CtαGλ,r0(u)(x).Proof.
(i) is just [12, Lemma 4.2(1)]. The proof of (ii) is similar to the proofs of (2) and (3) of [12, Lemma 4.2], but with [12, Lemma 4.1(2)] replaced by Lemma 3.2(ii). This finishes the proof of Lemma 3.5.We remark that by Lemma3.4, Tmf∈𝒮∞′(ℝn) when f∈Ḟp,qs,τ(ℝn) or Ḃp,qs,τ(ℝn) with all indices as in Lemma 3.4. Thus, Lemma 3.5 is also true for all f∈Ḟp,qs,τ(ℝn) or f∈Ḃp,qs,τ(ℝn) with all indices as in Lemma 3.4.Now we are ready to prove Theorems1.5, 1.7, 1.10, and 1.12.Proof of Theorem1.5.
LetΦ be as in Lemma 3.5.
(i)
By the assumption thatℓ>n[max(1/p,1/r)+1/2], there exists λ>n[max(1/p,1/r)] such that ℓ>λ+n/2. Then by Lemma 3.5(i), we see that for all x∈ℝn and t∈(0,∞),
(3.27)t-(α+γ)(Φt*(Tmf))λ(x)≲t-γuλ*(x,t),which yields the desired result in view of Theorem 2.1.
(ii)
By the assumption thatℓ>n(1/p+1/2), there exists λ>n/p such that ℓ>λ+n/2. Then by Lemma 3.5(i), we also see that for all x∈ℝn and t∈(0,∞), (3.27) holds, which yields the desired result in view of Theorem 2.1 and hence completes the proof of Theorem 1.5.Now we give the proof of Theorem1.7.Proof of Theorem1.7.
To prove the theorem, by the monotone embedding property on the parameterq of the spaces Ḟp*,qβ,τ(ℝn) (see [14, Proposition 3.1(i)]), namely, Ḟp*,q1β,τ(ℝn)⊂Ḟp*,q2β,τ(ℝn) if q1≤q2, it suffices to consider the case q∈(0,∞). We show the desired result in two cases for τ.Case 1 (τ∈[0,1/p)).
Assume first thatf∈Ḟp,r0,τ(ℝn) with r∈[2,∞]. By assumption that ℓ>n/2, we know that there exists λ>n/r such that ℓ>λ+n/2-n/r. Then from Lemma 3.5(ii), we deduce that for all x,y∈ℝn and t∈(0,∞) satisfying that |x-y|<t,
(3.28)|U(y,t)|≲tαGλ,r0(u)(x),
where and in what follows, U(x,t):=(Tmf*ψt)(x) for all x∈ℝn and t∈(0,∞).
If∥f∥Ḟp,r0,τ(ℝn)=0, by Theorem 2.7, we know that ∥Gλ,r0(u)∥Lτp(ℝn)=0, and hence Gλ,r0(u)(x)=0 for almost every x∈ℝn, which, together with (3.28), implies that U(y,t)=0 for all y∈ℝn. We then conclude that ∥Tmf∥Ḟp*,qβ,τ(ℝn)=0.
If∥f∥Ḟp,r0,τ(ℝn)>0, we know that ∥Gλ,r0(u)∥Lτp(ℝn)>0. Let P be a dyadic cube and t∈(0,ℓ(P)). Then, there exist 3n dyadic cubes {Pi}i=13n, with ℓ(Pi)=ℓ(P), such that
(3.29){y:dist⁡(y,P)<t}⊂⋃i=13nPi.
Then, raising (3.28) to the power p and integrating over the ball B(y,t), we see that(3.30){∫B(y,t)|U(y,t)|pdx}1/p≲tα{∫⋃i=13nPi|Gλ,r0(u)(x)|pdx}1/p≲tα∑i=13n{∫Pi|Gλ,r0(u)(x)|pdx}1/p,
which further implies that
(3.31)|U(y,t)|≲tα-n/p∑i=13n{∫Pi|Gλ,r0(u)(x)|pdx}1/p.
For any fixed x∈P and A:=A(x)∈(0,∞) which is determined later, by (3.28), (3.31), α>β and α-β-n/p=-n/p*, we see that
(3.32)∫0l(P)t-βq∫|y-x|<t|U(y,t)|qt-ndydtt=∫0A∫|y-x|<tχ(0,l(P))(t)t-βq|U(y,t)|qt-ndydtt+∫A∞∫|y-x|<t⋯dydtt≲[Gλ,r0(u)(x)]qA(α-β)q+∑i=13n{∫Pi|Gλ,r0(u)(x)|pdx}q/pA(α-β-n/p)q≲[Gλ,r0(u)(x)]qA(α-β)q+‖Gλ,r0(u)‖Lτp(Rn)q|P|τqA(α-β-n/p)q.
Take A such that
(3.33)A-n/p=Gλ,r0(u)(x)(|P|τ‖Gλ,r0(u)‖Lτp(Rn)).
Then we see that
(3.34){∫0l(P)t-βq∫|y-x|<t|U(y,t)|qt-ndydtt}1/q≲|P|τ(1-p/p*)‖Gλ,r0(u)‖Lτp(Rn)1-p/p*[Gλ,r0(u)(x)]p/p*.
Then, by Theorem 2.7 and λ>n/r, we conclude that
(3.35)‖Tmf‖Ḟp*,qβ,τ(Rn)=sup⁡P∈Q1|P|τ{∫P(∫0l(P)∫|y-x|≤t[t-β|U(y,t)|]qt-ndydtt)p*/qdx}1/p*≲{sup⁡P∈Q1|P|τ(∫P[Gλ,r0(u)(x)]pdx)1/p}p/p*‖Gλ,r0(u)‖Lτp(Rn)1-(p/p*)≲‖Gλ,r0(u)‖Lτp(Rn)~‖f‖Ḟp,r0,τ(Rn).
Whenf∈Ḟp,r0,τ(ℝn) with r∈(0,2), the desired conclusion is a direct consequence of the case r∈[2,∞], together with the the embedding Ḟp,r0,τ(ℝn)⊂Ḟp,20,τ(ℝn) (see [14, Proposition 3.1 (i)]).Case 2 (τ∈[1/p,∞)).
In this case, sincep*>p, we see that τ≥1/p>1/p*.
Ifτ∈(1/p,∞), by the assumption that ℓ>n/2, we know that there exists λ>0 such that ℓ>λ+n/2. Then from Remark 1.2, Theorem 2.3, Lemma 3.5(i) and the fact that Ḟ∞,∞s(ℝn)=Ḃ∞,∞s(ℝn), it follows that(3.36)‖Tmf‖Ḟp*,qβ,τ(Rn)~‖Tmf‖Ḃ∞,∞β+n(τ-1/p*)(Rn)~sup⁡t>0sup⁡x∈Rnt-β-n(τ-(1/p*))|Tmf*Φt(x)|≲sup⁡t>0sup⁡x∈Rntα-β-n(τ-(1/p*))|uλ*(x,t)|~sup⁡t>0sup⁡x∈Rnt-n(τ-(1/p))|uλ*(x,t)|~‖f‖Ḃ∞,∞n(τ-1/p)(Rn)~‖f‖Ḟp,r0,τ(Rn).
Ifτ=1/p, we only consider the case r=∞ in view of the embedding Ḟp,r0,τ(ℝn)⊂Ḟp,∞0,τ(ℝn) (see [14, Proposition 3.1(i)]). Then, similar to the above argument, we see that(3.37)‖Tmf‖Ḟp*,qβ,τ~sup⁡t>0sup⁡x∈Rnt-β-n(τ-1/p*)|Tmf*Φt(x)|≲‖f‖Ḟ∞,∞0(Rn)~‖f‖Ḟp,∞0,τ(Rn),
which completes the proof of Theorem 1.7.Now we give the proof of Corollary1.8.Proof of Corollary1.8.
The result follows from either a minor modification of the proof of Theorem1.7 or considering the symbols m̃(ξ):=m(ξ)|ξ|-γ for all ξ∈ℝn∖{0} and the lifting property. We omit the details.Next, we give the proof of Theorem1.10.Proof of Theorem1.10.
LetΦ be as in Lemma 3.5.
(i) Sinceℓ>n[max(1/p,1/r)+τ+1/2], there exists λ>n[max(1/p,1/r)+τ] such that ℓ>λ+n/2. Then by Lemma 3.5(i), we see that for all x∈ℝn and t∈(0,∞), (3.27) holds, which yields the desired result in view of Theorem 2.4.
(ii) Sinceℓ>n(1/p+τ+1/2), there exists λ>n(1/p+τ) such that ℓ>λ+n/2. Then by Lemma 3.5(i), we also see that for all x∈ℝn and t∈(0,∞), (3.27) holds, which yields the desired result in view of Theorem 2.5 and hence completes the proof of Theorem 1.10.Now we give the proof of Theorem1.12. We begin with a technical lemma proved in [15, Lemma 3.2], which reflects the geometrical properties of Hausdorff capacities.Lemma 3.6.
Letβ∈[1,∞),λ∈(0,∞), and ω be a nonnegative Borel measurable function on ℝ+n+1. Then there exists a positive constant C, independent of β,ω, and λ, such that
(3.38)Hd({x∈Rn:Nβω(x)>λ})≤CβdHd({x∈Rn:Nω(x)>λ}),
where Nβω(x):=sup|y-x|<βtω(y,t) for all x∈ℝn.Proof of Theorem1.12.
Since whenτ=0, the Triebel-Lizorkin-Hausdorff space is just the Triebel-Lizorkin space, we only give the proof for the case τ∈(0,min{1/(p∨r)′,1/(p*∨q)′}].
Assume first thatf∈𝒮∞(ℝn) and ∥f∥FḢp,r0,τ(ℝn)>0. Choose λ>n/r and ω̃ be a nonnegative function on ℝ+n+1 with
(3.39)∫Rn[Nω̃(x)](p∨r)′dHnτ(p∨r)′(x)≤1
such that
(3.40)‖f‖FḢp,r0,τ(Rn)≤‖Gλ,r0(ω̃,u)‖Lp(Rn)≤2‖f‖FḢp,r0,τ(Rn).
Then ∥Gλ,r0(ω̃,u)∥Lp(ℝn)>0.
Letφ be as in (2.1). Then there exists a Schwartz function ζ such that ζ̂ has compact support away from the origin and
(3.41)∫0∞φ̂(sξ)ζ̂(sξ)dss=1,ξ≠0,
see, for example, [30, 37]. By the Calderón reproducing formula, we know that for all y∈ℝn,
(3.42)(Tmf*ψt)(y)=∫0∞(f*φs*K*ζs*ψt)(y)dss.
Then, applying Hölder’s inequality, we conclude that for all nonnegative functions ω on ℝ+n+1 and x∈B(y,t),(3.43)|(Tmf*ψt)(y)|[ω(y,t)]-1≲Gλ,r0(ω̃,u)(x){∫0∞∫Rn(1+|z-x|s)λr′×|(K*ζs*ψt)(y-z)|r′[ω̃(z,s)ω(y,t)-1]r′sn(r′-1)dzdss{∫0∞∫Rn(1+|z-x|s)λr′(1+|z-x|s)λr′}1/r′≲Gλ,r0(ω̃,u)(x){∫0∞∫Rn(1+|z-y|s)λr′(1+ts)λr′×|(K*ζs*ψt)(y-z)|r′[ω̃(z,s)ω(y,t)-1]r′sn(r′-1)dzdss(1+|z-x|s)λr′{∫0∞∫Rn(1+|z-y|s)λr′(1+ts)λr′}1/r′=:Gλ,r0(ω̃,u)(x)h(y,t).
Raising this inequality to the power p and integrating over the ball B(y,t) with respect to x, we see that
(3.44)|(Tmf*ψt)(y)|[ω(y,t)]-1≲‖Gλ,q0(ω̃,u)‖Lp(Rn)t-n/ph(y,t).
Since∥f∥FḢp,r0,τ>0, then ∥Gλ,r0(ω̃,u)∥Lp(ℝn)>0. Thus, in this case, for any fixed x and D:=D(x)∈(0,∞) which is determined later, applying (3.43), (3.44), and the Aoki-Rolewicz theorem (see [35, 36]), we know that(3.45)‖Tmf‖FḢp*,qβ,τ(Rn)v=inf⁡ω‖{(∫0D+∫D∞)∫|y-⋅|<tt-βq|(Tmf*ψt)(y)|q[ω(y,t)]-qdydttn+1}1/q‖Lp*(Rn)v≲inf⁡ω‖{∫0Dt-βq∫|y-⋅|<t[Gλ,r0(ω̃,u)h(y,t)]qdydttn+1+∫D∞t-(β+(n/p))q∫|y-⋅|<t[‖Gλ,r0(ω̃,u)‖Lp(Rn)h(y,t)]qdydttn+1}1/q‖Lp*(Rn)v≲inf⁡ω‖{∫0Dt-βq∫|y-⋅|<t[∑j∞∑i∈Z∫s~2it∫|z-y|~̇2js(1+|z-y|s)λr′(1+ts)λr′×|(K*ζs*ψt)(y-z)|r′[ω̃(z,s)ω(y,t)-1]r′×sn(r′-1)dzdss∑j∞]q/r′dydttn+1[Gλ,r0(ω̃,u)]q+∫D∞t-βq-(n/p)q∫|y-⋅|<t[⋯]q/r′dydttn+1‖Gλ,r0(ω̃,u)‖Lp(Rn)q(1+|z-y|s)}1/q‖Lp*(Rn)v≲∑j=0∞∑i∈Zinf⁡ω‖{∫0Dt-βq∫|y-⋅|<t[∑j∞∫s~2it∫|z-y|~̇2js(1+|z-y|s)λr′(1+ts)λr′×|(K*ζs*ψt)(y-z)|r′[ω̃(z,s)ω(y,t)-1]r′×sn(r′-1)dzdss∑j∞]q/r′dydttn+1[Gλ,r0(ω̃,u)]q+∫D∞t-βq-(n/p)q∫|y-⋅|<t[⋯]q/r′dydttn+1‖Gλ,r0(ω̃,u)‖Lp(Rn)q}1/q‖Lp*(Rn)v,
where v is as in Remark 2.6, |z-y|~̇2js means 2j-1s≤|z-y|<2js for j∈ℕ and 0≤|z-y|<s for j=0, s~2it means 2i-1t≤s<2it for i∈ℤ.
For(y,t)∈ℝn×(0,∞), let(3.46)ωi,j(y,t):=2-(i+j)nτsup⁡{ω̃(ξ,δ):|ξ-y|<2j+1δ,2-i-1≤δt≤2i+1}.
Then by Lemma 3.6 and Remark 1.3, ωi,j satisfies that
(3.47)∫Rn[Nωi,j(x)](p*∨q)′dHnτ(p*∨q)≤1
modulo a positive constant.
Observing thatτ(p∨r)′≤τ(p*∨q)′ and p<p*, we know that r≥p*>p. We now show the desired conclusion in two cases for r and p.
Case 1 (r∈[2,∞) and p∈(1,∞)).
By (1.27), there exist λ>n/r and μ>n(τ+1/2-1/r) such that ℓ>λ+μ. Since r∈[2,∞), then by Hölder’s inequality and Lemma 3.2(ii), we have
(3.48)∫|z-y|~̇2js(1+|z-y|s)λr′|(K*ζs*ψt)(y-z)|r′dz=∫|z|~̇2js(1+|z|s)-μr′(1+|z|s)(λ+μ)r′|(K*ζs*ψt)(z)|r′dz≤[∫|z|~̇2js(1+|z|s)-μr′(2/(2-r′))dz](2-r′)/2×{∫|z|~̇2js(1+|z|s)2(λ+μ)|(K*ζs*ψt)(z)|2dz}r′/2≲2j(-μr′+n-n/2r′)sn(1-r′/2)(min⁡{t,s})((-n/2)+α)r′min⁡{1,(ts)kr′}(1+ts)-Nr′,
where k,N are arbitrary positive integers, which are determined later. Hence, choosing k>n(τ+1/2) and N>λ+|α|, we see that
(3.49)∫s~2it∫|z-y|~̇2js(1+|z-y|s)λr′(1+ts)λr′|(K*ζs*ψt)(y-z)|r′sn(r′-1)dzdss≲2j(-μr′+n-(n/2)r′)∫s~2its(n/2)r′(min⁡{t,s})(-n/2+α)r′min⁡{1,(ts)kr′}(1+ts)(λ-N)r′dss≲tαr′2j(-μr′+n-(n/2)r′)min⁡{2-i(k-n/2)r′,2-i(λ-N-α)r′}.
Thus, by choosing ω:=ωi,j, we conclude that
(3.50)‖Tmf‖FḢp*,qβ,τ(Rn)v≲∑j=0∞2j(nτ-μ+(n/r′)-n/2)v[∑i=0∞2i(nτ-k+(n/2))v+∑i=-∞-12i(nτ+α-λ+N)v]×‖{D(α-B)q[Gλ,r0(ω̃,u)]q+D(α-B-(n/p))q‖Gλ,r0(ω̃,u)‖Lp(Rn)q}1/q‖Lp*(Rn)v.
Take D such that
(3.51)D-n/p=Gλ,r0(ω̃,u)(x)‖Gλ,r0(ω̃,u)‖Lp(Rn).
We then see that
(3.52)‖Tmf‖FḢp*,qβ,τ(Rn)v≲∑j=0∞2j(nτ-μ+n/r′-n/2)v[∑i=0∞2i(nτ-k+n/2)v+∑i=-∞-12i(nτ+α-λ+N)v]‖Gλ,r0(ω̃,u)‖Lp(Rn)v≲‖Gλ,r0(ω̃,u)‖Lp(Rn)v,
which, together Theorem 2.9, implies that ∥Tmf∥FḢp*,qβ,τ(ℝn)≲∥f∥FḢp,r0,τ(ℝn) for all f∈𝒮∞(ℝn).Case 2 (1<p<r<2).
By the assumption thatℓ>n(1/r+1/2), there exists λ>n/r such that ℓ>λ+n/2. Then by Lemma 3.2(i), we see that
(3.53)(1+|z|s)λ|(K*ζs*ψt)(z)|≤∫Rn(1+|z|s)λ|(K*ψt)(z-y)||ζs(y)|dy≤∫Rn(1+|z-y|s)λ|(K*ψt)(z-y)|(1+|y|s)λ|ζs(y)|dy≤s-n{sup⁡y∈Rn(1+|y|)λ|ζ(y)|}∫Rn(1+|z-y|s)λ|(K*ψt)(z-y)|dy≲s-n(max⁡{1,ts})λ∫Rn(1+|z-y|t)λ|(K*ψt)(z-y)|dy≲s-ntα(max⁡{1,ts})λ.
From p<r and τ∈(0,min{1/(p∨r)′,1/(p*∨q)′}], it follows that τr′≤1. Thus, by ℓ>λ+n/2, there exists μ>nτr′/2 such that ℓ>λ+μ, which, together with Lemma 3.2(ii), implies that
(3.54)∫|z-y|~̇2js(1+|z-y|s)λr′|(K*ζs*ψt)(y-z)|r′dz≲2-2μj[s-ntα(max⁡{1,ts})λ]r′-2×∫|z-y|~̇2js(1+|z-y|s)2(λ+μ)|(K*ζs*ψt)(y-z)|2dz≲2-2μjs-n(r′-2)tα(r′-2)(max⁡{1,ts})λ(r′-2)(min⁡{t,s})-n+2α×min⁡{1,(ts)2k}(1+ts)-2N.
Then, choosing k>nτr′/2+n/2 and N>(λr′/2)+|α|, we see that
(3.55)∫s~2it∫|z-y|~̇2js(1+|z-y|s)λr′(1+ts)λr′|(K*ζs*ψt)(y-z)|r′sn(r′-1)dzdss≲2-2μjtα(r′-2)∫s~2itsn(max⁡{1,ts})λ(r′-2)(min⁡{t,s})-n+2α×min⁡{1,(ts)2k}(1+ts)-2N+λr′dss≲2-2μjtαr′min⁡{2-i(2k-n),22i(α+λ+N)}.
This, together with the fact that μ>nτr′/2 and an argument similar to Case 1, further implies that for all f∈𝒮∞(ℝn),
(3.56)‖Tmf‖FḢp*,qβ,τ(Rn)v≲∑j=0∞2j(nτ-2μ/r′)v[∑i=0∞2i(nτ-2k/r′+n/r′)v+∑i=-∞-12i(nτ+2(α+λ+N)/r′)v]‖Gλ,r0(ω̃,u)‖Lp(Rn)v≲‖f‖FḢp,r0,τ(Rn)v,
namely, ∥Tmf∥FḢp*,qβ,τ(ℝn)≲∥f∥FḢp,r0,τ(ℝn).
Next we assume thatf∈𝒮∞(ℝn) and ∥f∥FḢp,r0,τ(ℝn)=0. Then, for any ɛ∈(0,∞), there exists a nonnegative function ω̃ on ℝ+n+1 such that 0≤∥Gλ,r0(ω̃,u)∥Lp(ℝn)<ɛ. If ∥Gλ,r0(ω̃,u)∥Lp(ℝn)=0, then Gλ,r0(ω̃,u)(x)=0 for almost every x∈ℝn, which, together with an argument similar to (3.43), further implies that ∥Tmf∥FḢp*,qβ,τ(ℝn)=0. If ∥Gλ,r0(ω̃,u)∥Lp(ℝn) is positive, repeating the previous argument, we see that(3.57)‖Tmf‖FḢp*,qβ,τ(Rn)≲ɛ
for any ɛ∈(0,∞), and hence ∥Tmf∥FḢp*,qβ,τ(ℝn)=0. Thus, in this case, we also have ∥Tmf∥FḢp*,qβ,τ(ℝn)≲∥f∥FḢp,r0,τ(ℝn).
Finally, by the fact that𝒮∞(ℝn) is dense in FḢp,r0,τ(ℝn) (see [13, Lemma 5.3]), together with a density argument, we know that the inequality ∥Tmf∥FḢp*,qβ,τ(ℝn)≲∥f∥FḢp,r0,τ(ℝn) is true for all f∈FḢp,r0,τ(ℝn), which completes the proof of Theorem 1.12.


## 4. Applications to Sobolev Embeddings

As an application of Theorems1.7 and 1.12, we give new direct proofs for the following Sobolev embedding theorems (see also [14, Proposition 3.3] and [15, Proposition 2.2]).Theorem 4.1.
Letα,β∈ℝ,α>β, q,r∈(0,∞],p∈(0,∞), and τ∈[0,∞). If p*∈(0,∞) such that β-n/p*=α-n/p, then Ḟp,rα,τ(ℝn)↪Ḟp*,qβ,τ(ℝn).Proof.
If we takem(ξ):=|ξ|-α for all ξ∈ℝn∖{0} in Theorem 1.7 and then apply the lifting property (see [14, Proposition 3.5]), we immediately obtain the desired conclusion of Theorem 4.1, which completes the proof of Theorem 4.1.Theorem 4.2.
Letα,β∈ℝ,α>β, and p∈(1,∞). Assume that p*∈(1,∞) satisfies β-n/p*=α-n/p. Let r,q∈(1,∞), and τ∈[0,min{1/(p*∨q)′,1/(p∨r)′}] such that τ(p∨r)′≤τ(p*∨q)′. Then FḢp,rα,τ(ℝn)↪FḢp*,qβ,τ(ℝn).Proof.
If we takem(ξ):=|ξ|-α for all ξ∈ℝn∖{0} in Theorem 1.12 and then apply the lifting property which can be deduced directly from [15, Theorem 4.1], we immediately obtain the desired conclusion of Theorem 4.2, which completes the proof of Theorem 4.2.



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*Source: 431016-2012-02-26.xml*