# A Note on Weighted Besov-Type and Triebel-Lizorkin-Type Spaces

Authors: Canqin Tang
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

Lets,τ∈ℝ,q∈(0,∞], and ω be in the class A∞ of Muckenhoupt. We introduce the weighted Besov-type spaces B˙p,q,ωs,τ(ℝn) and weighted Triebel-Lizorkin-type spaces F˙p,q,ωs,τ(ℝn) for p∈(0,∞) and then establish the φ-transform characterizations of these new spaces in the sense of Frazier and Jawerth.


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## Body

## 1. Introduction

Function spaces have been a central topic in modern analysis and are now of increasing applications in areas such as harmonic analysis and partial differential equations.Since Besov spaces and Triebel-Lizorkin spaces were introduced in [1–3], these spaces became the focus of many scholars. Bui studied weighted Besov and Triebel-Lizorkin spaces in [4, 5]. In 2010, Yuan et al. gave a unified treatment of Morrey, Campanato, Besov, Lizorkin, and Triebel spaces in [6]. A series of research results on these topics can be found in [6–15]. They develop a theory of spaces of Besov-Triebel-Lizorkin type built on Morrey spaces. Based on their work, this note will generalize these spaces to the weighted cases.Let𝒮(ℝn) be the space of all Schwartz functions on ℝn. Let φ and ψ be functions on ℝn satisfying
(1)φ,ψ∈𝒮(ℝn),(2)suppφ^,ψ^⊂{ξ∈ℝn:12≤|ξ|≤2},(3)|φ^(ξ)|,|ψ^(ξ)|≥C>0if35≤|ξ|≤53,(4)∑j∈ℤ‍φ^(2jξ)-ψ^(2jξ)=1ifξ≠0,
where f^(ξ)≡∫ℝn‍f(x)e-ix·ξdx. Throughout the paper, for all j∈ℤ and x∈ℝn, we put φj(x)≡2jnφ(2jx).As in [16], we set
(5)𝒮∞(ℝn)≡{φ∈𝒮(ℝn):∫ℝn‍φ(x)xγdx=0forallmulti-indicesγ∈(ℕ∪{0})n∫ℝn‍}.
Let 𝒮∞′(ℝn) be the topological dual of 𝒮∞(ℝn), namely, the set of all continuous linear functionals on𝒮∞(ℝn). Endowed 𝒮∞′(ℝn) with the weak *-topology, then 𝒮∞′(ℝn) is complete; see [17].LetA∞=∪p≥1Ap. If ω∈A∞, then there exist independent positive constants Ci and αi, i=1,2, such that
(6)C1(|E||Q|)α1≤(ω(E)ω(Q))≤C2(|E||Q|)α2
for each cube Q and each subcube E⊃Q.Before stating our theorems on the weighted Triebel-Lizorkin-type spaces, we first give the definition of these spaces.Definition 1.
Letτ,s∈ℝ, q∈(0,∞], ω∈A∞ and let φ be a Schwartz function satisfying (1) through (3).(i)
Letp∈(0,∞]. The weighted Besov-type spaceB˙p,q,ωs,τ(ℝn) is defined to be the set of all f∈𝒮∞′(ℝn) such that ∥f∥B˙p,q,ωs,τ(ℝn)<∞, where
(7)∥f∥B˙p,q,ωs,τ(ℝn)≡supPdyadic1|P|τ×{∑j=jP∞‍[∫P‍(2js|φj*f(x)|)pω(x)dx]q/p}1/q
with suitable modifications made when p=∞ or q=∞.(ii)
Letp∈(0,∞). The weighted Triebel-Lizorkin-type spaceF˙p,q,ωs,τ(ℝn) is defined to be the set of all f∈𝒮∞′(ℝn) such that ∥f∥F˙p,q,ωs,τ(ℝn)<∞, where
(8)∥f∥F˙p,q,ωs,τ(ℝn)≡supPdyadic1|P|τ×{∫P‍[∑j=jP∞‍(2js|φj*f(x)|)q]p/qω(x)dx}1/p
with suitable modifications made when q=∞, where l(P) is the side length of dyadic cube P, jP≡-log2l(P), and the supremum is taken over all dyadic cubes P.Obviously, lettingτ=0, the above spaces are weighted Besov space B˙p,q,ωs(ℝn) and weighted Triebel-Lizorkin space F˙p,q,ωs(ℝn) in [5, 18]. If p=q∈(0,∞), then B˙p,q,ωs,τ(ℝn)=F˙p,q,ωs,τ(ℝn). For simplicity, in what follows, we useA˙p,q,ωs,τ(ℝn)to denote eitherB˙p,q,ωs,τ(ℝn)orF˙p,q,ωs,τ(ℝn). IfA˙p,q,ωs,τ(ℝn)meansF˙p,q,ωs,τ(ℝn), then the casep=∞is excluded. Below are the main results of the paper.Letφ and ψ satisfy (1) through (4). Recall that the φ-transform Sφ is defined to be the map taking each f∈𝒮∞′(ℝn) to the sequence Sφf={(Sφf)Q}Q, where (Sφf)Q≡〈f,φQ〉 for all dyadic cubes Q; the inverse φ-transform Tψ is defined to be the map taking a sequence t={tQ}Q to Tψt=∑Q‍tQψQ; see, for example, [19, 20]. The following theorem is about the φ-transform characterizations of the spaces A˙p,q,ωs,τ(ℝn).Theorem 2.
Lets∈ℝ, τ∈[0,∞), p,q∈(0,∞], and φ and ψ satisfy (1) through (4). Then the operators Sφ:A˙p,q,ωs,τ(ℝn)→a˙p,q,ωs,τ(ℝn) and Tψ:a˙p,q,ωs,τ(ℝn)→A˙p,q,ωs,τ(ℝn) are bounded. Furthermore, Tψ∘Sφ is the identity on A˙p,q,ωs,τ(ℝn).Next is the result of anɛ-almost diagonal operator (defined in Section 3) on a˙p,q,ωs,τ(ℝn).Theorem 3.
Letω∈A∞, p,q∈(0,∞], ɛ∈((n(α2-α1))/p,∞), s∈ℝ, and τ∈[0,(α1/p)+(ɛ/2n)). Then all ɛ-almost diagonal operators are bounded on a˙p,q,ωs,τ(ℝn).Remark 4.
Letα1=α2=1, then ɛ∈(0,∞) and τ∈[0,(1/p)+(ɛ/2n)). Then all ɛ-almost diagonal operators are bounded on a˙p,qs,τ(ℝn), which is just the results established by Yuan et al.; see [6, Theorem 3.1] and [10, Theorem 4.1].This paper is organized as follows. In Section2, we establish the φ-transform characterizations of the spaces A˙p,q,ωs,τ(ℝn). And the boundedness of almost diagonal operators on A˙p,q,ωs,τ(ℝn) is considered in Section 3.At the end of this section, we makesome conventions on notation. Throughout the paper, C denotes unspecified positive constants, possibly different at each occurrence; the symbolX≲Y means that there exists a positive constant C such that X≤CY, and X~Y means C-1Y≤X≤CY. For any φ∈𝒮(ℝn), we set φ~(x)≡φ(-x)- for all x∈ℝn. For k=(k1,…,kn)∈ℤn and j∈ℤ, Qjk denotes the dyadic cubeQjk≡{(x1,…,xn):ki≤2jxi<ki+1fori=1,…,n} and 𝒬≡{Qjk}j,k. We denote by xQ the lower left-corner2-jk of Q=Qjk. Throughout the paper, when dyadic cube Q appears as an index, such as ∑Q‍ and {·}Q, it is understood that Q runs over all dyadic cubes in ℝn. For each cube Q, we denote its side length by l(Q) and its center by cQ, and for r>0, we denote by rQ the cube concentric withQhaving the side lengthrl(Q). Let E be a set of ℝn. Denote by χE its characteristic function and E∘ its interior. Also, set ℕ≡{1,2,…} and ℤ+≡ℕ∪{0}.


## 2.φ-Transform Characterizations of A˙p,q,ωs,τ(ℝn)

In this section, we establish theφ-transform characterizations of the spaces A˙p,q,ωs,τ(ℝn). To this end, we introduce their corresponding sequence spaces as follows.Definition 5.
Letτ,s∈ℝ, q∈(0,∞], and ω∈A∞.(i)
Letp∈(0,∞]. The sequence spaceb˙p,q,ωs,τ(ℝn) is defined to be the set of all sequences {tQ}Q⊂ℂ such that ∥t∥b˙p,q,ωs,τ(ℝn)<∞, where
(9)∥t∥b˙p,q,ωs,τ(ℝn)≡supPdyadic1|P|τ×{∑j=jP∞‍2jsq[∫P‍(∑l(Q)=2-j‍|tQ|χ~Q(x))pω(x)dx]q/p}1/q
and χ~Q≡|Q|-1/2χQ.(ii)
Letp∈(0,∞). The sequence spacef˙p,q,ωs,τ(ℝn) is defined to be the set of all sequences {tQ}Q⊂ℂ such that ∥t∥f˙p,q,ωs,τ(ℝn)<∞, where
(10)∥t∥f˙p,q,ωs,τ(ℝn)≡supPdyadic1|P|τ×{∫P‍[∑Q⊂P‍(|Q|-s/n|tQ|χ~Q(x))q]p/qω(x)dx}1/p.Obviously, we have(11)∥t∥b˙p,q,ωs,τ(ℝn)=supPdyadic1|P|τ×{∑j=jP∞‍[∑l(Q)=2-jQ⊂P‍(|Q|-s/n-1/2ω(Q)1/p|tQ|)p]q/p}1/q.In a similar manner to considerA˙p,q,ωs,τ(ℝn), we usea˙p,q,ωs,τ(ℝn)to denote eitherb˙p,q,ωs,τ(ℝn)orf˙p,q,ωs,τ(ℝn). Ifa˙p,q,ωs,τ(ℝn)meansf˙p,qs,τ(ℝn), then the casep=∞is excluded.To prove Theorem2, we need some technical lemmas. The next lemma is a special case of [21, Lemma 2.11].Lemma 6.
Letδ∈ℝ, ω∈A∞. Then there exist positive constants L0 and C such that for all j∈ℤ,
(12)∑Q∈𝒬,l(Q)=2-j‍ω(Q)δ(1+|xQ|n/max{1,|Q|})L0≤C2n(2|δ|+1)|j|.For alls∈ℝ, τ∈[0,∞), p∈(0,∞), and q∈(0,∞], we see that b˙p,min{p,q},ωs,τ(ℝn)⊂f˙p,q,ωs,τ(ℝn)⊂b˙p,max{p,q},ωs,τ(ℝn) by Minkowski’s inequality. The following conclusions is easily verified similarly to the proof of lemma  2.7 in [6]. Here we omit the proof.Lemma 7.
Lets∈ℝ, τ∈[0,∞), p,q∈(0,∞], and ψ∈𝒮∞(ℝn), ω∈A∞. Then for all t∈a˙p,q,ωs,τ(ℝn), Tψt≡∑Q‍tQψQ converges in 𝒮∞′(ℝn); moreover, Tψ:a˙p,q,ωs,τ(ℝn)→𝒮∞′(ℝn) is continuous.For a sequencet={tQ}Q, r∈(0,∞], and a fixed λ∈(0,∞), set
(13)(tr,λ*)Q≡(∑{R:l(R)=l(Q)}‍|tR|r(1+l(R)-1|xR-xQ|)λ)1/r
and tr,λ*≡{(tr,λ*)Q}Q. We have the following estimates.Lemma 8.
Suppose0<a≤r<∞ and λ>nrα2/a. Let μ,v∈ℤ and μ≤v. Then for each x∈Q and l(Q)=2-v, we have
(14)(∑l(R)=2-μ‍|tR|r(1+l(R)-1|xR-xQ|)λ)1/r≤C[Mω(∑l(R)=2-μ‍|tR|aχk)(x)]1/a.Proof.
AssumexQ=0. Let
(15)A0={Rdyadic:l(R)=2-μ,|xR|l(R)≤1},Ak={Rdyadic:l(R)=2-μ,2k-1<|xR|l(R)≤2k}
for k=1,2,3,….
DenoteB~⊃∪R∈AkR, then
(16)∑R∈Ak‍|tR|r(1+|xR|/l(R))λ≤C2-kλ∑R∈Ak‍|tR|r≤C2-kλ(∑R∈Ak‍|tR|a)r/a≤C2-kλ(∫B~‍∑R∈Ak‍|tR|aω(R)χRω(x)dx)r/a≤C2-kλ(|B~||R|)rα2/a[Mω(∑R∈Ak‍|tR|aχR)]r/a≤C2-k(λ-nrα2/a)[Mω(∑R∈Ak‍|tR|aχR)]r/a.
Summing on k and taking the rth roots yield the result.Lemma 9.
Letτ=0, then
(17)∥tp∧q,λ*∥b˙p,q,ωs,0(ℝn)~∥t∥b˙p,q,ωs,0(ℝn),∥tp∧q,λ*∥f˙p,q,ωs,0(ℝn)~∥t∥f˙p,q,ωs,0(ℝn).Proof.
For allQ, we have |tQ|≤(tp∧q,λ*)Q. Then it is obvious that
(18)∥t∥f˙p,q,ωs,0(ℝn)≤C∥tp∧q,λ*∥f˙p,q,ωs,0(ℝn).
Letɛ=-1+(λ/nα2)>0, choose a=(p∧q)/(1+ɛ/2), then p/a,q/a>1. Using weighted Fefferman-Stein inequality, we have
(19)∥tp∧q,λ*∥f˙p,q,ωs,0(ℝn)=∥[∑Q(|Q|-s/n|(tp∧q,λ*)Q|χ~Q)q]1/q∥Lωp≤∥[∑Q(|Q|-s/n(∑{R:l(R)=l(Q)}|tR|p∧q(1+l(R)-1|xR-xQ|λ))1/p∧q×χ~Q(|Q|sn(∑{R:l(R)=l(Q)}|tR|p∧q(1+l(R)-1|xR-xQ|λ))1/p∧q)q]1/q∥Lωp≤C∥[∑ν∈ℤ(Mω(∑l(R)=2-ν|R|-s/n|tR|χ~R)a)q/a]a/q∥Lωp/a1/a≤C∥[∑ν∈ℤ(∑l(R)=2-ν|R|-s/n|tR|χ~R)q]1/q∥Lωp=∥t∥f˙p,q,ωs,0(ℝn).
Therefore, ∥tp∧q,λ*∥f˙p,q,ωs,0(ℝn)~∥t∥f˙p,q,ωs,0(ℝn). Similarly, we can verify that ∥tp∧q,λ*∥b˙p,q,ωs,0(ℝn)~∥t∥b˙p,q,ωs,0(ℝn).Lemma 10.
Letω∈A∞, s∈ℝ, τ∈[0,∞), p,q∈(0,∞], and λ∈(n+nα2,∞). Then there exists a constant C∈[1,∞) such that for all t∈a˙p,q,ωs,τ(ℝn),
(20)∥t∥a˙p,q,ωs,τ(ℝn)≤∥tmin{p,q},λ*∥a˙p,q,ωs,τ(ℝn)≤C∥t∥a˙p,q,ωs,τ(ℝn).Proof.
Notice that|tQ|≤(tr,λ*)Q holds for all dyadic cubes Q. This observation immediately implies that ∥t∥a˙p,q,ωs,τ(ℝn)≤∥tp∧q,λ*∥a˙p,q,ωs,τ(ℝn), where p∧q=min{p,q}.
To see the converse, fix a dyadic cubeP. Let rQ≡tQ if Q⊂3P and rQ≡0 otherwise, and let uQ≡tQ-rQ. Set r≡{rQ}Q and u≡{uQ}Q. Then for all dyadic cubes Q, we have
(21)(tp∧q,λ*)Qp∧q=(rp∧q,λ*)Qp∧q+(up∧q,λ*)Qp∧q.
Applying the fact of Lemma9 that for each sequence t={tQ}Q, ∥tp∧q,λ*∥b˙p,q,ωs,0(ℝn)~∥t∥b˙p,q,ωs,0(ℝn) and ∥tp∧q,λ*∥f˙p,q,ωs,0(ℝn)~∥t∥f˙p,q,ωs,0(ℝn), we then have
(22)IP≡1|P|τ×{∑j=jP∞‍[∑l(Q)=2-jQ⊂P‍[|Q|-s/n-1/2ω(Q)1/p(rp∧q,λ*)Q]p]q/p}1/q≤1|P|τ∥rp∧q,λ*∥b˙p,q,ωs,0(ℝn)≲1|P|τ∥r∥b˙p,q,ωs,0(ℝn)≲∥t∥b˙p,q,ωs,τ(ℝn)
and similarly,
(23)I~P≡1|P|τ{∫P‍[∑Q⊂P‍[|Q|-s/n-1/2(rp∧q,λ*)QχQ(x)]q]p/q×ω(x)dx∫P‍[∑Q⊂P‍[|Q|-s/n-1/2(rp∧q,λ*)QχQ(x)]q]p/q}1/p≲∥t∥f˙p,q,ωs,τ(ℝn).
On the other hand, letQ⊂P be a dyadic cube with l(Q)=2-il(P) for some i∈ℤ+. Suppose Q~ is any dyadic cube with l(Q~)=l(Q)=2-il(P) and Q~⊂P+kl(P)⊈3P for some k∈ℤn, where P+kl(P)≡{x+kl(P):x∈P}. Then |k|≥2 and 1+l(Q~)-1|xQ-xQ~|~2i|k|. For
(24)ω(Q)ω(Q~)=ω(Q)ω(P)ω(P)ω(P+kl(P))ω(P+kl(P))ω(Q~)≤Cknα22in(α2-α1),
we have
(25)JP≡1|P|τ×{∑i=0∞‍[∑l(Q)=2-il(P)Q⊂P[(up∧q,λ*)Q|Q|-s/n-1/2ω(Q)1/p‍×(up∧q,λ*)Q]p∑l(Q)=2-il(P)Q⊂P[|Q|-s/n-1/2ω(Q)1/p‍]q/p}1/q≲1|P|τ×{∑l(Q~)=2-il(P)Q~⊂P+kl(P)‍(|Q~|-s/n-1/2ω(Q~)1/p|tQ~|)p∧q]q/(p∧q)∑i=0∞‍2in(α2-α1)q/p-iλq/(p∧q)×[∑l(Q~)=2-il(P)Q~⊂P+kl(P)‍(|Q~|-s/n-1/2ω(Q~)1/p|tQ~|)p∧q]q/(p∧q)∑k∈ℤn|k|≥2‍|k|nα2-λ×∑l(Q~)=2-il(P)Q~⊂P+kl(P)‍(|Q~|-s/n-1/2ω(Q~)1/p×|tQ~||Q~|-s/n-1/2ω(Q~)1/p)p∧q[∑l(Q~)=2-il(P)Q~⊂P+kl(P)‍(|Q~|-s/n-1/2ω(Q~)1/p|tQ~|)p∧q]q/(p∧q)]q/(p∧q)}1/q.
When p≤q, by λ>n+nα2, we have
(26)JP≲∥t∥b˙p,q,ωs,τ(ℝn)×{∑i=0∞‍2in(α2-α1)q/p-iλq/p(∑k∈ℤn|k|≥2‍|k|nα2-λ)q/p}1/q≲∥t∥b˙p,q,ωs,τ(ℝn).
When p>q, by Hölder’s inequality and λ>n+nα2, we obtain
(27)JP≲1|P|τ×{∑l(Q~)=2-il(P)Q~⊂P+kl(P)‍(|Q~|-s/n-1/2ω(Q~)1/p|tQ~|)q]∑i=0∞‍2in(α2-α1)q/p-iλ×[∑l(Q~)=2-il(P)Q~⊂P+kl(P)‍(|Q~|-s/n-1/2ω(Q~)1/p|tQ~|)q∑k∈ℤn|k|≥2‍|k|nα2-λ×∑l(Q~)=2-il(P)Q~⊂P+kl(P)‍(|Q~|-s/n-1/2ω(Q~)1/p|tQ~|)q]}1/q≲∥t∥b˙p,q,ωs,τ(ℝn){∑i=0∞‍2in(α2-α1)-λ(∑k∈ℤn|k|≥2‍|k|nα2-λ)}1/q≲∥t∥b˙p,q,ωs,τ(ℝn).
Therefore, by (21), ∥tmin{p,q},λ*∥b˙p,q,ωs,τ(ℝn)≲supP∈𝒬(IP+JP)≲∥t∥b˙p,q,ωs,τ(ℝn).
To complete the proof, for anyi∈ℤ+, k∈ℤ+n, and dyadic cube P, set
(28)A(i,k,P)≡{Q~∈𝒬:l(Q~)=2-il(P),Q~⊂P+kl(P),Q~∩(3P)=∅}.
Recall that 1+l(Q~)-1|xQ-xQ~|~2i|k| for any Q⊂P and Q~∈A(i,k,P), and that for all d∈[0,1] and {aj}j⊂ℂ,
(29)(∑j‍|aj|)d≤∑j‍|aj|d.
Similarly to the proof of Lemma A.2 (see [18, Remark A.3]), we obtain that for all x∈P and a∈(0,p∧q],
(30)∑Q~∈A(i,k,P)‍(|Q~|-s/n-1/2|tQ~|)p∧q(1+l(Q~)-1|xQ-xQ~|)λ≲(2i|k|)-λ×[∫P+kl(P)‍∑‍(|Q~|-s/n-1/2|tQ~|)aω(Q~)χQ~ω(x)dx](p∧q)/a≲2-i(λ-nα2(p∧q)/a)|k|-λ×[Mω(∑l(Q~)=2-il(P)Q~⊂P+kl(P)‍(|Q~|-s/n|tQ~|χ~Q~)a)×(x+kl(P))(∑l(Q~)=2-il(P)Q~⊂P+kl(P)‍(|Q~|-s/n|tQ~|χ~Q~)a)](p∧q)/a,
where herein and in what follows, Mω denotes the Hardy-Littlewood maximal function on ℝn. Let a≡(2n(p∧q)α2)/(λ+nα2). Then a∈(0,p∧q). Applying Minkowski’s inequality, Fefferman-Stein’s weighted vector-valued inequality, and Hölder’s inequality, we have
(31)J~P≡1|P|τ×{∫P‍[∑Q⊂P‍[|Q|-(s/n)-(1/2)(up∧q,λ*)QχQ(x)]q]p/qdx}1/p≲1|P|τ×{∫P‍[∑i=0∞‍((x+kl(P))(∑l(Q~)=2-il(P)Q~⊂P+kl(P)‍(|Q~|-s/n|tQ~|χ~Q~)a)](p∧q)/a)q/(p∧q)]p/q∑k∈ℤn|k|≥2‍2-i(λ-nα2(p∧q)/a)|k|-λ×[Mω(∑l(Q~)=2-il(P)Q~⊂P+kl(P)‍(|Q~|-s/n|tQ~|χ~Q~)a)×(x+kl(P))(∑l(Q~)=2-il(P)Q~⊂P+kl(P)‍(|Q~|-s/n|tQ~|χ~Q~)a)](p∧q)/a(x+kl(P))(∑l(Q~)=2-il(P)Q~⊂P+kl(P)‍(|Q~|-s/n|tQ~|χ~Q~)a)](p∧q)/a)q/(p∧q)]p/q)q/(p∧q)]p/q×dx(x+kl(P))(∑l(Q~)=2-il(P)Q~⊂P+kl(P)‍(|Q~|-s/n|tQ~|χ~Q~)a)](p∧q)/a)q/(p∧q)]p/q}1/p≲∥t∥f˙p,q,ωs,τ(ℝn).
Therefore, by (21) again, ∥tmin{p,q},λ*∥f˙p,q,ωs,τ(ℝn)≲supP∈𝒬(I~P+J~P)≲∥t∥f˙p,q,ωs,τ(ℝn), which completes the proof of Lemma 10.Letφ satisfy (1) through (3). Since φ~(x)≡φ(-x)- also satisfy (1) through (3), we may take φ~ in place of φ in the definition of B˙p,q,ωs,τ(ℝn). For any f∈𝒮∞′(ℝn) and Q∈𝒬 with l(Q)=2-j, define the sequence sup(f)≡{supQ(f)}Q by setting supQ(f)≡|Q|1/2supy∈Q|φ~j*f(y)|, and for any γ∈ℤ+, the sequence infγ(f)≡{infQ,γ(f)}Q by setting infQ,γ(f)≡|Q|1/2max{infy∈Q~|φ~j*f(y)|:l(Q~)=2-γl(Q),Q~⊂Q}. Choosing λ>n+nα2 as in the proof of [6, Lemma 2.9], we have the following estimates.Lemma 11.
Lets∈ℝ, τ∈[0,∞), p,q∈(0,∞], and γ∈ℤ+ be sufficiently large. Then there exists a constant C∈[1,∞) such that for all f∈A˙p,q,ωs,τ(ℝn), C-1∥infγ(f)∥a˙p,q,ωs,τ(ℝn)≤∥f∥A˙p,q,ωs,τ(ℝn)≤∥sup(f)∥a˙p,q,ωs,τ(ℝn)≤C∥infγ(f)∥a˙p,q,ωs,τ(ℝn).Proof of Theorem2.
With Lemmas7, 10, and 11, the argument for Theorem 2 follows from the method pioneered by Frazier and Jawerth (see [18, pages 50-51]); see also the proof of [10, Theorem 3.1]. We omit the details.From Theorem2, we immediately deduce the following conclusion.Corollary 12.
With all the notations as in Definition1, then the spaces A˙p,q,ωs,τ(ℝn) are independent of the choice of φ.


## 3. Almost Diagonal Operators onA˙p,q,ωs,τ(ℝn)

As an application of Theorem2, we study boundedness of operators in A˙p,q,ωs,τ(ℝn) by first considering their boundedness in corresponding a˙p,q,ωs,τ(ℝn). In this section, we show that almost diagonal operators are bounded on a˙p,q,ωs,τ(ℝn) for appropriate indices, which generalize the classical results on b˙p,q,ωs,0(ℝn) and f˙p,q,ωs,0(ℝn); see [16, 18].Definition 13.
Lets∈ℝ, p,q∈(0,∞], J≡n/min{1,p} for the space b˙p,q,ωs,τ(ℝn), J≡n/min{1,p,q} for the space f˙p,q,ωs,τ(ℝn), and ɛ∈(0,∞). An operator A associated with a matrix {aQP}Q,P, namely, for all sequences t={tQ}Q⊂ℂ, At≡{(At)Q}Q≡{∑P‍aQPtP}Q, is called ɛ-almost diagonal ona˙p,q,ωs,τ(ℝn) if the matrix {aQP}Q,P satisfies
(32)supQ,P|aQP|ωQP(ɛ)<∞,
where
(33)ωQP(ɛ)≡(l(Q)l(P))s(1+|xQ-xP|max(l(P),l(Q)))-J-ɛ×min[(l(Q)l(P))(n+ɛ)/2,(l(P)l(Q))(n+ɛ)/2+J-n].We remark that anɛ-almost diagonal operator is also an almost diagonal operator introduced by Frazier and Jawerth in [18]. In [18, Section 9], Frazier and Jawerth showed that certain appropriate Calderón-Zygmund operators and certain classes of Fourier multiplier operators correspond to almost diagonal matrices, and hence, the φ-transform simultaneously “almost diagonalizes” these operators. Moreover, Yang and Yuan proved that all almost diagonal operators are bounded on f˙p,qs,τ(ℝn) and b˙p,qs,τ(ℝn); see [9, 10]. These results can be generalized into the weighted Besov- and Triebel-Lizorkin-type spaces. We turn to prove it.Proof of Theorem3.
Lett={tQ}Q∈a˙p,q,ωs,τ(ℝn) and let A be an ɛ-almost diagonal operator associated with the matrix {aQR}Q,R and ɛ∈(n(α1-α2)/p,∞). Without loss of generality, we may assume that s=0. Indeed, if the conclusion holds for s=0, let t~R≡l(R)-stR and let B be the operator associated with the matrix {bQR}Q,R, where bQR≡(l(R)/l(Q))saQR for all Q,R∈𝒬. Then we have ∥At∥a˙p,q,ωs,τ(ℝn)=∥Bt~∥a˙p,q,ω0,τ(ℝn)≲∥t~∥a˙p,q,ω0,τ(ℝn)~∥t∥a˙p,q,ωs,τ(ℝn), which deduces the desired conclusion.
We now consider the spaceb˙p,q,ω0,τ(ℝn) in the case min(p,q)>1. For all Q∈𝒬, we write A≡A0+A1 with (A0t)Q≡∑{R:l(R)≥l(Q)}‍aQRtR and (A1t)Q≡∑{R:l(R)<l(Q)}‍aQRtR. By Definition 13, we see that for all Q∈𝒬,
(34)|(A0t)Q|≲∑{R:l(R)≥l(Q)}‍(l(Q)l(R))(n+ɛ)/2×|tR|(1+l(R)-1|xQ-xR|)n+ɛ,
and therefore
(35)∥A0t∥b˙p,q,ω0,τ(ℝn)≲supPdyadic1|P|τ×{ω(x)dx(∑l(Q)=2-j‍∑{R:l(R)>l(P)}‍⋯)p]q/p∑j=jP∞‍[∫P‍(∑l(Q)=2-j‍∑{R:l(Q)≤l(R)≤l(P)}‍(l(Q)l(R))(n+ɛ)/2×|tR|χ~Q(x)(1+l(R)-1|xQ-xR|)n+ɛ(∑l(Q)=2-j‍∑{R:l(Q)≤l(R)≤l(P)}‍(l(Q)l(R))(n+ɛ)/2)p×ω(x)dx(∑l(Q)=2-j‍∑{R:l(Q)≤l(R)≤l(P)}‍(l(Q)l(R))(n+ɛ)/2]q/p}1/q+supPdyadic1|P|τ×{ω(x)dx(∑l(Q)=2-j‍∑{R:l(R)>l(P)}‍⋯)p]q/p∑j=jP∞‍[∫P‍(∑l(Q)=2-j‍∑{R:l(R)>l(P)}‍⋯)p×ω(x)dx(∑l(Q)=2-j‍∑{R:l(R)>l(P)}‍⋯)p]q/p}1/q≡I1+I2.
For alli∈ℤ and m∈ℕ, set U0,i≡{R∈𝒬:l(R)=2-i and |xQ-xR|<l(R)} and Um,i≡{R∈𝒬:l(R)=2-i and 2m-1l(R)≤|xQ-xR|<2ml(R)}. Then we have #Um,i≲2mn, where #Um,i denotes the cardinality of Um,i. Notice that |tR|≤|R|1/2+τω(R)-1/p∥t∥b˙p,q,ω0,τ(ℝn) for all R∈𝒬. Since 2m-1l(R)≤|xQ-xR|<2ml(R), l(Q)≤l(R),
(36)|x-xR|≤|x-xQ|+|xQ-xR|≤nl(Q)+2ml(R)≤(2m+n)l(R)≲C2ml(R)≤C2m+j-il(Q).
LetQ~=2m+j-iQ. If p≤1, τ<(ɛ/2n)+(α1/p), ɛ>(n(α2-α1)/p), then
(37)ℬ≡1|P|τ×{χQp(x)ω(x)dx∑j=jP∞‍[∫P‍∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛp/2]q/p∑j=jP∞‍[∫P‍(∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛ/2×|R|τω(R)-1/p2-m(n+ɛ)χQ(x)∑i=-∞jP-1‍)p×ω(x)dx∑i=-∞jP-1‍]q/p}1/q≤2jPnτ{χQp(x)ω(x)dx∑j=jP∞‍[∫P‍∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛp/2]q/p∑j=jP∞‍[∫P‍∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛp/2×ω(R)-1|R|τp2-(n+ɛ)mp×χQp(x)ω(x)dx∑j=jP∞‍[∫P‍∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛp/2]q/p}1/q≤2jPnτ{χQp(x)ω(x)dx∑j=jP∞‍[∫P‍∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛp/2]q/p∑j=jP∞‍[∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛp/2×l(R)nτp2-(n+ɛ)mpω(Q)ω(R)∑j=jP∞‍[∫P‍∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛp/2]q/p}1/q≤2jPnτ{χQp(x)ω(x)dx∑j=jP∞‍[∫P‍∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛp/2]q/p∑j=jP∞‍[∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛp/2×l(R)nτp2-(n+ɛ)mpω(Q~)ω(R)ω(Q)ω(Q~)∑j=jP∞‍[∫P‍∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛp/2]q/p}1/q≤C2jPnτ{χQp(x)ω(x)dx∑j=jP∞‍[∫P‍∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛp/2]q/p∑j=jP∞‍[∑j=jP∞‍[∫P‍∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛp/2∑i=-∞jP-1‍∑m=0∞‍2(iɛp/2)-inτp+inα12-(jɛp/2)-jnα1×2mn(α2-α1)2-mɛp)∑j=jP∞‍[∫P‍∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛp/2]q/p}1/q≤C,
here we use ω(Q~)/ω(R)≲2mnα2 and ω(Q)/ω(Q~)≲2-n(m+j-i)α1.
Ifp>1, using Minkowski’s inequality, we have
(38){∫P‍[∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛ/2×|R|τω(R)-1/p2-m(n+ɛ)χQ(x)∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛ/2]pω(x)dx}1/p≤∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍[l(Q)l(R)]ɛ/2×[ω(Q)ω(R)]1/p|R|τ2-m(n+ɛ)≤∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍[l(Q)l(R)]ɛ/2×[ω(Q~)ω(R)]1/p[ω(Q)ω(Q~)]1/p|R|τ2-m(n+ɛ).
Therefore,(39)ℬ≤C2jPnτ×{∑j=jP∞‍[∫P‍∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍2i((ɛ/2)+(nα1/p)-nτ)×2j(-(ɛ/2)-(nα1/p))2-m(n+ɛ)2nm(α2-α1)/p∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛ/2]q}1/q≤C.
Thus, by 0≤τ<(α1/p)+(ɛ/2n),
(40)I2≲∥t∥b˙p,q,ω0,τ(ℝn)supPdyadic1|P|τ×{χQp(x)ω(x)dx∑j=jP∞‍[∫P‍∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛp/2]q/p∑j=jP∞‍[∫P‍(∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))(n+ɛ)/2×|R|1/2+τω(R)-1/pχ~Q(x)(1+l(R)-1|xQ-xR|)n+ɛ)p×ω(x)dx∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛ/2]q/p}1/q≲∥t∥b˙p,q,ω0,τ(ℝn)supPdyadic1|P|τ×{χQp(x)ω(x)dx∑j=jP∞‍[∫P‍∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛp/2]q/p∑j=jP∞‍[∫P‍(∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛ/2×|R|τω(R)-1/p2-m(n+ɛ)χQ(x)∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛ/2)p×ω(x)dx∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛ/2]q/p}1/q≲∥t∥b˙p,qω0,τ(ℝn).
ForI1, let r and u be the same as in the proof of Lemma 10. We see that
(41)I1≲supPdyadic1|P|τ×{χQp(x)ω(x)dx∑j=jP∞‍[∫P‍∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛp/2]q/p∑j=jP∞‍[∫P‍(∑l(Q)=2-j‍∑i=jPj‍2(i-j)(n+ɛ)/2×∑l(R)=2-i‍|rR|χ~Q(x)(1+l(R)-1|xQ-xR|)n+ɛ)p×dx∑l(R)=2-i‍|rR|χ~Q(x)(1+l(R)-1|xQ-xR|)n+ɛ)p]q/p}1/q+supPdyadic1|P|τ×{dx∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛ/2]qp∑j=jP∞‍[∫P‍(∑l(Q)=2-j‍∑i=jPj‍2(i-j)(n+ɛ)/2×∑l(R)=2-i‍|uR|χ~Q(x)(1+l(R)-1|xQ-xR|)n+ɛ)p×dx∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛ/2]q/p}1/q≡J1+J2.
Applying Lemma 8 with a=1, for all x∈Q, we have
(42)∑l(R)=2-i‍|rR|(1+l(R)-1|xQ-xR|)n+ɛ≲Mω(∑l(R)=2-i‍|rR|χR)(x).
Hence Hölder’s inequality and the Lωp(ℝn) boundedness for p∈(1,∞] of the Hardy-Littlewood maximal operator yield
(43)J1≲supPdyadic1|P|τ×{dx∑l(Q)=2-j‍∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))ɛ/2]qp∑j=jP∞‍[∫P‍(∑i=jPj‍2((i-j)ɛ)/2Mω(∑l(R)=2-i‍|rR|χ~R)(x))p×ω(x)dx(∑i=jPj‍2((i-j)ɛ)/2Mω(∑l(R)=2-i‍|rR|χ~R)(x))p]q/p}1/q≲supPdyadic1|P|τ×{∑i=jP∞‍[∫3P‍(∑l(R)=2-i‍|tR|χ~R)p(x)ω(x)dx]q/p}1/q≲∥t∥b˙p,q,ω0,τ(ℝn),
where the last inequality follows from Minkowski’s inequality if q>p or (29) if q≤p.
To estimateJ2, we notice that if R∩(3P)=∅, then R⊂P+kl(P) and (P+kl(P))∩(3P)=∅ for some k∈ℤn with |k|≥2 and 1+l(R)-1|xQ-xR|~|k|l(P)/l(R). Therefore, by Hölder’s inequality,
(44)J2≲supPdyadic1|P|τ+1+ɛ/n×{∑j=jP∞‍2-jqɛ/2[∫P‍(∑l(Q)=2-j‍∑i=jPj‍2-i(n+ɛ)/2∑k∈ℤn|k|≥2‍|k|-n-ɛ×∑l(R)=2-iR⊂P+kl(P)‍|tR|χQ(x))p×ω(x)dx(∑l(Q)=2-j‍∑i=jPj‍2-i(n+ɛ)2×∑k∈ℤn|k|≥2‍|k|-n-ɛ]q/p}1/q≲∥t∥b˙p,q,ω0,τ(ℝn)supPdyadic1|P|ɛ/n×{∑j=jP∞‍2-jqɛ/2[∑i=jPj‍2-iɛ/2∑k∈ℤn|k|≥2‍|k|-n-ɛ]q}1/q≲∥t∥b˙p,q,ω0,τ(ℝn).
Hence,
(45)∥A0t∥b˙p,q,ω0,τ(ℝn)≲∥t∥b˙p,q,ω0,τ(ℝn).
The similar computations to I1 yield
(46)∥A1t∥b˙p,q,ω0,τ(ℝn)≲∥t∥b˙p,q,ω0,τ(ℝn).
For Triebel-Lizorkin-type spacesf˙p,q,ωs,τ(ℝn), we also have
(47)∥A0t∥f˙p,q,ω0,τ(ℝn)≲supPdyadic1|P|τ×{∫P‍[∑Q⊂P‍(∑{R:l(Q)≤l(R)≤l(P)}‍(l(Q)l(R))(n+ɛ)/2×|tR|χ~Q(x)(1+l(R)-1|xQ-xR|)n+ɛ(∑{R:l(Q)≤l(R)≤l(P)}‍(l(Q)l(R))(n+ɛ)/2)q]p/q×ω(x)dx(∑{R:l(Q)≤l(R)≤l(P)}‍(l(Q)l(R))(n+ɛ)/2}1/p+supPdyadic1|P|τ×{∫P‍[∑Q⊂P‍(∑{R:l(R)>l(P)}‍⋯)q]p/qω(x)dx}1/p≡I~1+I~2.
Similarly to the estimate for I2, applying the fact that |tR|≤|R|1/2+τω(R)-1/p∥t∥b˙p,q,ω0,τ(ℝn) and 0≤τ<(α1/p)+(ɛ/2n), we obtain
(48)I~2≲∥t∥f˙p,q,ω0,τ(ℝn)supPdyadic1|P|τ×{∫P‍[∑j=jP∞‍∑l(Q)=2-j‍×(∑i=-∞jP-1‍∑m=0∞‍∑R∈Um,i‍(l(Q)l(R))(n+ɛ)/2×|R|1/2+τω(R)-1/pχ~Q(x)(1+l(R)-1|xQ-xR|)n+ɛ(∑{R:l(Q)≤l(R)≤l(P)}‍(l(Q)l(R))(n+ɛ)/2)q]p/q×ω(x)dx(∑{R:l(Q)≤l(R)≤l(P)}‍(l(Q)l(R))(n+ɛ)/2}1/p≲∥t∥f˙p,q,ω0,τ(ℝn).
To estimateI~1, notice that
(49)I~1≲supPdyadic1|P|τ×{×∑l(R)=2-i‍|rR|χ~Q(x)(1+l(R)-1|xQ-xR|)n+ɛ)q]p/q∫P‍[∑j=jP∞‍∑l(Q)=2-j‍×(∑i=jPj‍2(i-j)(n+ɛ)/2×∑l(R)=2-i‍|rR|χ~Q(x)(1+l(R)-1|xQ-xR|)n+ɛ)q]p/q×ω(x)dx×∑l(R)=2-i‍|rR|χ~Q(x)(1+l(R)-1|xQ-xR|)n+ɛ)q]p/q}1/p+supPdyadic1|P|τ×{×∑l(R)=2-i‍|rR|χ~Q(x)(1+l(R)-1|xQ-xR|)n+ɛ)q]p/q∫P‍[∑j=jP∞‍∑l(Q)=2-j‍×(∑i=jPj‍2(i-j)(n+ɛ)/2×∑l(R)=2-i‍|uR|χ~Q(x)(1+l(R)-1|xQ-xR|)n+ɛ)q]p/q×ω(x)dx×∑l(R)=2-i‍|rR|χ~Q(x)(1+l(R)-1|xQ-xR|)n+ɛ)q]p/q}1/p≡J~1+J~2.
Similarly to the estimate of J1, by [18, Remark A.3], Hölder’s inequality and the Fefferman-Stein weighted vector-valued inequality, we obtain
(50)J~1≲supPdyadic1|P|τ×{×∑l(R)=2-i‍|rR|χ~Q(x)(1+l(R)-1|xQ-xR|)n+ɛ)q]p/q∫P((∑l(R)=2-i‍|rR|χ~R)(x)]q∑j=jP∞‍[∑i=jPj‍2(i-j)ɛ/2MHL‍×(∑l(R)=2-i‍|rR|χ~R)(x)]q)p/q×ω(x)dx×∑l(R)=2-i‍|rR|χ~Q(x)(1+l(R)-1|xQ-xR|)n+ɛ)q]p/q}1/p≲supPdyadic1|P|τ×{∫3P‍[∑i=jP∞‍(∑l(R)=2-i‍|tR|χ~R(x))q]p/qω(x)dx}1/p≲∥t∥f˙p,q,ω0,τ(ℝn).
For J~2, similarly to the estimate for J2, by Minkowski’s inequality and the Fefferman-Stein weighted vector-valued maximal inequality, we obtain
(51)J~2≲∑k∈ℤn|k|≥2‍|k|-(n+ɛ)supPdyadic1|P|τ+ɛ/n×{∫P‍(∑j=jP∞‍[(∑l(R)=2-iR⊂P+kl(P)‍|R|-12|tR|χR)∑i=jPj‍2(i-j)ɛ/22-iɛ×MHL(∑l(R)=2-iR⊂P+kl(P)‍|R|-1/2|tR|χR)×(x+kl(P))(∑l(R)=2-iR⊂P+kl(P)‍|R|-12|tR|χR)]q)p/q×ω(x)dx(∑l(R)=2-iR⊂P+kl(P)‍|R|-12|tR|χR)}1/p≲∥t∥f˙p,q,ω0,τ(ℝn).
Hence ∥A0t∥f˙p,q0,τ(ℝn)≲∥t∥f˙p,q,ω0,τ(ℝn).
Some similar estimates toI~1 also yield that ∥A1t∥f˙p,q,ω0,τ(ℝn)≲∥t∥f˙p,q,ω0,τ(ℝn). Thus, we obtain the desired conclusion for the case min{p,q}>1.
The casemin{p,q}≤1 is a simple consequence of the case min{p,q}>1. Indeed, choosing a δ∈(0,p∧q), then p/δ>1 and q/δ>1. Let A~ be an operator associated with the matrix {a~QP}Q,P≡{|aQP|δ(l(Q)/l(P))n/2-δn/2}Q,P. Then A~ is a ɛ~-almost diagonal operator with s=0 and ɛ~=δɛ.
Definet~≡{l(Q)n/2-δn/2|tQ|δ}Q. Then ∥t~∥a˙p/δ,q/δ,ω0,τδ(ℝn)1/δ=∥t∥a˙p,q,ω0,τ(ℝn). Since δ<1, we have ∥At∥a˙p,q,ω0,τ(ℝn)≲∥A~t~∥a˙p/δ,q/δ,ω0,τδ(ℝn)1/δ. Applying the conclusions for min{p,q}>1 yields ∥At∥a˙p,q,ω0,τ(ℝn)≲∥A~t~∥a˙p/δ,q/δ,ω0,τδ(ℝn)1/δ≲∥t~∥a˙p/δ,q/δ,ω0,τδ(ℝn)1/δ~∥t∥a˙p,q,ω0,τ(ℝn), which completes the proof of Theorem 3.Remark 14.
Recall thatb˙p,q,ωs,0(ℝn)=b˙p,q,ωs(ℝn) and f˙p,q,ωs,0(ℝn)=f˙p,q,ωs(ℝn). Thus, if τ=0, by Theorem 3, we see that for all ɛ∈(0,∞), all ɛ-almost diagonal operators are bounded on b˙p,q,ωs(ℝn) and f˙p,q,ωs(ℝn), which is just the classical results established by Frazier and Jawerth; see [18, Theorem 3.3] and [16, Theorem 6.20].



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*Source: 865835-2013-03-20.xml*