Received: 12 November 2010 / Accepted: 28 February 2011 / Published online: 16 March 2011 Springer Basel AG 2011

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1 Positivity (202) 6: DOI 0.007/s Positivity Duality of weighted anisotroic Besov and Triebel Lizorkin saces Baode Li Marcin Bownik Dachun Yang Wen Yuan Received: 2 November 200 / Acceted: 28 February 20 / Published online: 6 March 20 Sringer Basel AG 20 Abstract Let A be an exansive dilation on R n and w a Muckenhout A (A) weight. In this aer, for all arameters α R and, (0, ), the authors identify the dual saces of weighted anisotroic Besov saces Ḃ α, (A; w)and Triebel Lizorkin saces Ḟ, α (A; w) with some new weighted Besov-tye and Triebel Lizorkin-tye saces. The corresonding results on anisotroic Besov saces Ḃ α, (A; µ) and Triebel Lizorkin saces Ḟ, α (A; µ) associated with ρ A-doubling measure µ are also established. All results are new even for the classical weighted Besov and Triebel Lizorkin saces in the isotroic setting. In articular, the authors also obtain the ϕ-transform characterization of the dual saces of the classical weighted Hardy saces on R n. B. Li is suorted by the National Natural Science Foundation of China (Grant No ) and the Start-u Funding Doctor of Xinjiang University (Grant No. BS09009), M. Bownik is suorted by National Science Foundation of US (Grant No. DMS ) and D. Yang is suorted by the National Natural Science Foundation (Grant No ) of China and Program for Changjiang Scholars and Innovative Research Team in University of China. B. Li Deartment of Mathematics, Xinjiang University, Urumi , Peole s Reublic of China baodeli@mail.bnu.edu.cn M. Bownik Deartment of Mathematics, University of Oregon, Eugene, OR , USA mbownik@uoregon.edu D. Yang (B) W. Yuan Laboratory of Mathematics and Comlex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing 00875, Peole s Reublic of China dcyang@bnu.edu.cn W. Yuan wyuan@mail.bnu.edu.cn

2 24 B. Li et al. Keywords Exansive dilation Besov sace Triebel Lizorkin sace Muckenhout weight ϕ-transform Dual sace Mathematics Subject Classification (200) Primary 46A20; Secondary 42B25 42B35 42C40 46B0 46E39 Introduction There were many efforts on generalizing various secific function saces to alications of analysis such as PDEs, harmonic analysis and aroximation theory (see, for examle [ 3, 3,5,6,23 25,29,32 34,37,38]). This gave rise to the study of Besov and Triebel Lizorkin saces which form a unifying class of function saces containing many well-known classical function saces such as Lebesgue saces, Hardy saces and Hardy Sobolev saces. In articular, there were also several efforts to extending the classical function saces arising in harmonic analysis from Euclidean saces and isotroic settings to other domains and anisotroic settings. Calderón and Torchinsky [0 2] introduced and investigated Hardy saces associated with anisotroic dilations. A theory of anisotroic Hardy saces and their weighted counterarts were recently develoed by Bownik et al. in [,7]. Anisotroic Besov and Triebel Lizorkin saces including their weighted variants (more generally, associated with doubling measures) were also introduced and studied (see, for examle [2 5,5,25]). In these aers, the discrete wavelet transform, the atomic and molecular decomositions of these saces, and the dual saces of anisotroic Triebel Lizorkin saces without weights were established. However, the duality of weighted anisotroic Triebel Lizorkin saces in [4, Theorem 4.0] was obtained under an additional assumtion that the considered exansive dilations admit a Meyer-orthonormal wavelet. In this aer, we introduce some new weighted anisotroic Besov-tye and Triebel Lizorkin-tye saces and we identify the dual saces of weighted anisotroic Besov and Triebel Lizorkin saces with these new weighted saces. We oint out that our results are new even for the classical weighted Besov and Triebel Lizorkin saces in the isotroic setting. In articular, by relaxing the assumtion that w A (R n ) (the class of Muckenhout s weights) into w A (R n ), our results also imrove the results obtained by Bui in [9], Roudenko in [28] and Frazier and Roudenko in [9], which are resectively [9, Theorem 2.0] and the scalar versions of [28, Theorem A(3)] and [9, Theorem 5.9] on the dual saces of the matrix-weighted Besov saces. As a secial case of our results on the weighted Triebel Lizorkin saces in isotroic settings, we also obtain the ϕ-transform characterization of the dual saces of the classical weighted Hardy saces on R n, which also seems new. Recall that the classical weighted Hardy saces on R n and their dual saces were first studied by García- Cuerva in [20]. The wavelet characterizations of the weighted Hardy saces on R n and their dual saces were obtained in [2,27,40]. Let A be an exansive dilation on R n (see [] or Definition 2. below) and w a Muckenhout A (A) weight associated with A (see [5] or Definition 2.2

3 Duality of weighted anisotroic Besov and Triebel Lizorkin saces 25 below). In what follows, for any (0, ], denotes the conjugate index of, namely, when (0, ] and /( /) when (, ]. In this aer, for all arameters α R, τ [0, ) and, (0, ], we introduce new weighted anisotroic Besov-tye saces Ḃ α,τ,( A; w) and Triebel Lizorkin-tye saces Ḟ, α,τ ( A; w) (see Definitions 2.4 and 2.5 below). By establishing the duality results on their corresonding seuence saces, we rove that, for all α R and, (0, ), the dual sace of the weighted anisotroic Besov sace Ḃ α, (A; w) is Ḃ α,max{/,}, (A; w) (see Theorem 2.2 below), and the dual sace of the weighted anisotroic Triebel Lizorkin sace Ḟ, α (A; w) is Ḟ α,/+/, (A; w) when (0, ] or Ḟ α,0, (A; w) when (, ) (see Theorem 2. below). These results are also true for those anisotroic Besov saces Ḃ α, (A; µ) and Triebel Lizorkin saces Ḟ, α (A; µ) associated with ρ A-doubling measure µ (see Theorem 4. below). We remark that when w, for any α R, (0, ], (0, ) and τ 0 = /+/, Ḟ α,τ 0, (A; w) = Ḟ, α+/ ( A) with euivalent norms (see Corollary 2.(ii) below), which further shows the coincidence of our results on duality with existing known results in [4] in unweighted case. Moreover, if w and A 2 I n n, where I n n denotes the n n unit matrix, then the Triebel Lizorkin-tye saces Ḟ, α,τ (A; w) were introduced and studied in [4,42], and it was roved in [30,4 43] that they include several classical saces such as Triebel Lizorkin saces (see [32]), Q saces (see [4]), Morrey saces and art of Morrey Camanato saces; while, in this case, the Besov-tye saces Ḃ α,τ,(a; w) are just the Besov saces Ḃ α+τ, (A). This reflects the difference between Ḃ α,τ,(a; w) and Ḟ, α,τ (A; w). Two key ideas used in the roofs of Theorems 2. and 2.2 are that, differently from the roofs on the duality in [4,7,27,39], we adot the notion of the tents (see [3] or (2.) below) for dilated cubes and also introduce the notion of the seudo-maximal dilated cubes, which are used to subtly classify dilated cubes (see (3.2) below). In this sense, the roofs of Theorems 2. and 2.2 are uite geometrical. The organization of this aer is as follows. In Sect. 2, we resent some basic notions and the duality results on weighted anisotroic Besov and Triebel Lizorkin saces, whose roofs are given in Sect. 3. In Sect. 4, we rove that the duality results in Sect. 2 are also true for anisotroic Besov and Triebel Lizorkin saces associated with doubling measures. We oint out that all results of this aer are also true for inhomogeneous saces with slight modifications (see, for examle [32,43]). We omit the details. Finally, we make some conventions on symbols. Throughout the aer, we denote by C a ositive constant which is indeendent of the main arameters, but it may vary from line to line. Constants with subscrits, such as C 0, do not change in different occurrences. The symbol A B means that A CB and the symbol A B means that A B and B A. Denote by E the cardinality of the set E. We will use the convention that the conjugate exonent satisfies / + / = if < and = if 0 <. We also set N {, 2,...}, Z + {0} N and Z n + = (Z +) n. If E is a subset of R n, we denote by χ E the characteristic function of E.

4 26 B. Li et al. 2 Main results We begin with the notion of exansive dilations on R n (see []). Definition 2. A real n n matrix A is called an exansive dilation, shortly a dilation, if max λ σ(a) λ >, where σ(a) is the set of all eigenvalues of A.Auasi-norm associated with exansive matrix A is a Borel measurable maing ρ A : R n [0, ), for simlicity, denoted as ρ, such that (i) ρ(x) >0 for x = 0; (ii) ρ(ax) = bρ(x) for x R n, where b det A ; (iii) ρ(x + y) H[ρ(x) + ρ(y)] for all x, y R n, where H is a constant. Throughout the whole aer, we always let A be an exansive dilation on R n and b det A. The set Q of dilated cubes of R n is defined by Q {Q A j ([0, ) n + k) : j Z, k Z n }. For any Q A j ([0, ) n + k), let the symbol scale (Q) j and x Q A j k be the lower-left corner of Q. We see that for any fixed j Z, {Q A j ([0, ) n + k) : k Z n } is a artition of R n. For any P Q, let T (P) {Q Q : Q P =, scale (Q) scale (P)} (2.) be the tent of P (see [3, Definition 2.4]). We now recall the weight class of Muckenhout associated with A introduced in [5]. Definition 2.2 Let [, ), A be a dilation and w a non-negative and almost everywhere ositive measurable function on R n. A function w is said to belong to the weight class A (A) A (R n ; A) of Muckenhout, if there exists a ositive constant C such that when (, ), su su x R n k Z b k and when =, B ρ (x, b k ) su su x R n k Z b k w(y) dy B ρ (x, b k ) b k B ρ (x,b k ) [w(y)] /( ) dy { } w(y) dy esssu [w(y)] C, y B ρ (x,b k ) C, and the minimal constant C as above is denoted by C,A,n (w). Here, for all x R n and k Z, B ρ (x, b k ) {y R n : ρ(x y) <b k }. Define A (A) < A (A).

5 Duality of weighted anisotroic Besov and Triebel Lizorkin saces 27 For all (0, ) and w A (A), the weighted Lebesgue sace L w(r n ) is defined to be the sace of all measurable functions on R n such that f L w (R n ) { R n f (x) w(x) dx} / <. Denote by S(R n ) the set of all Schwartz functions on R n and S (R n ) its toological dual sace. As in [8], we set S (R n ) φ S(Rn ) : R n φ(x)x α dx = 0, α Z n +. We consider S (R n ) as a subsace of S(R n ), including the toology. Thus, S (R n ) is a comlete metric sace (see, for examle [3,. 2, (3.7)]). Let S (Rn ) be the toological dual sace of S (R n ) with the weak- toology. Definition 2.3 Let A be an exansive dilation and A its transose. Define S (R n ) to be the set of all ϕ S(R n ) such that (i) su ϕ [ π, π] n \{0}, (ii) su j Z ϕ((a ) j ξ) > 0 for all ξ R n \{0}. Obviously, S (R n ) S (R n ). Now let us first recall the notion of the weighted anisotroic Triebel Lizorkin saces in [5] and then introduce some new weighted anisotroic Triebel Lizorkin-tye saces. In what follows, for all Q Q, let j Q scale (Q) and χ Q χ Q Q /2. Definition 2.4 Let w A (A), ϕ S (R n ), α R, (0, ), (0, ] and τ [0, ). (i) The weighted anisotroic Triebel Lizorkin sace Ḟ, α (A; w) is defined to be the set of all f S (Rn ) such that f Ḟα, (A;w) ( Q α ϕ jq f χ Q ) L w (R n ) <, where and in what follows, for all j Z and x R n, ϕ j (x) b j ϕ(a j x). The corresonding discrete weighted anisotroic Triebel Lizorkin sace f, α (A; w) is defined to be the set of all comlex-valued seuences s {s Q } such that s f, α (A;w) ( Q α s Q χ Q ) L w (R n ) <.

6 28 B. Li et al. (ii) The weighted anisotroic Triebel Lizorkin-tye sace Ḟ α,τ, (A; w) is defined to be the set of all f S (Rn ) such that f Ḟα,τ, (A;w) su P Q [w(p)] τ <. P ( Q α ϕ jq f (x) χ Q (x) Q ) w(q) w(x) dx Its corresonding discrete weighted anisotroic Triebel Lizorkin-tye sace f, α,τ (A; w) is defined to be the set of all comlex-valued seuences s {s Q } such that s f, α,τ (A;w) su P Q [w(p)] τ <. P ( Q α s Q χ Q (x) Q ) w(q) w(x) dx It is understood that the above definitions need the usual modification when =. Remark 2. (i) Integrating the norm of Ḟα, (A;w) over cubes Q with fixed scale j yields a familiar euivalent form / f Ḟα, (A;w) b jα ϕ j f. j Z Lw (R n ) (ii) The weighted anisotroic Triebel Lizorkin sace Ḟ, α (A; w) and its discrete variant f, α (A; w) were first introduced in [5]. Moreover, when, (0, ), by the ϕ-transform characterization of Ḟ, α (A; w) (see [5, Theorem 3.5]) and the fact that seuences with finite suort are dense in f, α (A; w) (see [5,. 452]), we know that S (R n ) is dense in Ḟ, α (A; w). (iii) We oint out that in Definition 2.4(ii), when w and A 2 I n n, the sace Ḟ, α,τ (R n ) and its corresonding discrete seuences saces were introduced in [30,4,42] (see also [43] for inhomogeneous versions). The following is the main theorem of this aer. Theorem 2. Let α R,, (0, ), τ 0 = / + / and w A (A). Then, (i) { ( ) f, α (A; w) f α,τ 0 =, (A; w), (0, ], f α,0, (A; w), (, ).

7 Duality of weighted anisotroic Besov and Triebel Lizorkin saces 29 More recisely, l is a bounded linear functional on form f α, (A; w) if and only if l is of the l(λ) = λ, t λ Q t Q, where λ {λ Q } f, α (A; w), (2.2) for some seuence t {t Q } C such that (ii) l ( f α, (A;w) ) t f α,τ 0, (A;w) t f α,0, (A;w), (0, ],, (, ). (Ḟα, (A; w)) = { Ḟ α,τ 0, (A; w), (0, ], Ḟ α,0, (A; w), (, ) in the following sense. For each g Ḟ α,τ 0, (A; w) when (0, ] or g Ḟ α,0, (A; w) when (, ), the ma l( f ) = f, g f (x)g(x) dx, (2.3) R n defined initially for all f S (R n ), has a bounded linear extension to Ḟ, α (A; w). Conversely, any bounded linear functional l on Ḟ, α (A; w) is of the form (2.3) and g Ḟ α,τ0, (0, ], l ( ), (A;w) Ḟ, α (A;w) g Ḟ α,0, (, )., (A;w) Observe that Theorem 2. when w includes [4, Theorem 4.8] which briefly states as follows: ) Ḟ (Ḟα α+, (A) =, (A), (0, ), Ḟ α, (A), [, ). Indeed, let w,, (0, ) and τ 0 = / + /. By definitions of these saces, we immediately have that when (, ), Ḟ, α,0 (A) = Ḟ, α (A), and when =, Ḟ α,τ 0, (A; w) = Ḟ α, (A). By the following Corollary 2.(ii), when (0, ), we also have Ḟ α,τ 0, (A; w) = Ḟ, α+/ (A). This shows the above claim. Moreover, Theorem 2. when w and A 2I n n coincides with the corresonding classical results in [7, Section 5]. As a conseuence of Theorem 2. and [4, Theorem 4.2], we have the following result.

8 220 B. Li et al. Corollary 2. Let α R, (, ] and τ (/, ). (i) If w A (A), then the sace f, α (A; w) is isomorhic with the sace f, α,τ (A; w) via the ma {s Q } { [w(q)]τ /+ Q s Q }. That is, for all {s Q } (A; w), f α, {s Q } f α, (A;w) (ii) If w, then f, α,τ (A; w) = { [w(q)] τ /+ } s Q Q f, α+τ / f, α,τ (A;w) (A; w) with euivalent norms. The same conclusions are true for the saces Ḟ α,τ, (A; w). The roof of Corollary 2. is given in Sect. 3. We oint out that art (ii) of Corollary 2. may not be true if w. We give a counter-examle on -dimensional Euclidean sace R as follows. Examle 2. Let α R, (, ],τ (/, ), A = 2,l (0, ) and w(x) = x l A (A) (see [22,. 286, Examle 9..7]). In this case, we know that Q is the set of all classical dyadic cubes in R. Now, we construct a seuence s {s Q } such that s s f, α+τ / f, α+τ / (A;w) = but s f, α,τ (A;w) =. Since (A;w) su Q (α+τ /) s Q Q /2 (see [4, (2.7)]),. we set s {s Q } with s Q Q α+τ /+/2 for all Q Q. Then s. On the other hand, s f, α,τ (A;w) = su P Q = su P Q [w(p)] τ [w(p)] τ [ ] P τ+ / su. P Q w(p) f, α+τ / (A;w) Q α s Q Q /2 Q [w(q)] w(q) [ ] Q Q τ w(q) Let Q {Q [2 k, 2 k+ ) : k Z}. Then, for any P Q and w(x) = x l, we have w(p) = 2 k+ 2 k x l dx 2 k(l+). Combining this, the above estimate, l> 0 and τ (/, ), we obtain that s f, α,τ (A;w) 2 kl(τ+ /). By letting k, we further obtain that s f, α,τ (A;w) =, which imlies that the saces f, α+τ / (A; w) and f, α,τ (A; w) are not the same saces with euivalent norms. / /

9 Duality of weighted anisotroic Besov and Triebel Lizorkin saces 22 We also have the corresonding duality theory for weighted anisotroic Besov saces. Let us begin with recalling the notion of the weighted anisotroic Besov saces introduced in [2] and then introduce some new weighted anisotroic Besov-tye saces. Definition 2.5 Let w A (A), ϕ S (R n ), α R,, (0, ] and τ [0, ). (i) The weighted anisotroic Besov sace Ḃ α, (A; w) is defined to be the set of all f S (Rn ) such that f Ḃα, (A;w) j Z <. scale (Q)= j R n Q α ϕ j f (x)χ Q (x) w(x) dx The corresonding discrete weighted anisotroic Besov sace ḃ α, (A; w) is defined to be the set of all comlex-valued seuences s {s Q } such that s ḃα, (A;w) j Z scale (Q)= j R n ( Q α s Q χ Q (x) ) w(x) dx <. (ii) The weighted anisotroic Besov-tye sace Ḃ α,τ,(a; w) is defined to the set of all f S (Rn ) such that f Ḃα,τ, (A;w) j Z <. R n scale (Q) = j Q α ϕ j f (x) Q χ Q(x) [w(q)] τ w(x) dx The corresonding discrete weighted anisotroic Besov-tye sace ḃ α,τ,(a; w) is defined to be the set of all comlex-valued seuences s {s Q } such that s ḃα,τ, (A;w) j Z R n scale (Q)= j Q α s Q Q χ Q(x) [w(q)] τ w(x) dx <. It is understood that the above definitions need the usual modifications when = or =.

10 222 B. Li et al. Remark 2.2 (i) Integrating the norm of Ḃα, (A;w) over cubes Q with fixed scale j yields a familiar euivalent form f Ḃα, (A;w) = / b jα ϕ j f (x) w(x) dx j Z R n (ii) The weighted anisotroic Besov sace Ḃ α, (A; w) and its discrete counterart were first introduced in [2]. Moreover, when, (0, ), by the ϕ-transform characterization of Ḃ α, (A; w) (see [2, Theorem 3.5]) and the fact that seuences with finite suort are dense in ḃ α, (A; w)(see [2,. 553]), we know that S (R n ) is dense in Ḃ α, (A; w). (iii) Observe that when w, the Besov sace Ḃ α,τ,( A; w) coincides with Ḃ α+τ, (A); see Proosition 2.. This is in contrast with Ḟ, α,τ (A; w), where the arameter τ lays a significant role; see Remark 2.(iii). From Definition 2.5, we can immediately deduce the following result. Proosition 2. Let w,α R,, (0, ] and τ [0, ). Then, ḃ α,τ, (A; w) = ḃα+τ, (A) and Ḃ α,τ,(a; w) = Ḃ α+τ (A) with euivalent norms., Examle 2.2 In general, Proosition 2. may not be true when w. For examle, letting the dimension n =,α = 0, = =, A = 2 and s {s Q } with s Q Q τ /2 for all Q Q, we see that and s ḃτ, (A) su Q /2 τ s Q = s ḃ0,τ, (A;w) su s Q Q /2 /[w(q)] τ = su [ Q /w(q)] τ. Choose w(x) x for all x R. Then, for all j Z, s ḃ0,τ, (A;w) = su [ Q /w(q)] τ Letting j, we obtain s ḃ0,τ, (A;w) ḃ, τ (A) and ḃ0,τ, (A; w) are not the same. [ 2 j 2 j+ 2 j x dx ] τ 2 jτ. / =, which imlies that the saces We have the following duality results on weighted anisotroic Besov saces, which is another main theorem of this aer..

11 Duality of weighted anisotroic Besov and Triebel Lizorkin saces 223 Theorem 2.2 Let α R,, (0, ) and w A (A). Then in the sense of (2.2), and in the sense of (2.3). (ḃ α, (A; w)) = ḃ α,max{/,}, (A; w) (Ḃ α, (A; w)) = Ḃ α,max{/,}, (A; w) Remark 2.3 (i) We oint out that the duality results obtained in Theorem 2.2 when w and A 2I n n generalize the classical results on Besov saces in [7,32]. (ii) Theorem 2.2 when A 2I n n and w A max{,} (R n ) (the class of Muckenhout s weights) coincides with the scalar versions of [28, Theorem A(3)] ( [, )) and [9, Theorem 5.9] ( (0, )). We finish this section by giving a coule of euivalent descritions of anisotroic weighted Besov-tye saces and Triebel Lizorkin-tye saces for certain arameters. Definition 2.6 Let w A (A), α R, (0, ), τ 0 / + / and τ max{/, }. (i) The sace F α,τ 0, (A; w) is defined to be the set of all f S (Rn ) such that f F α,τ 0, (A;w) su P Q < [w(p)] τ 0 P Q α ϕ jq f (x) χ Q(x) Q [w(q)] dx with the usual modification made when =. (ii) The sace B α,τ, (A; w) is defined to the set of all f S (Rn ) such that f B α,τ, (A;w) < j Z R n scale (Q)= j Q α ϕ j f (x) Q χ Q (x) [w(q)] τ dx with the usual modification made when =. Comaring with the definitions of Ḟ α,τ 0, (A; w) and Ḃ α,τ, (A; w), we find that in the definitions of F α,τ 0, (A; w) and B α,τ, (A; w), the integrals are not weighted. However, the two coules of saces are euivalent as follows. Corollary 2.2 Let w A (A), α R, (0, ), τ 0 / + / and τ max{/, }. Then Ḟ α,τ 0, (A; w) = F α,τ 0, (A; w) and Ḃ α,τ, (A; w) = B α,τ, (A; w) with euivalent norms.

12 224 B. Li et al. By adating the roof of Lemma 3. below, we show that Lemma 3. also holds with Ḟ α,τ 0, (A; w) and Ḃ α,τ, (A; w) relaced, resectively, by F α,τ 0, (A; w) and B α,τ, (A; w) albeit with the same seuence saces f α,τ 0, (A; w)and ḃ α,τ, (A; w). Once this is shown, Corollary 2.2 follows immediately. We omit the details. 3 Proofs of Theorems 2. and 2.2 Let us begin with recalling some notation. For all functions ϕ on R n, x R n, j Z, k Z n and Q A j ([0, ) n +k), let ϕ Q (x) Q 2 ϕ j (x x Q ), where means the Lebesgue measure on R n. Let ϕ S (R n ). For all f S (Rn ), recall that the ϕ-transform S ϕ is defined by S ϕ ( f ) {(S ϕ ( f )) Q } { f,ϕ Q }, and the inverse ϕ-transform T ϕ is defined by T ϕ (t) t Qϕ Q initially for finitely suorted seuences t {t Q } C; see [3]. In what follows, for simlicity, we use the symbol Ȧ α, (A; w) to denote either the sace Ḃ α, (A; w) or the sace Ḟα, (A; w), and use the symbol ȧα, (A; w) to denote the corresonding seuence saces. Likewise we introduce the symbols Ȧ α,τ,(a; w) and ȧ α,τ,(a; w). The ϕ-transform characterizations for weighted anisotroic Besov and Triebel Lizorkin saces in Definitions 2.4 and 2.5 are resented as follows. Lemma 3. Let w A (A), ϕ, ψ S (R n ), α R,, (0, ) and τ 0 = / + /. Then, the following hold. (i) The ϕ-transform S ϕ is bounded, resectively, from the saces Ȧ α, (A; w), Ḟ α,τ 0, (A; w), Ḟ α,0, (A; w) and Ḃ α,max{/,}, (A; w) to the corresonding discrete saces with the same arameters. (ii) The inverse ϕ-transform T ψ is bounded, resectively, from the saces ȧ α, (A; w), f α,τ 0, (A; w), f α,0, (A; w), and ḃ α,max{/,}, (A; w) to the corresonding continuous saces with the same arameters. (iii) Assume that ϕ and ψ additionally satisfy j Z ϕ((a ) j ξ) ψ((a ) j ξ) = for all ξ R n \{0}, where A denotes the transose of A. Then, T ψ S ϕ is the identity on Ȧ α, (A; w), Ḟα,τ 0, (A; w), Ḟ α,0, (A; w) and Ḃ α,max{/,} (A; w). We first oint out that Lemma 3. may be true for Ȧ α,τ,(a; w) and their corresonding saces of seuences with full indices. However, to limit the length of this aer, we only indicate how to show Lemma 3. in these secial indices described therein, which is enough for alications of this aer. The results in Lemma 3. associated with Besov saces Ḃ α, (A; w) and Triebel Lizorkin saces Ḟ, α (A; w) were, resectively, obtained in [2,3]. The results in Lemma 3. associated with the saces Ȧ α,τ,(a; w) can be obtained by a modification of the roofs for [3, Theorem 3.2] with (0, ) and [2, Theorem.]; see also [26, Lemma 3.9]. We give some details only on the sace Ḟ α,τ 0, (A; w) and its seuence sace f α,τ 0, (A; w).,

13 Duality of weighted anisotroic Besov and Triebel Lizorkin saces 225 Using Lemma 3., by reeating the roofs of [5, Corollary 3.7] and [2, Corollary 3.7], we have the following conclusion. We omit the details. Corollary 3. Let α, w,, and τ 0 be as in Lemma 3.. Then the saces Ȧ α, (A; w), Ḟα,τ 0, (A; w), Ḟ α,0, (A; w) and Ḃ α,max{/,}, (A; w) are indeendent of the choices of ϕ. For any w A (A) with w inf{l [, ) : w A l (A)}, λ, r (0, ) satisfying some additional conditions, the seuence s {s Q } f α,τ 0, (A; w) and its majorant seuence sr,λ {(s r,λ ) Q} defined by (sr,λ ) Q P Q scale (P)= scale (Q) s P r [ + Q ρ(x Q x P )] λr by following the roofs of [3, Lemma 3.0] and [26, Lemma 3.9], we see that the key of the roof of Lemma 3. in this case is to show s f α,τ 0, (A;w) s r,λ f α,τ 0, (A;w) ; once this is done, the other details are similar to those of the roof of [3, Theorem 3.2]. Now let us show this conclusion. Lemma 3.2 Let w A (A), α R,, (0, ), τ 0 / + /, r [, ] and λ (/ + w max{/, / }, ). Then there exists a ositive constant C such that for all seuences s {s Q } f α,τ 0, (A; w) and their majorant seuences sr,λ {(s,λ ) Q}, s f α,τ 0, (A;w) s r,λ f α,τ 0, (A;w) C s f α,τ 0, (A;w). Proof The first ineuality is obvious, and we only need to rove the second ineuality. For all α R,w A (A),, (0, ) and τ 0 / + /, we have s f α,τ 0, (A;w) su P Q [w(p)] ( )+ ( ) Q α+/2 s Q [w(q)] ( ) /r,. (3.) By similarity, we only give the roof for the case that (, ). For any P Q, by [3, Lemma 2.9], there exists a ositive integer c 0 such that Q B ρ (x P, b c 0+ scale (P) ) and B ρ (c P, b c 0+ scale (P) ) P, (3.2) where c P is the center of P. Then for any fixed P Q, let B P B ρ (x P, 3H 2 b c 0+ scale (P) ),

14 226 B. Li et al. where H is as in Definition 2.. Let U BP { P Q : scale (P ) = scale (P), P B P = } and Ũ BP P U BP P. Thus, by the fact that {P Q : scale (P ) = scale (P)} is a artition of R n, we have that R Q scale (R)= scale (Q) = P U BP R T (P ) scale (R)= scale (Q) + P Ũ BP = scale (P )= scale (P) R T (P ),R Ũ BP = scale (R)= scale (Q) which, together with the well-known ineuality that for all γ (0, ) and {a j } j C,, γ j a j j a j γ (3.3) and Q = R when scale (Q) = scale (R), further imlies that and [w(p)] ( )+ [w(p)] ( )+ + [w(p)] ( )+ I + J. Ste. Prove I s ( Q α+ 2 (s r,λ ) Q ) [w(q)] P U BP f α,τ 0, (A;w) R T (P ) scale (R)= scale (Q) P Ũ BP = scale (P )= scale (P). For any R Q, let R ( 2 α) s R [w(q)] [ + Q ρ(x Q x R )] λ R T (P ),R Ũ BP = scale (R)= scale (Q) M R,0 {Q T (P) : scale (Q) = scale (R), Q ρ(x Q x R )<b}, M R,l {Q T (P) : scale (Q) = scale (R), b l Q ρ(x Q x R )<b l+ } for all l N. Then we have I = [w(p)] ( )+ P U BP R T (P ) l Z + R ( 2 α) s R [w(q)] [ + Q. ρ(x Q M Q x R )] λ R,l

15 Duality of weighted anisotroic Besov and Triebel Lizorkin saces 227 Since λ> / + w ( / ), we choose ( w, ) sufficiently close to w such that λ> / + ( / ). For any Q M R,l and P U BP, by [3, Lemma 2.9] and [7, Proosition 2.6(i)], we have w(r) b l w(q) and w(p) w(p ). Moreover, by an elementary lattice counting lemma (see [6, Lemma 2.8]), M R,l b l and U BP. From the above estimates, λ> / + ( / ) and (3.), it follows that I P U BP [w(p )] ( )+ s f α,τ, 0, (A;w) R T (P ) ( R 2 α s R ) [w(r)] l Z + b l[ ( )+ λ ] which is the desired ineuality. Ste 2. Prove J s f α,τ. For any fixed P Q, Q T (P), 0, (A;w) P Q with P Ũ BP = and scale (P ) = scale (P), and any R T (P ) with R Ũ BP = and scale (R) = scale (Q), by (3.2) and B P B ρ (x P, 3H 2 b c 0+ scale (P) ) U BP, we obtain ρ(x P x P ) H 2 [ρ(x P x Q ) + ρ(x Q x R ) + ρ(x R x P )] which, together with H 2 [2b c 0+ scale (P) + ρ(x Q x R )], ρ(x R x Q ) ρ(x R x P ) ρ(x Q x P ) 3Hb c 0+ scale (P) b c 0+ scale (P) H 2Hb c 0+ scale (P), imlies that ρ(x P x P ) 2H 2 ρ(x Q x R ). (3.4) Moreover, by P Ũ BP = and scale (P ) = scale (P), we have {P Q : P Ũ BP =, scale (P ) = scale (P)} {P Q : ρ(x P x P ) 3H 2 b c 0+ scale (P), scale (P ) = scale (P)} = j Z + {P Q : 3H 2 b c0+ scale (P)+ j ρ(x P x P )<3H 2 b c0+ scale (P)+ j+, scale (P ) = scale (P)} j Z + V P, j. (3.5) Since λ> / + w max{/, / }, we choose ( w, ) sufficiently close to w such that λ> / + max{/, / }. Notice that for any j Z +

16 228 B. Li et al. and P V P, j, by P B ρ (x P, 4H 3 b c 0+ scale (P)+ j+ ), w(b ρ (x P, b c 0+ scale (P) )) w(b ρ (c P, b c 0+ scale (P) )), B ρ (c P, b c 0+ scale (P) ) P and [7, Proosition 2.6(i)], we have w(p ) w(b ρ (x P, 4H 3 b c 0+ scale (P)+ j+ )) b j w(b ρ (x P, b c 0+ scale (P) )) b j w(b ρ (c P, b c 0+ scale (P) )) b j w(p). (3.6) Symmetrically, we also have w(p) b j w(p ). (3.7) Moreover, for any j Z +, k Z +, Q T (P) with scale (P) = scale (Q)+k, P V P, j, R T (P ) with R U BP = and scale (R) = scale (Q), by (3.4), we obtain b j+k+ scale (R) ρ(x P x P ) ρ(x Q x R ). (3.8) Furthermore, for any j Z +, k Z +, P V P, j, Q T (P), R T (P ) with R Ũ BP = and scale (R) = scale (Q) = scale (P) k, by R B ρ (x P, 4H 3 b c 0+ scale (P)+ j+ ) B ρ (c Q, 5H 4 b c 0+ scale (P)+ j+ ), B ρ (c Q, b c 0+ scale (Q) ) Q and [7, Proosition 2.6(i)], we have w(r) w(b ρ (c Q, 5H 4 b c 0+ scale (P)+ j+ )) b ( j+k) w(b ρ (c Q, b c 0+ scale (Q) )) b ( j+k) w(q). (3.9) Thus, for any (0, ), (, ) and P Q, using (3.5) through (3.9), {Q T (P) : scale (Q) + k = scale (P)} b k and V P, j b j (see [6, Lemma 2.8]), (3.2) and λ> / + max{/, / }, we obtain J [w(p)] ( )+ R T (P ),R U BP = scale (R)= scale (Q) k Z + scale (Q)+k= scale (P) j Z + P V P, j R ( 2 α) s R [w(q)] [ + Q ρ(x Q x R )] λ

17 Duality of weighted anisotroic Besov and Triebel Lizorkin saces 229 k Z + b k s s j Z + P V P, j R T (P ) j [ b ( )+] [w(p )] ( )+ R ( 2 α) s R [w(r)] b ( j+k)( ) b λ ( j+k) f α,τ 0 b k[ ( )+ λ ] b j (+ λ ), (A;w) k Z + j Z + f α,τ, 0, (A;w) which is also the desired ineuality. Combining the estimates of I and J, by the arbitrariness of P Q, we have sr,λ f α,τ 0, (A;w) s f α,τ 0, (A;w), which comletes the roof of Lemma 3.2. Now we are ready to rove Theorem 2.. Proof of Theorem 2. Let τ 0 = / + / and w A (A). We rove Theorem 2. in three stes. Ste. Proof of ( f, α (A; w)) f α,0, (A; w) with (, ) (, ) (0, ). We first rove f α,0, (A; w) ( f, α (A; w)). For any t {t Q } f α,0, (A; w), define a linear functional l t on f, α (A; w) by l t(s) s Q t Q for all s f, α (A; w). By alying Hölder s ineuality twice when (, ), or by the imbedding f, α (A; w) (A; w) when (0, ], we have l t (s) R n f α, Q α s Q χ Q (x) Q α Q t Q w(q) χ Q(x)w(x) dx s f, α (A; w) t f α,0, (A;w), which yields l t ( f, α (A; t w)) f α,0, and hence f α,0, (A;w), (A; w) ( f, α (A; w)). Let us rove the converse by referring some ideas from [7,. 78]. Since seuences with finite suort are dense in f, α (A; w), each bounded linear functional l ( f, α (A; w)) must be of the form l(s) s Q t Q for some t {t Q } C. It suffices to show that t f α,0, (A;w) l ( f, α (A; w)).

18 230 B. Li et al. For all, (0, ], let Lw(l ) be the sace of all seuences { f j } j Z of functions on R n such that / { f j } j Z Lw (l ) f j <. j Z Lw (R n ) By [4, Proosition 4.3], we know that (Lw(l )) = Lw (l ) for all (, ) and (0, ). Notice that the ma In : f, α (A; w) L w(l ) defined by setting, for all s f, α (A; w), In(s) { f j} j, where f j scale (Q)= j Q α s Q χ Q for all j Z, is a linear isometry onto a subsace of Lw(l ). When (, ) and [, ), by the Hahn Banach theorem, there exists an l (Lw(l )) with l ( f, α (A; = l w)) ( f, α (A; such that l In = l. w)) In other words, there exists g {g j } j Z Lw (l ) with g L w (l ) l ( f, α (A; such that for all s f α w)), (A; w), s Q t Q = f j (x)ḡ j (x)w(x) dx. R n j Z By taking s Q = 0 for all but one dilated cube, we obtain t Q = Q α /2 g j (x)w(x) dx (3.0) Q for all cubes Q with scale (Q) = j. For any f L loc (Rn ; w), which denotes the sace of all locally integrable functions on the measure w(x) dx, define the weighted anisotroic Hardy Littlewood maximal function of f by M w ( f )(x) su x w(q) Q f (y) w(y) dy. Then by [3, Lemma 2.9 and Theorem 2.8] and the fact that w(x) dx is a ρ A -doubling measure (see Sect. 4 below), we have the vector-valued maximal ineuality that for all (, ), (, ] and functions { f i } i Lw (l ), ( i M w f i ) / L w (R n ) which together with (3.0) yields that ( i f i ) / L w (R n ) t f α,0, (A;w) {M w(g j )} j Z g L w (l ) Lw l (l ) ( f, α (A; w)),, and hence ( f, α (A; w)) (, ) and [, ). f α,0, (A; w). This finishes the roof of Ste when

19 Duality of weighted anisotroic Besov and Triebel Lizorkin saces 23 To comlete the roof of Ste, it suffices to rove ( f, α (A; w)) f α,0, (A; w) when (, ) and (0, ). We need to use Verbitsky s method from [35,36] (see also [4]). In fact, since l ( f, α (A; w)) is of the form l(s) s Q t Q for some t {t Q } C, we know that there exists a ositive constant C such that for all s (A; w), f α, l(s) = s Q t Q C s f, α (A; w). Define Q {Q Q : t Q = 0}, u Q s Q t Q for all Q Q and c Q Q α /2 t Q for all Q Q. Then the above ineuality can be rewritten as / {u Q } l C u Q (c Q ) χ Q. Lw (R n ) Then alying [4, Theorem 4.4(ii)] with 0 < < r = < <, we obtain that t f = α,0, (A;w) R n = R n [ su,x Q Q α /2 Q t Q w(q) ] w(x) dx su [c Q w(q)] w(x) dx <,,x Q which imlies that ( f, α (A; w)) f α,0, (A; w), and hence comletes the roof of Ste. Ste 2. Proof of ( f, α (A; w)) = f α,τ 0, (A; w) for (, ) (0, ] (, ). For any t f α,τ 0(A; w), observe that t f α,τ 0,, (A;w) su P Q [w(p)] τ 0 su P Q [w(p)] τ 0 P ( Q α Q t Q w(q) χ Q(x) ( Q α 2 Q tq w(q) ) ) w(x) dx w(q), (3.) where T (P) is the tent of P defined in (2.).

20 232 B. Li et al. For any s {s Q } f, α (A; w), (, ), α R and k Z, let k x Rn : [ Q α s Q χ Q (x)] / > 2 k, k {x R n : M w (χ k )(x) >/2} and R k {Q Q : w(q k ) > w(q)/2, w(q k+ ) w(q)/2}. Then, we see that (i) for all k Z, k+ k and k k ; (ii) for any k, j Z with k = j, R k R j = ; (iii) R n = k Z Q Rk Q; (iv) Q Rk Q k ; (v) w( k ) w( k ). We oint out that (v) holds by the L 2 w (Rn )-boundedness of M w. Moreover, we say that Q Q is seudo-maximal in R k if there is no other P R k such that scale ( Q) < scale (P) and Q P =. Notice that the seudo-maximal cubes in R k are disjoint with each other. Then, we obtain a classification for R k associated with seudomaximal cubes in R k such that any Q R k belongs to one and only one tent of seudo-maximal cubes. Precisely, ick any seudo-maximal cube Q in R k, denoted by Q (), and set T k ( Q () ) {Q R k : Q Q () =, scale (Q) scale ( Q () )}. Then, we ick another seudo-maximal cube P in R k, denoted by Q (2), and set T k ( Q (2) ) {Q R k \ T k ( Q () ) : Q Q (2) =, scale (Q) scale ( Q (2) )}. Inductively, for any j N, ick any seudo-maximal cube R in R k, denoted by Q ( j+), and set T k ( Q ( j+) ) {Q R k \ j l= T k ( Q (l) ) : Q Q ( j+) =, scale (Q) scale ( Q ( j+) )}. Thus, R k = j N T k ( Q ( j) ). For simlicity, let R k be the set of all seudo-maximal cubes in R chosen as above. Then R k = Q R k T k ( Q). (3.2) Furthermore, by the fact that the seudo-maximal cubes in R k are disjoint with each other, (iv) and (v), we have

21 Duality of weighted anisotroic Besov and Triebel Lizorkin saces 233 Q R k w( Q) w( k ) w( k ). (3.3) Now let us first rove that f α,τ 0, (A; w) ( f, α (A; w)). For any t f α,τ 0, (A; w), define a linear functional l t on f, α (A; w) by l t(s) s Q t Q for any s f, α (A; w). Then for all α R, (, ), (0, ], by (iii), (3.2), Hölder s ineuality with, (3.), (3.3), Hölder s ineuality for / and (3.3), we have l t (s) k Z k Z t Q R k Q T k ( Q) Q R k Q T k ( Q) f α,τ 0, (A;w) t Q T k ( Q) f α,τ 0, (A;w) Q T k ( Q) Q α 2 sq [w(q)] Q α t Q Q 2 ( Q α 2 sq ) w(q) ( ) Q α t Q Q 2 Q w(q) w(q) [w( Q)] Q R k k Z ( Q α 2 sq ) w(q) w( Q) Q R k k Z Q R k Q T k ( Q) ( Q α 2 sq ) w(q) Q [w(q)] t f α,τ 0, (A;w) [w( k )] k Z ( Q α 2 sq ) w(q) Q R k. (3.4) Moreover, notice that for any k Z and Q R k, we have w(q k+ ) w(q)/2, which imlies that w(q ( k+ ) ) > w(q)/2. By this and Q k, we obtain that

22 234 B. Li et al. w(q ( k \ k+ )) w(q)/2, which, together with (iii) and the definition of k+, yields that ( Q α 2 sq ) w(q) Q R k k \ k+ ( Q α s Q χ Q (x)) w(x) dx Q R k 2 k [w( k )]. Combining this and (3.4) yields that l t (s) t further imlies that l t ( f, α (A; w)) ( f, α (A; w)). t f α,τ 0 f α,τ 0,, (A;w), and hence (A;w) s f, α (A; w), which f α,τ 0, (A; w) Conversely, since seuences with finite suort are dense in f, α (A; w), each l ( f, α (A; w)) must be of the form l(s) s Q t Q for some t {t Q } C. It suffices to show that t f α,τ 0, (A;w) l ( f, α (A; w)). For any P Q, define a measure ν by ν(q) w(q)/w(p) if Q P = and scale (Q) scale (P) or else ν(q) 0. Then, for any (, ) (0, ] (, ), by (3.), we have t f α,τ 0, (A;w) su P Q su P Q su P Q su P Q [w(p)] τ 0 ( Q α 2 Q tq w(q) ) w(q) ( ) Q α Q w(q) [w(p)] 2 tq w(q) w(p) { } Q α Q [w(p)] 2 tq w(q) l (ν) su [w(p)] Q α Q 2 tq s l (ν) w(q) s w(q) Q w(p) { } su Q α+ 2 P Q [w(p)] s l (ν) w(p) s Q l ( f, α (A; su w)) where and in what follows, for s {s Q } and (, ), s l (ν) s Q ν(q) /., f, α (A; w)

23 Duality of weighted anisotroic Besov and Triebel Lizorkin saces 235 Notice that by [3, Lemma 2.9], there exist ositive constants c 0 and c such that Q B ρ (c P, b c 0 P ) and B ρ (c P, b c P ) P, where c P is the center of P. Thus, for any fixed > w, by [8, Proosition 2.5] with w A (A), we have w Q w(b ρ (c P, b c 0 P )) which, together with Hölder s ineuality, yields that { } Q α+ 2 w(p) s Q w(p) w(p) b (c 0+c ) w(b ρ (c P, b c P )) w(p), (3.5) f, α (A; w) Q [w(p)] s l (ν). Combining these estimates yields that t ( s Q χ Q (x)) s Q w(q) w f α,τ 0, w(x) dx Q (A;w) l ( f, α (A; w)), which further imlies that f α,τ 0, (A; w) ( f, α (A; w)). This finishes the roof of Ste 2. Ste 3. Proof of ( f, α (A; w)) = f α,τ 0, (A; w) for (, ) (0, ] (0, ]. For any (, ) (0, ] (0, ] and α R, by (3.3), we obtain that f, α (A; w) f, α (A; w), and hence ( f, α (A; w)) ( f, α (A; w)). Thus, to rove that ( f, α (A; w)) f α,τ 0, (A; w), we only need to show ( f, α (A; w)) f α,τ 0, (A; w). For any k Z and α R, set k x Rn : Q α s Q χ Q (x) >2 k, and k, R k and Q as in Ste 2. Then, by an argument similar to that of Ste 2, we obtain ( f, α (A; w)) f α,τ 0, (A; w).

24 236 B. Li et al. Conversely, notice that for any l ( f α, (A; w)),l must be of the form l(s) s Q t Q for some t {t Q } C. Then, it suffices to rove that t f, α,/ (A; w) l ( f, α (A; w)). For any fixed Q Q, define a seuence s Q {(s Q ) R } R Q by (s Q ) R if R = Q or else (s Q ) R 0. Then, we have t f α,, = su P Q = su P Q (A;w) [w(p)] [w(p)] l ( f, α (A; su w)) su su P Q Q α 2 Q tq w(q) R α 2 tr (s Q R ) R w(r) R Q [w(p)] su { R α+ 2 (s Q ) R [w(r)] } R Q f, α (A; w). By (3.5), we know that for any (0, ] and fixed P Q, su { R α+ 2 (s Q ) R [w(r)] } R Q f, α (A; w) = su [w(q)] w Q [w(p)]. By this, we finally obtain t f, α,/ (A; w) l ( f, α (A; w)). Combining Ste through Ste 3, we obtain that the desired duality results for the discrete Triebel Lizorkin saces f, α (A; w). Alying Lemma 3. and similarly to the roof of [7, Theorem 5.3], we also obtain the corresonding duality results for Triebel Lizorkin saces, which comletes the roof of Theorem 2.. Next we give the roof of Corollary 2.. Proof of Corollary 2. Let α R, (, ], τ > / and w A (A). Define (0, ) such that τ = /. By [4, Theorem 4.2] we have that the dual of the sace f α, (A; w) can be identified with f, α (A; w), albeit with a different airing than the standard scalar roduct

25 Duality of weighted anisotroic Besov and Triebel Lizorkin saces 237 airing (2.2). That is, λ {λ Q } λ, t w, [w(q)] max(,/) λ Q t Q. Q, On the other hand, Theorem 2.(i) states that the dual of f α, (A; w) is f α,τ 0, (A; w), where τ 0 = / + / = τ + /. Therefore, the saces f, α (A; w) and f α,τ+/ ( A; w) are isomorhic. The isomorhism ma is given by the multilier oerator {s Q } { [w(q)]/ Q s Q }. Conseuently, in the unweighted case w we have the identification f, α+τ (A; w) = f, α,τ+/ (A; w). The same conclusions for the continuous saces Ḟ, α,τ (A; w) follow as a conseuence of Theorem 2.. This finishes the roof of Corollary 2.. We finally give the roof of Theorem 2.2. Proof of Theorem 2.2 Let α R,τ 0 = / + / and w A (A). Since the roof is similar to that of Theorem 2., we only rove Theorem 2.2 under the cases of (, ) (0, ] (, ) and (, ) (, ) (0, ). Ste. Proof of (ḃ α, (A; w)) = ḃ α,/, (A; w) for (, ) (0, ] (, ). We first rove (ḃ α, (A; w)) ḃ α,/, (A; w). For any t ḃ α,τ 0, (A; w), define a linear functional l t on ḃ α, (A; w) by l t(s) s Q t Q for all s ḃ α, (A; w). We only need to show that l t (ḃ α, (A; t w)) ḃ α,/, (A;w). For any (, ) (0, ] (, ), by Hölder s ineuality and (3.3), we have l t (s) j Z j Z [ scale (Q)= j scale (Q)= j Q α 2 sq [w(q)] Q [w(q)] Q α 2 tq w(q) Q α 2 sq [w(q)] su Q α 2 sq Q [w(q)] scale (Q)= j [ s ḃα, (A; w) j Z = s ḃα, (A; w) t ḃ α,/, (A;w), su Q α 2 sq Q [w(q)] scale (Q)= j ] ] which imlies that l t (ḃ α, (A; t w)) ḃ α,/, (A;w). Conversely, for any l (ḃ α, (A; w)),lmust be of the form l(s) s Q t Q for some t {t Q } C. Then, it is left to show t ḃ α,/, (A;w) l (ḃ α, (A; w)).

26 238 B. Li et al. Notice that t ḃ α,/, (A;w) = su P Q j= scale (P) [ su t Q Q α+ 2 scale (Q)= j, [w(q)] ]. Since for any P Q, there are finitely many cubes Q in T (P) with scale (Q) = j, then for each j scale (P), there exists a cube Q j satisfying that scale (Q j ) = j and Q j T (P) such that su t Q Q α+/2 scale (Q)= j, [w(q)] / = t Q j Q j α+/2 [w(q j )] /, and hence t ḃ α,/, (A;w) = su P Q = su j= scale (P) su P Q {s Q j } j scale (P) l l (ḃ α, (A; w)) su su P Q {s Q j } j scale (P) l t Q j Q j α+ 2 [w(q j )] t Q j α+ 2 Q j [w(q j )] { s Q j s Q Q α+ 2 [w(q)] } ḃα Q {Q j : j scale (P)}, (A; w). However, { s Q Q α+/2 [w(q)] /} {s ḃα Q j } j scale (P) l. Q {Q j : j scale (P)}, (A; w) Thus, we obtain t ḃ α,/, (A;w) l (ḃ α, (A; w)). This finishes the roof of Ste. Ste 2. Proof of (ḃ α, (A; w)) = ḃ α,, (A; w) for (, ) (, ) (0, ]. To show ḃ α,, (A; w) (ḃα, (A; w)), for any t ḃ α,, (A; w), we define a linear functional l t on ḃ α, (A; w) by l t(s) s Q t Q for any s ḃ α, (A; w). We only need to rove that l t (ḃ α, (A; t w)) ḃ α, (A;w). Indeed, for any (, ),

27 Duality of weighted anisotroic Besov and Triebel Lizorkin saces 239 and (0, ], by Hölder s ineuality and (3.3), we have l t (s) j Z j Z scale (Q)= j su j Z scale (Q)= j scale (Q)= j s ḃα, (A; w) t ḃ α,, (A;w), Q α 2 sq [w(q)] Q α 2 Q tq w(q) [w(q)] ( Q α 2 sq ) w(q) ( Q α 2 Q tq w(q) ) w(q) which imlies that l t (ḃ α, (A; w)) t ḃ α,, (A;w). Conversely, for any l (ḃ α, (A; w)),l must be of the form l(s) s Q t Q for some t {t Q } C. Then, to comlete the roof of Theorem 2.2, it suffices to show that Indeed, we have t ḃ α,, (A;w) l (ḃ α, (A; w)). t ḃ α,, (A;w) = su j Z = su scale (Q)= j ( ) Q α 2 Q tq [w(q)] su Q α 2 Q t Q (s j ) Q [w(q)] j Z {(s j ) Q } Q {: scale (Q)= j} l scale (Q)= j (A; w)) su su j Z {(s j ) Q } Q {: scale (Q)= j} l l (ḃ α, { } Q α 2 Q (s j ) Q [w(q)] l (ḃ α, (A; w)), ḃα Q {: scale (Q)= j}, (A; w) where {(s j ) Q } l Q {: scale (Q)= j} scale (Q)= j (s j ) Q /.

28 240 B. Li et al. From this, we deduce that t ḃ α,, (A;w) l (ḃ α, (A; w)) and then comlete the roof for the case that (, ) (, ] (0, ]. Now combining the existing roved results for discrete Besov saces and Lemma 3., by a similar roof to that of [7, Theorem 5.3], we obtain the desired result for Besov saces, which comletes the roof of Theorem Duality of Besov and Triebel Lizorkin saces associated with doubling measures This section focuses on a more general setting involving anisotroic Besov and Triebel Lizorkin saces associated with ρ A -doubling measures. We show that Theorems 2. and 2.2 are still true with A (A) weights relaced by ρ A -doubling measures. Recall that ρ A -doubling measures are first introduced in [2]. Definition 4. A non-negative Borel measure µ on R n is called a ρ A -doubling measure if there exists a nonnegative constant β β(µ) such that for all x R n and r > 0, µ(b ρa (x, br)) b β µ(b ρa (x, r)). We oint out that for any w A (A), dµ(x) w(x) dx (with resect to a uasidistance ρ A ) also defines a ρ A -doubling measure albeit with a ositive constant C (see [7, Proosition 2.6(i)]). Definition 4.2 Let α R,, (0, ) and µ be a ρ A -doubling measure. The saces Ȧ α, (A; µ) and Ȧα,τ,( A; µ), and their corresonding seuence saces are defined as in Definitions 2.4 and 2.5 with w(x) dx, w(p) and w(q) relaced, resectively, by dµ(x), µ(p) and µ(q). The saces Ḟ, α (A; µ), Ḃα, (A; µ) and their corresonding seuences saces mentioned above were introduced, resectively, in [3,2]. Similarly to the roof of Theorems 2. and 2.2, we have the following conclusion. Theorem 4. Let µ be a ρ A -doubling measure,,,τ 0 and α the same as in Theorems 2. and 2.2. Then Theorems 2. and 2.2 still hold with those mentioned saces associated A weight relaced by the corresonding saces associated with ρ A -doubling measures as in Definition 4.2. To rove Theorem 4., we first oint out that, with a similar roof, Lemma 3. is also true for the saces associated with ρ A -doubling measures. With this, the roof of Theorem 4. is nearly the same as those of Theorems 2. and 2.2. We give some details for the secial cases for the reader s convenience to show their differences. Proof of Theorem 4. Since the roof is nearly verbatim reetition of the roofs of Theorems 2. and 2.2, we only give details for ( f, α (A; µ)) = f α,0, (A; µ) in the case (, ) (, ) (0, ). We first rove that f α,0, (A; µ) ( f, α (A; µ)). For any t {t Q } f α,0, (A; µ), define a linear functional l t on f, α (A; µ) by l t(s) s Q t Q

29 Duality of weighted anisotroic Besov and Triebel Lizorkin saces 24 for all s f, α (A; µ). By alying Hölder s ineuality twice for, while for (0, ), using the imbedding f, α (A; µ) f, α (A; µ), we have l t (s) R n s Q α s Q χ Q (x) Q α t Q Q µ(q) χ Q(x) dµ(x) f, α (A;µ) t f α,0, (A;µ), which yields that l t ( f, α t (A;µ)) f α,0, and hence f α,0, (A;µ), (A; µ) ( f, α (A; µ)). Conversely, since seuences with finite suort are dense in f, α (A; µ), each bounded linear functional l ( f, α (A; µ)) must be of the form l(s) s Q t Q for some t {t Q } C. It suffices to show that t f α,0, (A;µ) l ( f, α (A;µ)). Let Lµ(l ) be the sace of all seuences { f j } j Z of functions on R n such that / { f j } L j Z µ (l ) f j <. j Z Lµ (R n ) We know that (Lµ(l )) = Lµ (l ) for (, ) and (0, ). Notice that the ma In : f, α (A; µ) L µ(l ) defined by In(s) { f j } j, where f j scale (Q)= j Q α s Q χ Q for all j Z, is a linear isometry onto a subsace of Lµ(l ). When (, ) and [, ), by the Hahn Banach theorem, there exists an l (Lµ(l )) with l ( f, α = l (A;µ)) ( f, α such that l In = l. In other (A;µ)) words, there exists g {g j } j Z Lµ (l ) with g l L µ (l ) ( f, α (A;µ)) such that for all s f, α (A; µ), s Q t Q = f j (s)g j (x) dµ(x). (4.) R n j Z By taking s Q =0 for all but one dilated cube, we obtain t Q = Q Q α /2 g j (x) dµ(x) for all cubes Q with scale (Q) = j. For any f L loc (Rn ; µ), define the anisotroic Hardy Littlewood maximal function of f by M µ ( f )(x) su x µ(q) Q f (y) dµ(y). Then by [4, Proosition 4.3], we know that the vector-valued maximal ineuality holds for M µ. Then, by (4.) and the vector-valued maximal ineuality for M µ with (, ) and (, ], we obtain that t f α,0, (A;µ) {M µ(g j )} j Z g L µ (l ) Lµ l (l ) ( f, α (A;µ)), which imlies that ( f, α (A; µ)) f α,0, (A; µ).

30 242 B. Li et al. When (, ) and (0, ), similarly to Ste of the roof for Theorem 2., we also obtain ( f, α (A; µ)) f α,0, (A; µ), and hence comlete the roof of Theorem 4.. We oint out that when dilation A admits a Meyer-tye wavelet, Bownik [4, Theorem 4.0] determined the dual saces of Triebel Lizorkin saces with ρ A doubling measures under the airing f, g f,ψ Q ψ Q, g [µ(q)]max{, }. (4.2) Q ψ Theorem 4. identifies the dual saces of Ḟ, α (A; µ) for arbitrary dilation A under the airing f, g R n f (x)g(x) dx. Since the airings used in [4] and here are different, the dual saces also aear differently. A uestion osed in [4,. 55] asks whether the duality (4.2) holds without the assumtion on the existence of Meyer-tye wavelets. While our aer does not answer this uestion it rovides the duality result without this extra assumtion. The duality in Theorem 4. for anisotroic Besov saces with doubling measures is new. As an alication of Theorem 4., let us discuss a articular class of Triebel Lizorkin saces associated with Hardy saces. There are several euivalent definitions of Hardy saces. The weighted Hardy saces associated with A (A) defined via maximal functions or the atomic decomosition were studied in [7]. We define two kinds of Hardy saces with ρ A -doubling measure µ via the Littlewood Paley g-function sace and via the suare function. Definition 4.3 Let (0, ), ϕ S (R n ) and µ be a ρ A -doubling measure. (i) Define the anisotroic Hardy sace H (A; µ) with a ρ A -doubling measure µ via the Littlewood Paley g-function by H (A; µ) { f S (R n ) : f H (A;µ) g ϕ( f ) L µ (R n ) < }, where the anisotroic Littlewood Paley g-function g ϕ ( f ) of f is defined by /2 g ϕ ( f ) ϕ j f 2 ; j Z (ii) Define the anisotroic Hardy sace H (A; µ) with a ρ A -doubling measure µ via the suare function by H (A; µ) { f S (R n ) : f H (A;µ) S ϕ ( f ) L µ (R n ) < },

31 Duality of weighted anisotroic Besov and Triebel Lizorkin saces 243 where for any x R n, the anisotroic suare function S ϕ ( f ) of f is defined by S ϕ ( f )(x) k Z b k B ρ (x,b k ) f ϕ k (y) 2 dy Corollary 4. Let (0, ) and µ be a ρ A -doubling measure. Then, (i) H (A; µ) = H (A; µ) = Ḟ,2 0 (A; µ) with euivalent norms; (ii) (H (A; µ)) = Ḟ 0,/ /2 2,2 (A; µ) in the sense of (2.3). A weighted anisotroic roduct version of the first euality in Corollary 4.(i) has been obtained in [26, Theorem 2.2]. With an obvious modification on its roof therein, namely, via relacing the weighted roduct measure in the roof of [26, Theorem 2.2] by dµ(x) here and then reeating the roof therein, we have H (A; µ) = H (A; µ) with euivalent norms. The saces H (A; µ) = Ḟ,2 0 (A; µ) with euivalent norms follow directly from their definitions, which comletes the roof of Corollary 4.(i). Corollary 4.(ii) is a simle corollary of Corollary 4.(i) and Theorem 4.. We omit the details. Acknowledgments The authors would like to exress their gratitude to the anonymous referee who has brought their attention to the work by Bui [9]. /2. References. Bownik, M.: Anisotroic Hardy saces and wavelets. Mem. Am. Math. Soc. 64(78), vi+22 (2003) 2. Bownik, M.: Atomic and molecular decomositions of anisotroic Besov saces. Math. Z. 250, (2005) 3. Bownik, M.: Anisotroic Triebel Lizorkin saces with doubling measures. J. Geom. Anal. 7, (2007) 4. Bownik, M.: Duality and interolation of anisotroic Triebel Lizorkin saces. Math. Z. 259, 3 69 (2008) 5. Bownik, M., Ho, K.-P.: Atomic and molecular decomositions of anisotroic Triebel Lizorkin saces. Trans. Am. Math. Soc. 358, (2006) 6. Bownik, M., Lemvig, J.: Affine and uasi-affine frames for rational dilations. Trans. Am. Math. Soc. (20, in ress) 7. Bownik, M., Li, B., Yang, D., Zhou, Y.: Weighted anisotroic Hardy saces and their alications in boundedness of sublinear oerators. Indiana Univ. Math. J. 57, (2008) 8. Bownik, M., Li, B., Yang, D., Zhou, Y.: Weighted anisotroic roduct Hardy saces and boundedness of sublinear oerators. Math. Nachr. 283, (200) 9. Bui, H.Q.: Weighted Besov and Triebel saces: interolation by the real method. Hiroshima Math. J. 2, (982) 0. Calderón, A.-P.: An atomic decomosition of distributions in arabolic H saces. Adv. Math. 25, (977). Calderón, A.-P., Torchinsky, A.: Parabolic maximal functions associated with a distribution. Adv. Math. 6, 64 (975) 2. Calderón, A.-P., Torchinsky, A.: Parabolic maximal functions associated with a distribution. II. Adv. Math. 24, 0 7 (977) 3. Coifman, R.R., Weiss, G.: Extensions of Hardy saces and their use in analysis. Bull. Am. Math. Soc. 83, (977)