MOLECULAR CHARACTERIZATIONS AND DUALITIES OF VARIABLE EXPONENT HARDY SPACES ASSOCIATED WITH OPERATORS

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1 Annales Academiæ Scieniarum Fennicæ Mahemaica Volumen 41, 2016, MOLECULAR CHARACTERIZATIONS AND DUALITIES OF VARIABLE EXPONENT HARDY SPACES ASSOCIATED WITH OPERATORS Dachun Yang and Ciqiang Zhuo Beijing Normal Universiy, School of Mahemaical Sciences Laboraory of Mahemaics and Complex Sysems, Minisry of Educaion Beijing , P. R. China; Beijing Normal Universiy, School of Mahemaical Sciences Laboraory of Mahemaics and Complex Sysems, Minisry of Educaion Beijing , P. R. China; Absrac. Le L be a linear operaor on L 2 (R n ) generaing an analyic semigroup {e L } 0 wih kernels having poinwise upper bounds and : R n (0,1] be a variable exponen funcion saisfying he globally log-hölder coninuous condiion. In his aricle, he auhors inroduce he variable exponen Hardy space associaed wih he operaor L, denoed by H L (Rn ), and he BMO-ype space BMO,L (R n ). By means of en spaces wih variable exponens, he auhors hen esablish he molecular characerizaion of H L (Rn ) and a dualiy heorem beween such a Hardy space and a BMO-ype space. As applicaions, he auhors sudy he boundedness of he fracional inegral on hese Hardy spaces and he coincidence beween H L (Rn ) and he variable exponen Hardy spaces H (R n ). 1. Inroducion In recen years, funcion spaces wih variable exponens arac much aenions (see, for example, [4, 14, 16, 18, 19, 20, 37, 44, 50, 51, 54, 55] and heir references). The variable exponen Lebesgue space L (R n ), wih an exponen funcion : R n (0, ), which consiss of all measurable funcions f such ha R n f(x) p(x) dx <, is a generalizaion of he classical Lebesgue space. The sudy of variable exponen Lebesgue spaces can be raced back o Birnbaum Orlicz [6] and Orlicz [40] (see also Luxemburg [34] and Nakano [38, 39]), bu he modern developmen sared wih he aricles [31] of Kováčik and Rákosník as well as [13] of Cruz-Uribe and [17] of Diening. The variable funcion spaces have been widely used in he sudy of harmonic analysis; see, for example, [14, 18]. Apar from heoreical consideraions, such funcion spaces also have ineresing applicaions in fluid dynamics [1, 42], image processing [9], parial differenial equaions and variaional calculus [2, 26, 43]. Paricularly, Nakai and Sawano [37] inroduced Hardy spaces wih variable exponens, H (R n ), and esablished heir aomic characerizaions which were furher applied o consider dual spaces of such Hardy spaces. Laer, in [44], Sawano exended he aomic characerizaion of he space H (R n ) in [37], which also improves he doi: /aasfm Mahemaics Subjec Classificaion: Primary 42B35; Secondary 42B30, 35K08, 47D03. Key words: Hardy space, BMO space, variable exponen, operaor, hea kernel, molecule. Corresponding auhor. This projec is suppored by he Naional Naural Science Foundaion of China (Gran Nos and ), he Specialized Research Fund for he Docoral Program of Higher Educaion of China (Gran No ) and he Fundamenal Research Funds for Cenral Universiies of China (Gran Nos. 2013YB60 and 2014KJJCA10).

2 358 Dachun Yang and Ciqiang Zhuo corresponding resul in [37], and gave ou more applicaions including he boundedness of he fracional inegral operaor and he commuaors generaed by singular inegral operaors and BMO funcions, and an Olsen s inequaliy. Afer ha, Zhuo e al. [55] esablished heir equivalen characerizaions via inrinsic square funcions, including he inrinsic Lusin area funcion, he inrinsic g-funcion and he inrinsic gλ -funcion. Independenly, Cruz-Uribe and Wang [16] also invesigaed he variable exponen Hardy space wih some slighly weaker condiions han hose used in [37]. Recall ha he heory of classical Hardy spaces H p (R n ) wih p (0,1] and heir duals are well sudied and cerainly play an imporan role in harmonic analysis as well as parial differenial equaions; see, for example, [11, 25, 36, 46]. On he oher hand, in recen years, he sudy of funcion spaces, especially on Hardy spaces associaed wih differen operaors, has also inspired grea ineress (see, for example, [5, 22, 23, 24, 29, 30, 48, 33] and heir references). Paricularly, le L be a linear operaor on L 2 (R n ) and generae an analyic semigroup {e L } 0 wih kernel having poinwise upper bounds, whose decay is measured by θ(l) (0, ]. Then, by using he Lusin area funcion, Auscher, Duong and McInosh [5] iniially inroduced he Hardy space HL 1(Rn ) associaed wih he operaor L and esablished is molecular characerizaion. Based on his, Duong and Yan [23, 24] inroduced he BMO-ype space BMO L (R n ) associaed wih L and proved ha he dual space of HL 1(Rn ) is jus BMO L (R n ), where L denoes he adjoin operaor of L in L 2 (R n ). Laer, Yan [48] furher generalized hese resuls o he Hardy spaces H p L (Rn ) wih p (n/[nθ(l)],1] and heir dual spaces. Moreover, Jiang e al. [30] invesigaed he Orlicz-Hardy space and is dual space associaed wih such an operaor L. Le : R n (0,1] be a variable exponen funcion saisfying he globally log- Hölder coninuous condiion. Moivaed by [37, 48], in his aricle, we inroduce he variable exponen Hardy space associaed wih he operaorl, denoed by H L (Rn ). More precisely, for all f L 2 (R n ) and x R n, le { S L (f)(x) := Γ(x) m Le ml (f)(y) 2 dyd n1 }1 2, where m is a posiive consan appearing in he poinwise upper bound of he hea kernel (see (2.2) below) and Γ(x) := {(y,) R n (0, ): y x < }. The Hardy spaces H L (Rn ) is defined o be he compleion of he se {f L 2 (R n ): S L (f) L (R n )} wih respec o he quasi-norm { [ ] p(x) f := S SL (f)(x) H L (Rn ) L(f) L (R n ) := inf λ (0, ): dx 1}. λ We hen esablish he molecular characerizaion of H L (Rn ) via variable exponen en spaces. Using his molecular characerizaion, we furher prove ha he dual space of H L (Rn ) is he BMO-ype space BMO,L (R n ), which is also inroduced in his aricle. As more applicaions, we sudy he boundedness of he fracional inegral L γ (γ (0, n ) wih m as in Assumpion (A) below) from H m L (Rn ) o H q( ) L (Rn ) wih 1 := 1 mγ and he coincidence beween H q( ) n L (Rn ) and variable exponen Hardy spaces H (R n ) inroduced in [37]. A novel aspec of his aricle is o give a non-rivial combinaion of funcion spaces wih variable exponens and he heory of operaors including heir funcional calculi and semigroups, and hese new funcion spaces prove necessary in he sudy R n

3 Molecular characerizaions and dualiies of variable exponen Hardy spaces 359 of he boundedness of he associaed operaors (for example, fracional inegrals L γ wih γ (0, n)). m This aricle is organized as follows. In Secion 2, we firs recall some noaion and definiions abou variable exponen Lebesgue spaces, holomorphic funcional calculi of operaors and semigroups, also including some basic assumpions on he operaor L considered in his aricle and he domain of he semigroup {e L } 0. Via he Lusin area funcion S L (f), we hen inroduce he variable exponen Hardy space associaed wih L, denoed by H L (Rn ). In Secion 3, we mainly esablish a molecular characerizaion of he space H L (Rn ) (see Theorem 3.13 below). To his end, we firs esablish an aomic characerizaion of he variable exponen en space T ) (see Corollary 3.7 below). Then he molecular characerizaion of H L (Rn ) is obained by using a projec operaor π L corresponding o L, which is proved o be bounded from T ) o H L (Rn ). We poin ou ha [44, Lemma 4.1] of Sawano (a sligh weaker varian of his lemma was early obained by Nakai and Sawano [37, Lemma 4.11]), which is re-saed in Lemma 3.5 below, plays a key role in he proof of Theorem 3.13 Secion 4 is devoed o proving a dualiy heorem. Indeed, in Theorem 4.3 below, we show ha he dual space ofh L (Rn ) is jus he BMO-ype space BMO,L (R n ), which is also inroduced in his secion. To show Theorem 4.3, we rely on several key esimaes relaed o BMO-ype spaces and -Carleson measures (see Proposiions 4.5, 4.6 and 4.7, and Lemma 4.9 below), and he dualiy of he variable exponen en space (see Proposiion 4.8 below). The main difficuly o esablish hese esimaes is ha he quasi-norm L (R n ) has no he ranslaion invariance, namely, for any cube Q(x,r) R n, wih x R n and r (0, ), and z R n, χ Q(x,r) L (R n ) may no equal o χ Q(xz,r) L (R ). To overcome his difficuly, we n make full use of Lemma 3.14 below, which is jus [55, Lemma 2.6] and presens a relaion beween wo quasi-norms L (R n ) corresponding o wo cubes. As applicaions of he molecular characerizaion ofh L (Rn ) from Theorem 3.13, in Secion 5, we invesigae he boundedness of fracional inegrals on H L (Rn ) (see Theorem 5.9 below) and show ha he spaces H L (Rn ) and H (R n ) coincide wih equivalen quasi-norms under some addiional assumpions on L (see Theorem 5.3 below). 2. Preliminaries In his secion, we firs recall some noaion and noions on variable exponen Lebesgue spaces and some knowledge abou holomorphic funcional calculi as well as semigroups. Then we inroduce he variable exponen Hardy spaces associaed wih operaors, denoed by H L (Rn ), which generalize he Hardy spaces H p L (Rn ) sudied in [23, 48]. We begin wih some noaion which will be used in his aricle. Le N := {1,2,...} and Z := N {0}. We denoe by C a posiive consan which is independen of he main parameers, bu may vary from line o line. We use C (α,...) o denoe a posiive consan depending on he indicaed parameers α,... The symbol A B means A CB. If A B and B A, hen we wrie A B. If E is a subse of R n, we denoe by χ E is characerisic funcion and by E he se R n \E. For a R, a denoes he larges ineger m such ha m a. For all x R n and r (0, ), denoe by Q(x,r) he cube cenered a x wih side lengh r, whose sides

4 360 Dachun Yang and Ciqiang Zhuo are parallel o he axes of coordinaes. For each cube Q R n and a (0, ), we use x Q o denoe he cener of Q and l(q) o denoe he side lengh of Q, and denoe by aq he cube concenric wih Q having he side lengh al(q) Variable exponen Lebesgue spaces. In wha follows, a measurable funcion : R n (0, ) is called a variable exponen. For any variable exponen, le (2.1) p := essinf x R n p(x) and p := esssup x R n p(x). Denoe by P(R n ) he collecion of variable exponens : R n (0, ) saisfying 0 < p p <. For a measurable funcion f on R n and a variable exponen P(R n ), he modular (f) of f is defined by seing (f) := R n f(x) p(x) dx and he Luxemburg quasi-norm f L (R n ) := inf { λ (0, ): (f/λ) 1 }. Then he variable exponen Lebesgue space L (R n ) is defined o be he se of all measurable funcionsf such ha (f) < equipped wih he quasi-norm f L (R n ). For more properies on he variable exponen Lebesgue spaces, we refer he reader o [14, 18]. Remark 2.1. Le P(R n ). (i) If p [1, ), hen L (R n ) is a Banach space (see [18, Theorem 3.2.7]). In paricular, for all λ C and f L (R n ), λf L (R n ) = λ f L (R n ) and, for all f,g L (R n ), f g L (R n ) f L (R n ) g L (R n ). (ii) For any non-rivial funcion f L (R n ), i holds rue ha (f/ f L (R n )) = 1; see, for example, [14, Proposiion 2.21]. (iii) If R n [ f(x) /δ] p(x) dx c for some δ (0, ) and some posiive consan c independen of δ, hen i is easy o see ha f L (R n ) Cδ, where C is a posiive consan independen of δ, bu depending on p (or p ) and c. Recall ha a measurable funcion g P(R n ) is said o be locally log-hölder coninuous, denoed by g C log loc (Rn ), if here exiss a posiive consan C log (g) such ha, for all x, y R n, g(x) g(y) C log (g) log(e1/ x y ), and g is said o saisfy he globally log-hölder coninuous condiion, denoed by g C log (R n ), if g C log loc (Rn ) and here exis a posiive consan C and a consan g R such ha, for all x R n, g(x) g Remark 2.2. Le n = 1 and, for all x R, C log(e x ). p(x) := max { 1 e 3 x,min ( 6/5,max { 1/2,3/2 x 2})}.

5 Molecular characerizaions and dualiies of variable exponen Hardy spaces 361 Then C log (R); see [37, Example 1.3]. Wih a sligh modificaion, anoher example was obained in [49, Example 2.20] as follows. For all x R, le p(x) := max { 1 e 3 x, min(6/5,max(1/2,k x 1/2 k)) }, where k := 7/[10( 3/10 1)]. Then C log (R). For all r (0, ), denoe by L r loc (Rn ) he se of all locally r-inegrable funcions on R n and, for any measurable se E R n, by L r (E) he se of all measurable funcions f such ha { 1/r f L r (E) := f(x) dx} r <. E Recall ha he Hardy Lilewood maximal operaor M is defined by seing, for all f L 1 loc (Rn ) and x R n, 1 M(f)(x) := sup f(y) dy, B x B B where he supremum is aken over all balls B of R n conaining x. Remark 2.3. Le C log (R n ) and 1 < p p <. Then here exiss a posiive consan C such ha, for all f L (R n ), M(f) L (R n ) C f L (R n ); see, for example, [18, Theorem 4.3.8] Holomorphic funcional calculi. Here, we firs recall some noions of he bounded holomorphic funcional calculus, which were inroduced by McInosh [35], and hen make wo assumpions on L required in his aricle. For wo normed linear spaces X and Y, le L(X,Y) be he collecion of coninuous linear operaors from X o Y and, for any T L(X,Y), T X Y is operaor norm. Le v (0,π). Define he closed secor S v by S v := {z C: argz v} {0} and denoe by Sv 0 he inerior of S v. Le H(Sv) 0 be he se of all holomorphic funcions on Sv 0, { } and H (S 0 v) := b H(Sv): 0 b := sup b(z) < z Sv 0 Ψ(S 0 v) := {ψ H(S 0 v): s, C (0, ) such ha ψ(z) C z s (1 z 2s ) 1, z S 0 v }. Given v (0,π), a closed operaor L L(L 2 (R n ),L 2 (R n )) is said o be of ype v if σ(l) S v, where σ(l) denoes he specra of L, and, for all γ (v,π), here exiss a posiive consan C such ha, for all λ / S γ, (L λi) 1 L 2 (R n ) L 2 (R n ) C λ 1. Le θ (v,γ) and Σ be he conour {ξ = re ±iθ : r [0, )} parameerized clockwise around S v. Then, for ψ Ψ(Sv) 0 and L being of ype v, he operaor ψ(l) is defined by ψ(l) := 1 (L λi) 1 ψ(λ)dλ, 2πi Σ where he inegral is absoluely convergen in L(L 2 (R n ),L 2 (R n )) and, by he Cauchy heorem, he above definiion is independen of he choices of v and γ saisfying θ (v,γ). If L is a one-o-one linear operaor having dense range and b H (Sγ 0), hen define an operaorb(l) byb(l) := [ψ(l)] 1 (bψ)(l), whereψ(z) := z(1z) 2 for

6 362 Dachun Yang and Ciqiang Zhuo all z S 0 γ. I was proved in [35] ha b(l) is well defined on L 2 (R n ). The operaor L is said o have a bounded H funcional calculus onl 2 (R n ) if, for all γ (v,π), here exiss a posiive consan C such ha, for all b H (S 0 γ), b(l) L(L 2 (R n ),L 2 (R n )) and b(l) L 2 (R n ) L 2 (R n ) C b. Le L be a linear operaor of ype v on L 2 (R n ) wih v (0, π ). Then i generaes 2 a bounded holomorphic semigroup {e zl } z Dv, where D v := {z C: 0 arg(z) < π 2 v} and, for all z C, arg(z) ( π,π] is he argumen of z; see, for example, [41, Theorem 1.45]. In his aricle, we make he following wo assumpions on he operaor L. Assumpion (A). Assume ha, for each (0, ), he disribuion kernel p of e L belongs o L (R n R n ) and saisfies ha, for all x, y R n, ( ) (2.2) p (x,y) n x y m g, where m is a posiive consan and g is a posiive, bounded and decreasing funcion saisfying ha, for some ε (0, ), (2.3) lim r nε g(r) = 0. r Assumpion (B). Assume ha he operaor L is one-o-one, has dense range in L 2 (R n ) and a bounded H funcional calculus on L 2 (R n ). Remark 2.4. (i) If {e L } 0 is a bounded analyic semigroup on L 2 (R n ) whose kernels {p } 0 saisfy (2.2) and (2.3), hen, for any k N, here exiss a posiive consan C (k), depending on k, such ha, for all (0, ) and almos every x, y R n, (2.4) p (x,y) k k k C ( ) (k) x y n/mg k. 1/m Here, i should be poined ou ha, for all k N, he funcion g k may depend on k bu always saisfies (2.3); see [41, Theorem 6.17] and [12]. (ii) Le v (0,π). Then L has a bounded H funcional calculus on L 2 (R n ) if and only if, for any γ (v,π) and nonzero funcion ψ Ψ(Sγ 0 ), L saisfies he following square funcion esimae: here exiss a posiive consan C such ha, for all f L 2 (R n ), { C 1 f L 2 (R n ) ψ (L)f 2 L 2 (R n ) where ψ (ξ) := ψ(ξ) for all (0, ) and ξ R n ; see [35]. 0 1 m } 1/2 d C f L2(Rn), 2.3. An acing class of semigroups {e L } 0. For all β (0, ), le M β (R n ) be he se of all funcions f L 2 loc (Rn ) saisfying } {Rn f(x) 2 1/2 f Mβ (R n ) := dx <. 1 x nβ We poin ou ha he space M β (R n ) was inroduced by Duong and Yan in [24] and i is a Banach space under he norm Mβ (R n ). For any given operaor L saisfying Assumpions (A) and (B), le (2.5) θ(l) := sup{ε (0, ): (2.2) and (2.3) hold rue}

7 and Molecular characerizaions and dualiies of variable exponen Hardy spaces 363 M(R n ) := { M θ(l) (R n ), if θ(l) <, β (0, ) M β(r n ), if θ(l) =. Le s Z. For any f M(R n ) and (x,) R n1 := R n (0, ), le (2.6) P s, f(x) := f(x) (I e L ) s1 f(x) and Q s, f(x) := (L) s1 e L f(x), and, paricularly, le (2.7) P f(x) := P 0, f(x) = e L f(x) and Q f(x) := Q 0, f(x) = Le L f(x). Here, we poin ou ha hese operaors in (2.6) were inroduced by Blunck and Kunsmann [7] and Holfmann and Marell [27]. Remark 2.5. (i) For all f M(R n ), he operaors P s, f and Q s, f are well defined. Moreover, he kernels p s, of P s, and q s, of Q s, saisfy ha here exiss a posiive consan C such ha, for all (0, ) and x, y R n, ( ) x y (2.8) p s, m(x,y) q s, m(x,y) C n g, where he funcion g saisfies he condiions as in Assumpion (A); see, for example, [48]. (ii) A ypical example of L saisfying θ(l) = is ha he kernels {p } 0 of {e L } 0 have he poinwise Gaussian upper bound, namely, here exiss a posiive consan C such ha, for all (0, ) and x, y R n, p (x,y) C Obviously, if := n 2 i=1 2 e x y n/2. x i is he Laplacian operaor and L =, hen he hea kernels have he poinwise Gaussian upper bound. There are several oher operaors whose hea kernels have he poinwise Gaussian upper bound; see, for example, [48, p. 4390, Remarks]. (iii) Le s Z and p (1, ). Then, by (i) of his remark, we easily conclude ha here exiss a posiive consan C such ha, for all (0, ) and f L p (R n ), P s, m(f) L p (R n ) C f L p (R n ) Definiion of Hardy spaces H L (Rn ). For all funcions f L 2 (R n ), define he Lusin area funcion S L (f) by seing, for all x R n, { S L (f)(x) := Q mf(y) 2 dyd Γ(x) n1 } 1/2, here and hereafer, for all x R n, Γ(x) := {(y,) R n1 : y x < } and Q is defined as in (2.7). In [5], Auscher e al. proved ha, for any p (1, ), here exiss a posiive consan C (p), depending on p, such ha, for all f L p (R n ), (2.9) C 1 (p) f L p (R n ) S L (f) L p (R n ) C (p) f L p (R n ); see also Duong and McInosh [21] and Yan [47]. We now inroduce he variable exponen Hardy spaces associaed wih operaors. Definiion 2.6. Le L be an operaor saisfying Assumpions (A) and (B), and C log (R n ) saisfy p (0,1]. A funcion f L 2 (R n ) is said o be in H L (Rn ) if S L (f) L (R n ); moreover, define { [ ] p(x) f := S SL (f)(x) H L (Rn ) L(f) L (R n ) := inf λ (0, ): dx 1}. λ R n

8 364 Dachun Yang and Ciqiang Zhuo Then he variable Hardy space associaed wih operaor L, denoed by H L (Rn ), is defined o be he compleion of H L (Rn ) in he quasi-norm. H L (Rn ) Remark 2.7. (i) By he heorem of compleion of Yosida [53, p. 65], we find ha H L (Rn ) is dense in H L (Rn ), namely, for any f H L (Rn ), here exiss a Cauchy sequence {f k } k N in H L (Rn ) such ha lim k f k f = 0. H L (Rn ) Moreover, if {f k } k N is a Cauchy sequence in H L (Rn ), hen here exiss an unique f H L (Rn ) such ha lim k f k f = 0. Moreover, H L (Rn ) L2 (R n ) H L (Rn ) is dense in H L (Rn ). (ii) We poin ou ha smooh funcions wih compac suppors do no necessarily belong o H L (Rn ); see [48] and also Remark 4.4 below for more deails. (iii) Observe ha, when p (0, ), L (R n ) = L p (R n ). If 1, hen H L (Rn ) = HL 1(Rn ), which was inroduced by Auscher e al. [5]; see also Duong n and Yan [23]. If p (,1), hen he space H nθ(l) L (Rn ) is jus he space H p L (Rn ) inroduced by Yan [48]. (iv) Differen from he space H p L (Rn ) which is jus L p (R n ) when p (1, ) (see, for example, [48, p. 4400]), since i is sill unclear wheher (2.9) holds rue or no wih L p (R n ) replaced by L (R n ) when p (1, ), i is also unclear wheher H L (Rn ) and L (R n ) (or H (R n )) coincide or no. We will no push his issue in his aricle due o is lengh. We end his secion by comparing he variable exponen Hardy spaces associaed wih operaors in his aricle wih he Musielak Orlicz Hardy spaces associaed wih operaors saisfying reinforced off-diagonal esimaes in [8]. Indeed, in general, hese wo scales of Hardy-ype spaces do no cover each oher. Remark 2.8. Le ϕ: R n [0, ) [0, ) be a growh funcion in [32] and L an operaor saisfying reinforced off-diagonal esimaes in [8]. Then Bui e al. [8] inroduced he Musielak Orlicz Hardy space associaed wih operaor L via he Lusin area funcion, denoed by H ϕ,l (R n ). Recall ha he Musielak Orlicz space L ϕ (R n ) is defined o be he se of all measurable funcions f on R n such ha { } f L ϕ (R n ) := inf λ (0, ) : ϕ(x, f(x) /λ)dx 1 <. R n Observe ha, if (2.10) ϕ(x,) := p(x) for all x R n and [0, ), hen L ϕ (R n ) = L (R n ). However, a general Musielak Orlicz funcion ϕ saisfying all he assumpions in [32] (and hence [8]) may no have he form as in (2.10) (see [32]). On he oher hand, i was proved in [49, Remark 2.23(iii)] ha here exiss a variable exponen funcion C log (R n ), bu is no a uniformly Muckenhoup weigh, which was required in [32] (and hence [8]). Thus, Musielak Orlicz Hardy spaces associaed wih operaors in [8] and variable exponen Hardy spaces associaed wih operaors in his aricle do no cover each oher. Moreover, in Theorem 5.3 below, we show ha, under some addiional assumpions on L, he spaces H L (Rn ) coincide wih he variable exponen Hardy spaces H (R n ) which can no cover and also can no be covered by Musielak Orlicz Hardy spaces in [32] based on he same reason as above.

9 Molecular characerizaions and dualiies of variable exponen Hardy spaces Molecular characerizaions of H L (Rn ) In his secion, we aim o obain he molecular characerizaions of H L (Rn ). To his end, we firs give ou some properies of he en spaces wih variable exponens including heir aomic characerizaions, which are hen applied o esablish he molecular characerizaions ofh L (Rn ) by using a projec operaorπ L corresponding o L Aomic characerizaions of en spaces T ). We begin wih he definiion of he en space wih variable exponen. Le P(R n ). For all measurable funcions g on R n1 and x R n, define { T(g)(x) := g(y,) 2 dyd } 1/2 Γ(x) n1 and { Q 1/2 C (g)(x) := sup g(y,) 2 dyd } 1/2, Q x χ Q L (R n ) Q where he supremum is aken over all cubes Q of R n conaining x and Q denoes he en over Q, namely, Q := {(y,) R n1 : B(y,) Q}. Definiion 3.1. Le P(R n ). (i) Le q (0, ). Then he en space T q 2(R n1 ) is defined o be he se of all measurable funcions g on R n1 such ha g T q 2 (Rn1 ) := T(g) L q (R n ) <. (ii) The en space wih variable exponen T ) is defined o be he se of all measurable funcions g on R n1 such ha g := T(g) T ) L (R n ) <. (iii) The space T 2, (R n1 ) is defined o be he se of all measurable funcions g on R n1 such ha g T 2, (Rn1 ) := C (g) L (R n ) <. Remark 3.2. (i) We poin ou ha he spaces T q 2 (Rn1 ) and T ) were inroduced in [10] and [55], respecively. Moreover, if q (0, ), hen T ) = T q 2 (Rn1 ). (ii) Ifg T2 2(Rn1 ), hen we easily see ha g T 2 ) = { g(x,) 2 dxd } 2. 1 R n1 Le q (1, ) and P(R n ). Recall ha a measurable funcion a on R n1 is called a (,q)-aom if a saisfies (i) supp a Q for some cube Q R n ; (ii) a T q 2 (Rn1 ) Q 1/q χ Q 1. L (R n ) Furhermore, if a is a (,q)-aom for all q (1, ), hen a is call a (, )- aom. We poin ou ha he (, )-aom was inroduced in [55]. For any P(R n ), {λ j } C and cubes {Q j } of R n, le { }1 [ ] (3.1) A({λ j },{Q j } ) := λ j χ p p Qj Q j, L (R n ) here and hereafer, we le (3.2) p := min{1,p } wih p as in (2.1). L (R n )

10 366 Dachun Yang and Ciqiang Zhuo The following aomic decomposiion of T ) was proved in [55, Theorem 2.16]. Lemma 3.3. Le C log (R n ). Then, for all f T ), here exis {λ j } C and a sequence {a j } of (, )-aoms such ha, for almos every (x,) R n1, (3.3) f(x,) = λ j a j (x,); moreover, he series in (3.3) converges absoluely for almos all (x,) R n1 and here exiss a posiive consan C such ha, for all f T ), (3.4) B({λ j a j } ) := A({λ j },{Q j } ) C f T ), where, for each j N, Q j denoes he cube such ha supp a j Q j. By Lemma 3.3, we have he following conclusion. Corollary 3.4. Le C log (R n ). Assume ha f T ), hen he decomposiion (3.3) also holds rue in T ). To prove Corollary 3.4, we need he following useful lemma, which is jus [44, Lemma 4.1]. Lemma 3.5. Le C log (R n ) and q [1, ) (p, ) wih p as in (2.1). Then here exiss a posiive consan C such ha, for all sequences {Q j } of cubes of R n, numbers {λ j } C and funcions {a j } saisfying ha, for each j N, supp a j Q j and a j L q (R n ) Q j 1/q, ( )1 p ( )1 p λ j a j p C λ j χ Qj p, j=1 where p is as in (3.2). L (R n ) j=1 L (R n ) Proof of Corollary 3.4. Le f T ). Then, by Lemma 3.3, we may assume ha f = λ ja j almos everywhere on R n1, where {λ j } C and {a j } is a sequence of (, )-aoms such ha, for each j N, supp a j Q j wih some cube Q j R n, and (3.5) A({λ j },{Q j } ) f T ). Le q [1, ) (p, ). Then, by he definiion of (, )-aoms, we see ha, for all j N, T(a j ) L q (R n ) = a j T q 2 (Rn1 ) Q j 1/q. χ Qj L (R n ) From his, Lemma 3.5 and he fac ha, for all θ (0,1] and {ξ j } C, (3.6) ( ) θ ξ j ξ j θ,

11 Molecular characerizaions and dualiies of variable exponen Hardy spaces 367 we deduce ha, for all N N, ( N T f λ j a j) λ j T (a j ) j=1 L (R n ) j=n1 (3.7) { [ λ j χ Qj χ Qj L (R n ) j=n1 L (R n ) }1 ] p p L (R n ) This, combined wih (3.5) and he dominaed convergence heorem (see [14, Theorem 2.62]), implies ha ( N { }1 lim N T f λ j a j) j=1 lim [ ] λ j χ p p Qj N χ Qj = 0. L (R n ) j=n1 L (R n ) L (R n ) Therefore, (3.3) holds rue in T ), which complees he proof of Corollary 3.4. Remark 3.6. I was proved in [29, Proposiion 3.1] ha, if f T q 2 (Rn1 ) wih q (0, ), hen he decomposiion (3.3) also holds rue in T q 2 (Rn1 ). Using Corollary 3.4 and an argumen similar o ha used in he proof of (3.7), we obain he following aomic characerizaion oft ), he deails being omied. Corollary 3.7. Le C log (R n ) saisfy p (0,1]. Then f T ) if and only if here exis {λ j } C and a sequence {a j } of (, )-aoms such ha, for almos every (x,) R n1, f(x,) = λ ja j (x,) and R n { [ λ j χ Qj χ Qj L (R n ) ] p }p(x) p dx <, where, for each j, Q j denoes he cube appearing in he suppor of a j ; moreover, for all f T ), f A({λ T 2 (R j} n1 ),{Q j } ) wih he implici posiive consans independen of f. The following remark plays an imporan role in he proof of Theorem 4.3. Remark 3.8. Le P(R n ) saisfy p (0,1]. Then, by [37, Remark 4.4], we know ha, for any {λ j } C and cubes {Q j } of R n, λ j A({λ j },{Q j } ). In wha follows, le T 2,c (Rn1 ) and T q 2,c (Rn1 ) wih q (0, ) be he ses of all funcions, respecively, in T ) and T2(R q n1 ) wih compac suppors. Proposiion 3.9. Le C log (R n ). Then T 2,c (R n1 ) T 2 2,c (Rn1 ) as ses. Proof. By [29, Lemma 3.3(i)], we know ha, for any q (0, ), T q 2,c (Rn1 ) T2,c(R 2 n1 ). Thus, o prove his proposiion, i suffices o show ha T 2,c (R n1 ) T q 0 2,c (Rn1 ) for some q 0 (0, ). To his end, suppose ha f T 2,c (Rn1 ) and supp f K, where K is a compac se in R n1. Le Q be a cube in R n such ha.

12 368 Dachun Yang and Ciqiang Zhuo K Q. Then supp T(f) Q. From his and he fac ha p p(x) for all x R n, we deduce ha [T(f)(x)] p dx [T (f)(x)] p dx R n {x Q: T (f)(x)<1} {x Q: T (f)(x) 1} Q [T (f)(x)] p(x) dx <, R n which implies ha T 2,c (R n1 ) T p of Proposiion ,c(R n1 ) as ses and hence complees he proof 3.2. Molecular characerizaions of H L (Rn ). In his subsecion, we esablish he molecular characerizaions of H L (Rn ). We begin wih some noions. In wha follows, for any q (0, ), le L q (R n1 ) be he se of all q-inegrable funcions on R n1 and L q loc (Rn1 ) he se of all locally q-inegrable funcions on R n1. For any P(R n ), le (3.8) s 0 := (n/m)(1/p 1), namely, s 0 denoes he larges ineger smaller han or equal o n m ( 1 p 1). Le m be as in (2.2) and s [s 0, ). Le C (m,s) be a posiive consan, depending on m and s, such ha (3.9) C (m,s) 0 m(s2) e 2m (1 e m ) s 01 d = 1. Le q (0, ). Recall ha he operaor π L is defined by seing, for all f T q 2,c (Rn1 ) and x R n, π L (f)(x) := C (m,s) 0 d Q s, m(i P s0,m)(f(,))(x). Moreover, π L is well defined and π L (f) L 2 (R n ) for all f T q 2,c(R n1 ) (see [48, p.4395]). Remark Le f L 2 (R n ). Then, by [48, (3.10)], we know ha f = C (m,s) 0 Q s, m(i P s0, m)q d mf, where he inegral converges in L 2 (R n ); see also [3, 35]. Definiion Le C log (R n ) and s [s 0, ) wih s 0 as in (3.8). A measurable funcionαonr n is called a(,s,l)-molecule if here exiss a(, )- aom a suppored on Q for some cube Q R n such ha, for all x R n, α(x) := π L (a)(x). When i is necessary o specify he cube Q, hen a is called a (,s,l)-molecule associaed wih Q. Remark Le C log (R n ) wih p (, ), where p nθ(l) and θ(l) are as in (2.1) and (2.5), respecively. Then, by Proposiion 3.17(ii) below, we see ha he (,s,l)-molecule is well defined. Indeed, if a is a (, )-aom, by Corollary 3.7, we hen know a T ), which, ogeher wih Proposiion 3.17(ii) below, implies ha π L (a) H L (Rn ). Thus, he (,s,l)-molecule is well defined. The molecular characerizaion of H L (Rn ) is saed as follows. n

13 Molecular characerizaions and dualiies of variable exponen Hardy spaces 369 Theorem Le C log (R n ) saisfy p (0,1] and p (,1], and nθ(l) s [s 0, ) wih p, p, θ(l) and s 0, respecively, as in (2.1), (2.5) and (3.8). (i) If f H L (Rn ), hen here exis {λ j } C and a sequence {α j } of (,s,l)-molecules associaed wih cubes{q j } such haf = λ jα j in H L (Rn ) and B({λ j α j } ) := A({λ j },{Q j } ) C f H L (Rn ) wih C being a posiive consan independen of f. (ii) Suppose ha {λ k } k N C and {α k } k N is a family of (,s,l)-molecules saisfying B({λ k α k } k N ) <. Then k N λ kα k converges in H L (Rn ) and k α k CB({λ k α k } k N ) k Nλ H L (Rn ) wih C being a posiive consan independen of {λ k α k } k N. The proof of Theorem 4.3 srongly depends on several auxiliary esimaes and will be presened laer. The following Lemma 3.14 is jus [55, Lemma 2.6] (For he case when p > 1, see also [28, Corollary 3.4]). Lemma Le C log (R n ). Then here exiss a posiive consan C such ha, for all cubes Q 1 and Q 2 saisfying Q 1 Q 2, ( ) 1/p C 1 Q1 χ ( ) Q 1 1/p L (R n ) Q1 C, Q 2 χ Q2 L (R n ) Q 2 where p and p are as in (2.1). The following Fefferman Sein vecor-valued inequaliy of he Hardy Lilewood maximal operaor M on he space L (R n ) was obained in [15, Corollary 2.1]. Lemma Le r (1, ) and C log (R n ). If p (1, ) wih p as in (2.1), hen here exiss a posiive consan C such ha, for all sequences {f j } j=1 of measurable funcions, { } 1/r ( ) 1/r [M(f j )] r C f j r. j=1 L (R n ) j=1 L (R n ) Remark Le k N and C log (R n ). Then, by Lemma 3.15 and he fac ha, for all cubes Q R n, r (0,p ), χ 2 k Q 2 kn/r [M(χ Q )] 1/r, we conclude ha here exiss a posiive consan C such ha, for any {λ j } C and cubes {Q j } of R n, A({λ j },{2 k Q j } ) C2 kn(1 r 1 p ) A({λj },{Q j } ), where p and p are as in (2.1). Proposiion Le C log (R n n ) wih p (, ) and q (1, ), nθ(l) where p and θ(l) are as in (2.1) and (2.5), respecively. Then (i) he operaor π L is a bounded linear operaor from T q 2 (Rn1 ) o L q (R n ); (ii) he operaor π L, well defined on he space T 2,c (R n1 ), exends o a bounded linear operaor from T ) o H L (Rn ). n

14 370 Dachun Yang and Ciqiang Zhuo Proof. To prove his proposiion, i suffices o show (ii), since (i) is jus [48, Lemma 3.4(a)]. Noicing ha, due o Corollary 3.7, T 2,c (Rn1 ) is a dense subse of T ), o prove (ii), we only need o show haπ L mapst 2,c (R n1 ) coninuously ino H L (Rn ). To his end, le f T 2,c (Rn1 ). Then, by Proposiion 3.9, we know ha f T2,c(R 2 n1 ) and hence π L is well defined on T 2,c (R n1 ) by (i). This, combined wih Lemma 3.3, Corollary 3.4 and Remark 3.6, implies ha here exis sequences {λ j } C and{a j } of(, )-aoms such ha, for eachj N, supp a j Q j wih some cube Q j R n, f = λ ja j in boh T ) and T2(R 2 n1 ), and A({λ j },{Q j } ) f T ). Thus, i follows from (i) ha, for all N N, ( N π L as N ; furhermore, (3.10) π L (f) = lim N f λ j a j) j=1 L 2 (R n ) = N π L (λ j a j ) j=1 λ j C (m,s) 0 j=n1λ j a j T 2 2 (Rn1 ) 0 Q s, m(i P s0, m)(a j(,)) d =: λ j α j in L 2 (R n ), where s 0 is as in (3.8) and s [s 0, ). Nex, we prove S L (π L (f)) L (R n ) f T ). Observe ha, for almos every x R n, ( ) S L (π L (f))(x) = S L λ j α j (x) L (λ j α j )(x) S due o (3.10), he Faou lemma and he fac has L is bounded onl 2 (R n ) (see (2.9)). Then, by Remark 2.1(i) and he Faou lemma of L (R n ) (see [14, Theorem 2.61]), we see ha ( 1 )1 p p (3.11) S L (π L (f)) L (R n ) p [ ] p SL (λ j α j )χ Ui (Q j ), i=0 L (R n ) wherepis as in (3.2) and, for anyj N,U 0 (Q j ) := 4Q j andu i (Q j ) := 2 i2 Q j \(2 i1 Q j ) for all i N. By (2.9), (i) and Lemma 3.14, we find ha, for all q (1, ), S L (α j ) L q (4Q j ) α j L q (R n ) π L (a j ) L q (R n ) a j T q 2 (Rn1 ) Q j 1 q χ4qj 1. L (R n ) From his, Lemma 3.5 and Remark 3.16, we deduce ha ( )1 p [ ] p λj S L (α j )χ 4Qj (3.12) L (R n ) A({λ j },{4Q j } ) A({λ j },{Q j } ) f T ).

15 Molecular characerizaions and dualiies of variable exponen Hardy spaces 371 n Since p (, ), we choose ε (0,θ(L)) such ha p nθ(l) ( n i N, by [30, (4.12)], we know ha, for all x (4Q j ), nε, ). For S L (α j )(x) (r Qj ) εn 2 x xqj (nε) a j T 2 ). Then, by his, he Hölder inequaliy and Lemma 3.14, we furher find ha, for any q (p, ) [2, ) wih p as in (2.1), (3.13) S L (α j ) L q (U i (Q j )) 2 i(n q n ε) Q j 1 q 1 2 aj T 2 ) 2i(n q n ε) a j T q 2 (Rn1 ) 2 i(nε) 2 i Q j 1 q χqj 1 L (R n ). Observe ha, for all r ( n,p nε ), χ 2 i Q j 2 n r i [M(χ Qj )] 1 r. Thus, from his, (3.13) and Lemmas 3.5 and 3.15, we deduce ha ( )1 p [ ] p λj S L (α j )χ Ui (Q j ) L (R n ) { }1 (3.14) [ ] λ 2 i(nε) j χ p 2 i p Q j χ Qj L (R n ) L (R n ) 2 i(nε n/r) A({λ j },{Q j } ) 2 i(nε n/r) f T ). Combining (3.11), (3.12) and (3.14), ogeher wih r > n, we conclude ha nε S L (π L (f)) L (R n ) { i=0 )}1/p 2 i(nε n r )p f T f T which implies ha π L is a bounded linear operaor from T ) o H L (Rn ) and hence complees he proof of Proposiion We now urn o he proof of Theorem Proof of Theorem We firs prove (i). Le C (m,s) be he consan as in (3.9) and f H L (Rn ) L 2 (R n ). Then, by Remark 3.10, we see ha (3.15) f = C (m,s) Q s, m(i P s0, m)q d mf = π L(Q mf) 0 inl 2 (R n ), where s 0 is as in (3.8) ands [s 0, ). Since f H L (Rn ), i follows ha Q mf T ). Thus, by Lemma 3.3 and Corollary 3.4, we find ha Q mf = λ ja j in he sense of boh poinwise and in T ), where {λ j } C and {a j } are (, )-aoms saisfying ha, for each j N, supp a j Q j wih some cube Q j R n, and A({λ j },{Q j } ) Q mf T ) f H L (Rn ). For any j N, le α j := π L (a j ). Then α j is a (,s,l)-molecule and, by (3.15) and Proposiion 3.17, we conclude ha f = π L (Q mf) = λ j π L (a j ) =: λ j α j ), in boh L 2 (R n ) and H L (Rn ).

16 372 Dachun Yang and Ciqiang Zhuo Now, for any f H L (Rn ), since H L (Rn ) L 2 (R n ) is dense in H L (Rn ), i follows ha here exiss a sequence {f k } k N [H L (Rn ) L 2 (R n )] such ha, for all k N, Le f 0 := 0. Then f f k H L (Rn ) 2 k f H L (Rn ). (3.16) f = (f k f k 1 ) in H L (Rn ). k N From he above argumen, we deduce ha, for each k N, here exis {λ k j } k N C and a sequence {α k j } of (,s,l)-molecules such ha (3.17) f k f k 1 = λ k j αk j in H L (Rn ) and A({λ k j },{Q k j } ) f k f k 1 H L (Rn ) 2 k f H L (Rn ), where, for any k, j N, Q k j denoes he cube appearing in he definiion of he (,s,l)-moleculeαj k. By his, he Minkowski inequaliy, (ii) and (iii) of Remark 2.1 and (3.6), we see ha R n { [ k N R n k N k N λ k j χ Q k j f χ H L (Rn ) Q k j L (R n ) [ ( λ k j χ Q k j R n 2 kp(x) ] p } p(x) p f H L (Rn ) χ Q k j L (R n ) [ ( λ k j χ Q k j dx ) p ] p(x) 1 p 2 k f H L (Rn ) χ Q k j L (R n ) dx ) p ] p(x) p p dx 1 p ( ) 1/p 2 kp2 <, k N which implies ha B ( {λ k j αk j } j,k N) f H we conclude ha f = k N L (Rn ) λ k j αk j. Moreover, by (3.16) and (3.17), in H L (Rn ) and hence he proof of (i) is compleed. Nex, we show (ii). Wihou loss of generaliy, we may assume ha, for each k N, α k := π L (a k ), where a k is a (, )-aom suppored on Q k for some cube Q k R n. Then, from Proposiion 3.17(ii) and Corollary 3.7, we deduce ha, for all

17 Molecular characerizaions and dualiies of variable exponen Hardy spaces 373 N 1, N 2 N wih N 1 < N 2, ( N 2 N2 ) λ k α k π N 2 L λ k a k λ k a k k=n 1 H L (Rn ) k=n 1 H L (Rn ) k=n 1 { }1 N2 [ ] p p λ k χ Qk χ Qk, L (R n ) k=n 1 L (R n ) T ) which ends o zero as N 1, N 2 due o he dominaed convergence heorem. Thus, k N λ kα k converges in H L (Rn ) and, by he Faou lemma of L (R n ), Proposiion 3.17(ii) and Corollary 3.7, we furher know ha ( N k α kh liminf k α kh = liminf N N k=1λ L (Rn ) k=1λ π N L k a k)h L (Rn ) k=1λ L (Rn ) { }1 N N [ ] p p liminf k a kt liminf λ k χ Qk N N k=1λ χ ) Qk k=1 L (R n ) B({λ k α k } k N ), which complees he proof of Theorem L (R n ) In wha follows, for all s [s 0, ) wih s 0 as in (3.8) and P(R n ), denoe by H L,fin (Rn ) he se of finie linear combinaions of (,s,l)-molecules. For any f H L,fin (Rn ), he quasi-norm is given by { N } f H := inf B({λ L,fin (Rn ) j α j } N j=1 ): N N, f = λ j α j, where he infimum is aken over all finie molecular decomposiions of f. Corollary Le C log (R n ) saisfy p (0,1] and p ( n nθ(l),1], where p, p and θ(l) are, respecively, as in (2.1) and (2.5). Then H L,fin (Rn ) is dense in H L (Rn ). Proof. Le f H L (Rn ) L 2 (R n ). Then Q mf T ) and hence, by Lemma 3.3, we have Q mf = k λ ka k in T ), where {λ k } k N C and {a k } k N are (, )-aoms. For every k N, le α k := π L (a k ). Then {α k } k N are (, s, L)-molecules. Thus, by Proposiion 3.17(ii), we conclude ha, for all N N, N ( f N k α k = k=1λ π L Q mf k a k) H L (Rn ) k=1λ H L (Rn ) N Q mf j=1 k a k 0, k=1λ T ) as N, which implies ha H L,fin (Rn ) is dense in H L (Rn ) L 2 (R n ) and hence in H L (Rn ). This finishes he proof of Corollary 3.18.

18 374 Dachun Yang and Ciqiang Zhuo 4. BMO-ype spaces and he dualiy of H L (Rn ) In his secion, we mainly consider he dualiy of H L (Rn ). To his end, moivaed by [30], we inroduce he space BMO s,l(r n ) associaed wih he operaor L and he variable exponen. Definiion 4.1. Le L saisfy Assumpions (A) and (B), C log (R n ) wih p (0,1] and s [s 0, ), where p and s 0 are, respecively, as in (2.1) and (3.8). Then he BMO-ype space BMO s,l (Rn ) is defined o be he se of all funcions f M(R n ) such ha f BMO s,l (R n ) <, where { Q 1/2 }1 2 f BMO s,l (R n ) := sup f(x) P s,(rq ) Q R n χ Q mf(x) 2 dx L (R n ) Q and he supremum is aken over all cubes Q of R n. Remark 4.2. (i) The space (BMO s,l (Rn ), BMO s,l (R n )) is a vecor space wih he semi-norm vanishing on he space K (L,s) (R n ) which is defined by K (L,s) (R n ) :={f M(R n ): P s, f(x) = f(x) for almos every x R n and all (0, )}. In his aricle, he space BMO s,l (Rn ) is undersood o be modulo K (L,s) (R n ); see [23, Secion 6] for a discussion ofk (L,0) (R n ) when L is a second order ellipic operaor of divergence form or a Schrödinger operaor. (ii) If 1 and s = 0, hen BMO s,l(r n ) is jus BMO L (R n ) inroduced by Duong and Yan [23]. If P(R n ) is defined by 1 := α 1 for some consan 2 α (0, θ(l) ), hen n BMOs,L(R n ) becomes he space L L (α,2,s) sudied in [48]. Now we sae he main resul of his secion as follows. Theorem 4.3. Le C log (R n n ) saisfy p (0,1] and p (,1] wih nθ(l) p, p and θ(l), respecively, as in (2.1) and (2.5). Le s 0 be as in (3.8) and L denoe he adjoin operaor of L. Then (H L (Rn )) coincides wih BMO s 0,L (R n ) in he following sense: (i) If g BMO s 0,L (R n ), hen he linear mapping l, which is iniially defined on H L,fin (Rn ) by (4.1) l g (f) := f(x)g(x)dx, R n exends o a bounded linear funcional on H L (Rn ) and l g (H L (Rn )) C g BMO s 0,L (R n ), where C is a posiive consan independen of g. (ii) Conversely, le l be a bounded linear funcional on H L (Rn ). Then l has he form as in (4.1) wih a unique g BMO s 0,L (R n ) for all f H L,fin (Rn ) and g BMO s 0,L (R n ) C l (H L (Rn )), where C is a posiive consan independen of l.

19 Molecular characerizaions and dualiies of variable exponen Hardy spaces 375 Remark 4.4. Le, s 0, L, L and θ(l) be as in Theorem 4.3. (i) Iff H L (Rn ), hen, from Theorem 4.3, we deduce haf saisfies he cancelaion f(x)g(x)dx = 0 for all g K condiion R n L,s 0 (R n ), since, if g K L,s 0 (R n ), hen g s BMO 0,L (R n ) = 0. Observe ha, if g K L,s 0 (R n ), hen g is no necessary o be zero almos everywhere and hence, if f is a smooh funcion wih compac suppor, f(x)g(x)dx may no equal zero. Therefore, smooh funcions wih hen R n compac suppors are no necessary o be in H L (Rn ); see also [23, p.962]. (ii) Observe ha, by he proof of Corollary 3.18, we see ha H L,fin (Rn ) [H L (Rn ) L 2 (R n )] andh L,fin (Rn ) is dense inh L (Rn ) L 2 (R n ). From his, i follows ha, if we require ha (4.1) holds rue for all f H L (Rn ) L 2 (R n ) insead of all f H L,fin (Rn ), hen all conclusions of Theorem 4.3 also hold rue. To prove Theorem 4.3, we need some preparaions. Proposiion 4.5. Le C log (R n ) saisfy p (0,1] and p ( n nθ(l),1], and s [s 0, ), where p, p, θ(l) and s 0 are, respecively, as in (2.1), (2.5) and (3.8). Then here exiss a posiive consan C such ha, for all (0, ), K (1, ), f BMO s,l (Rn ) and x R n, when p = 1, (4.2) P s, f(x) P s,k f(x) L C(1log 2 K) χ 1 Q(x,(K) m) Q(x,(K) 1 m ) 1 f BMO s (R n ),L (R n ) and, when p (0,1), (4.3) P s, f(x) P s,k f(x) L Cχ 1 Q(x,(K) m) Q(x,(K) 1 m ) 1 f BMO s (R n ),L (R n ). Proof. Wihou loss of generaliy, we may assume ha f BMO s,l (R n ) = 1. We claim ha, for all, v (0, ) wih v 2, and x 2 Rn, L (4.4) P s, f(x) P s,v f(x) χ 1 Q(x,m) Q(x, 1 m ) 1. (R n ) If his claim holds rue, hen, by Lemma 3.14, we see ha l 1 P s, f(x) P s,k f(x) Ps,2 i f(x) P s,2 i1 f(x) Ps,2 f(x) P l s,k f(x) i=0 l 1 χ Q(x,(2 i ) 1 m) L (R n ) χ 1 Q(x,(K) m) L (R n ) i=0 Q(x,(2 i ) m) 1 Q(x,(K) m) 1 [ ] l 1 Q(x,(2 i ) m) p χ 1 Q(x,(K) m) L 1 (R n ), Q(x,(K) m) 1 Q(x,(K) m) 1 i=0 where l := log 2 K. By his, we furher conclude ha, when p = 1, L P s, f(x) P s,k f(x) (1log 2 K) χ 1 Q(x,(K) m) Q(x,(K) 1 m ) 1 (R n )

20 376 Dachun Yang and Ciqiang Zhuo and, when p (0,1), P s, f(x) P s,k f(x) L χ 1 Q(x,(K) m) Q(x,(K) 1 m ) 1, (R n ) which implies ha (4.2) and (4.3) hold rue. Therefore, o complee he proof of his proposiion, i remains o prove he above claim. By he commuaive properies of semigroups, we have (4.5) P s, f P s,v f = P s, (f P s,v f) P s,v (f P s, f). Since θ(l) (n[ 1 p 1], ) and p (0,1], i follows ha here exiss ε (0,θ(L)) such ha ( ) ( 1 1 (4.6) ε > n 1 > n 1 ). p p p From (2.8), Assumpion (A), he Hölder inequaliy, Lemma 3.14 and he fac ha 2 v 2, we deduce ha, for all x Rn, (4.7) P s, (f P s,v f)(x) n m { v n m R n g v n m ( x y Q(x,v 1 m) i=1 1/m ) f(y) P s,v f(y) dy f(y) P s,v f(y) 2 dy S i g ( x y 1 m } 1/2 ) f(y) P s,v f(y) dy L χ 1 Q(x,m) Q(x, 1 m ) 1 f BMO s (R n ),L (R n ) 2 nε m i v n m f(y) P s,v f(y) dy, i=1 Q(x,(2 i v) 1 m) where, for eachi N,S i := Q(x,(2 i v) m)\q(x,(2 1 i 1 v) m). 1 Noice ha, for anyi N, here exiss a collecion {Q i,j } N i j=1 of cubes wih N i 2 ni/m such ha l(q i,j ) = v 1/m and Q(x,(2 i v) 1/m ) N i j=1 Q i,j. Thus, by he Hölder inequaliy and Lemma 3.14, we find ha Q(x,(2 i v) 1 m) N i f(y) P s,v f(y) dy f(y) P s,v f(y) dy Q i,j j=1 }1 N i { 2 f(y) P s,v f(y) 2 dy Q i,j 1 2 Q i,j j=1 f BMO s,l (R n ) N i j=1 χ Qi,j L (R n ) 2 n m i 2 n m ( 1 1 )i p p L χq(x,v 1 m), (R n )

21 Molecular characerizaions and dualiies of variable exponen Hardy spaces 377 which, ogeher wih Lemma 3.14 again and (4.6), implies ha i=1 v n m 2 inε m v n m i=1 Q(x,(2 i v) 1 m) f(y) P s,v f(y) dy 2 n m [ ε n 1 1 ]i p p L L χq(x, 1 m) χ 1 (R n ) Q(x,m) Q(x, 1 m ) 1. (R n ) By his and (4.7), we furher conclude ha, for all x R n, (4.8) P s, (f P s,v f)(x) L χ 1 Q(x,m) Q(x, 1 m ) 1. (R n ) By an argumen similar o ha used in he proof of (4.8), we also see ha, for all x R n, L P s,v (f P s, f)(x) χ 1 Q(x,m) Q(x, 1 m ) 1, (R n ) which, combined wih (4.5) and (4.8), implies ha (4.4) holds rue. This finishes he proof of Proposiion 4.5. Proposiion 4.6. Le and s be as in Proposiion 4.5. Then, for any δ (n[ 1 p 1], ), here exiss a posiive consan C such ha, for allf BMO s,l(r n ), (0, ) and x R n, f(y) P s,f(y) dy R n ( 1 C m x y ) nδ nδ m Proof. For all (0, ) and x R n, we wrie f(y) P s,f(y) dy = R n ( 1 m x y ) nδ Q(x, 1 m) χ Q(x, 1 m) L (R n ) f BMO s,l (R n ). f(y) P s, f(y) dy ( 1 m x y ) =: I 1 I 2. nδ [Q(x,m)] 1 Obviously, by he Hölder inequaliy, we easily see ha I 1 nδ m f(y) P s, f(y) dy nδ L m χ 1 Q(x,m) BMO s (R ) f n,l (R n ). (4.9) Q(x, 1 m) To esimae I 2, we firs noice ha I 2 k=1 k=1 (2 k ) nδ m (2 k ) nδ m k=1 (2 k ) nδ m =: I 2,1 I 2,2. Q(x,(2 k ) 1 m) Q(x,(2 k ) 1 m) sup y Q(x,(2 k ) 1 m) f(y) P s, f(y) dy f(y) P s,2 k f(y) dy P s, f(y) P s,2 k f(y) Q(x,(2 k ) 1 m )

22 378 Dachun Yang and Ciqiang Zhuo By he Hölder inequaliy, Lemma 3.14 and he fac ha δ > n( 1 p 1), we find ha I 2,1 (2 k ) nδ m BMO s L (R ) f n,l (R n ) (4.10) k=1 nδ m nδ m χ Q(x,(2 k ) 1 m) χ Q(x, 1 m) L (R n ) f BMO s,l (R n ) χ Q(x, 1 m) L (R n ) f BMO s,l (R n ). k=1 2 k(nδ n p ) For I 2,2, by Proposiion 4.5 and Lemma 3.14, we know ha I 2,2 k(2 k ) nδ m sup BMO s L (R ) f n,l (R n ) k=1 k=1 nδ m k(2 k ) nδ m y Q(x,(2 k ) 1 m) χ Q(x,(2 k ) 1 m) χ Q(y,(2 k ) 1 m) BMO s L (R ) f n,l (R n ) χ Q(x, 1 m) L (R n ) f BMO s,l (R n ). This, ogeher wih (4.9) and (4.10), implies ha I 2 nδ m which complees he proof of Proposiion 4.6. χ Q(x, 1 m) L (R n ) f BMO s,l (R n ), Le P(R n ). Recall ha a measure dµ on R n1 is called a -Carleson measure if { Q 1/2 dµ := sup Q R n χ Q L (R n ) Q d µ } 1/2 <, where he supremum is aken over all cubes Q of R n and Q denoes he en over Q; see [55]. Proposiion 4.7. Le,s 0 andsbe as in Proposiion 4.5. Iff BMO s 0,L (Rn ), hen he measure dxd dµ f (x,) := Q s, m(i P s0,m)f(x) 2, (x,) R n1 is a -Carleson measure on R n1 and here exiss a posiive consan C, independen of f, such ha dµ f C f BMO s 0,L (R n ). Proof. Since s s 0 = n( 1 m p 1) and θ(l) (n[ 1 p 1], ) wih θ(l) as in (2.5), i follows ha here exissε (n[ 1 p 1],θ(L)) such ham(s1) > ε. To prove his proposiion, by definiion, i suffices o show ha, for any cube R := R(x R,r R ) wih some x R R n and r R (0, ), { R 1/2 dxd (4.11) Q s, m(i P s0,m)f(x) 2 χ R L (R n ) R Observe ha } 1/2 f BMO s0,l (Rn ). I P s0, m = (I P s 0, m)[ I P s0,(r R ) m ] Ps0,(r R ) m(i P s 0, m).

23 Molecular characerizaions and dualiies of variable exponen Hardy spaces 379 Then he esimae (4.11) is a direc consequence of { R 1/2 Qs, m(i P s0, χ R m)[ ] I P s0,(r R ) m f(x) 2 dxd (4.12) L (R n ) R f s BMO 0,L (R n ) and (4.13) { R 1/2 Q s, mp s0,(r χ R R ) m(i P dxd s 0,m)f(x) 2 L (R n ) R f s BMO 0,L (R n ). Nex we prove (4.12) and (4.13), respecively. To show (4.12), le b 1 := [ I P s0,(r R ) m ] fχ2r } 1/2 and b 2 := [ I P s0,(r R ) m ] fχr n \(2R). By [30, (4.25)], we know ha { (4.14) J := Q s, m(i P s0, m)b 1(x) 2 dxd } 1/2 b 1 L2(Rn), R } 1/2 which, combined wih Proposiion 4.5 and Lemma 3.14, implies ha { J [ }1 ] I P s0,(2r R ) m f(x) 2 2 dx R 1/2 sup P s0,(r R ) mf(x) P s 0,(2r R ) mf(x) x 2R 2R R 1/2 χ R L (R n ) f BMO s 0,L (R n ). For b 2, we wrie (4.15) Q s, m(i P s0, m)b 2 = Q s, mb 2 Q s, mp s0, mb 2. Le (x,) R. Then, by (2.3), (2.8) and Proposiion 4.6, we have ε [ ] Q s, mb 2 (x) I x y nε Ps0,(r R ) m f(y) dy which implies ha { (4.16) R R n \(2R) Rn ε (r R x y ) nε [ I Ps0,(r R ) m ] f(y) dy (/r R ) ε R 1 χ R L (R n ) f BMO s 0,L (R n ), Q s, mb 2 (x) 2 dxd } 1/2 R 1/2 χ R L(Rn) f BMO s0,l (Rn ). On he oher hand, for all k {1,...,s 0 1},, v (0, ), f M(R n ) and x R n, le ( ) d (4.17) Ψ k,v f(x) := [kvm m ] s1 s1 P η dη s1 f (x). kv m m Then, by Assumpion (A) and (2.4), we conclude haψ k,v, he kernel of Ψk,v, saisfies ha, for all, v (0, ) and x, y R n, (4.18) ψ k,v (x,y) v ε (v x y ) nε.

24 380 Dachun Yang and Ciqiang Zhuo From his, Proposiion 4.6 and he fac ha s 0 1 (4.19) P s0,v mf = ( 1) k1 Cs k 0 1 e kvml f, k=1 where Cs k 0 1 := (s 01)! k!(s 0, we deduce ha, for all (x,) R, 1 k)! s Q s, mp s0, mb 0 1 2(x) = ( 1) k C k m(s1) s 0 1 [(k 1) m ],b s1ψk 2 (x) k=1 ε [ ] I R n \(2R) ( x y ) nε Ps0,(r R ) m f(y) dy (4.20) ( ) εrn (r R ) ε [ ] I (r R x y ) nε Ps0,(r R ) m f(y) dy which furher implies ha { (4.21) R r R Q s, mp s0, mb 2(x) 2 dxd (/r R ) ε χ R L (R n ) f BMO s 0,L (R n ), } 1/2 R 1/2 χ R L(Rn) f BMO s0,l (Rn ). Combining (4.14), (4.15), (4.16) and (4.21), we conclude ha (4.12) holds rue. Similarly, by (4.17), (4.18), (4.19) and Proposiion 4.6, we also see ha, for all (x,) R, Qs, mp s0,(r R ) m(i P s 0, m)(f)(x) s 0 1 = ( 1) k Cs k m(s1) 0 1 [k(r R ) m m ] s1ψk,r R (I P s0, m)(f)(x) k=1 m(s1) Rn (r R ) ε [k(r R ) m m ] s1 (r R x y ) nε (I P s 0, m)(f)(y) dy ( ) m(s1) εrn ε r R ( x y ) nε (I P s 0, m)(f)(r) dy ( ) m(s1) ε n χ Q(x,) r L(Rn) f BMO s,l ), (Rn R which, ogeher wih Lemma 3.14, implies ha, for all (x,) R, ( ) m(s1) ε Qs, mp s0,(r R ) m(i P s 0, m)(f)(x) n χ R L(Rn) f BMO s ).,L (Rn By his and he fac ha m(s1) > ε, we furher conclude ha (4.13) holds rue. This finishes he proof of Proposiion 4.7. r R Proposiion 4.8. (i) Le q (1, ) and q := q. Then, for all f q 1 T2(R q n1 ) and g T q ), R n1 f(y,)g(y,) dyd T (f)(x)t(g)(x)dx. R n

25 Molecular characerizaions and dualiies of variable exponen Hardy spaces 381 (ii) Le C log (R n ) saisfy p (0,1]. Then he dual space of T T 2, (R n1 ) in he following sense: for any h T (4.22) l h (f) := h(y,)f(y,) dyd R n1 2, (R n1 ) is ), he mapping is a bounded linear funcional on T ); conversely, if l is a bounded linear funcional ont ), henlhas he form as in (4.22) wih a unique h T 2, (Rn1 ). Moreover, h l T h 2, (Rn1 ) (T wih he implici )) posiive consans independen of h. Proof. To prove his proposiion, i suffices o show (ii), since (i) was already proved in [10, p.316, Theorem 2]. We firs show ha T2, (R n1 ) (T )). Le h T 2, (Rn1 ). Then, by he Hölder inequaliy and Remark 3.2(ii), we find ha, for any (, )-aom a suppored on Q wih some cube Q R n, h(x,)a(x,) dxd { Q 1/2 h(x,) 2 dxd } 1/2 (4.23) R n1 χ Q L (R n ) Q C (h) L (R n ) = h T ). 2, (Rn1 For anyf T ), by Lemma 3.3, we know ha, for almos every(x,) R n1, f(x,) = λ ja j (x,), where {λ j } and {a j } are as in Lemma 3.3 saisfying (3.4). From his, (4.23) and Remark 3.8, we deduce ha l h (f) λ j R n1 h(x,) a j (x,) dxd λ j h h T 2, (Rn1 ) T ) f 2, (Rn1 T ), which implies ha l h is a bounded linear funcional on T ) and l h (T )) h T 2, (Rn1 ). Nex, we prove ha (T )) T2, (R n1 ). Le l (T )). For all k N, le Õk := {(x,) R n1 : x k,1/k k}. Then {Õk} k N is a family of compac ses of R n1 and R n1 = k NÕk. Observe ha, for each k N, if f L ) wih supp f Õk, hen supp T (f) Ok := {x Rn : x 2k}. I follows, from he Hölder inequaliy, ha O k T(f)(x)dx O k 1/2 { O k 1/2 { O k Γ(x) f(y,) 2dyd f(y,) 2dyd Õ k n1 dx By his and he fac ha p (0,1], we furher find ha [ ] p(x) [ T (f)(x) dx R n O k 1 2 f L 2 (Õk) O k } 1/2 } 1/2 O k 1/2 f L 2 (Õk). 1 O k 1 2T (f)(x) f L 2 (Õk) ] p(x) dx O k,