Intrinsic Square Function Characterizations of Hardy Spaces with Variable Exponents

Transcription

1 Inrinsic Square Funcion Characerizaions of Hardy Spaces wih Variable Exponens Ciqiang Zhuo, Dachun Yang and Yiyu Liang School of Mahemaical Sciences, Beijing Normal Universiy, Laboraory of Mahemaics and Complex Sysems, Minisry of Educaion, Beijing 875, People s Republic of China cqzhuo@mail.bnu.edu.cn, dcyang@bnu.edu.cn, yyliang@mail.bnu.edu.cn Absrac Le p( ) : (, ) be a measurable funcion saisfying some decay condiion and some locally log-hölder coninuiy. In his aricle, via firs esablishing characerizaions of he variable exponen Hardy space H p( ) ( ) in erms of he Lilewood-Paley g-funcion, he Lusin area funcion and he gλ-funcion, he auhors hen obain is inrinsic square funcion characerizaions including he inrinsic Lilewood-Paley g-funcion, he inrinsic Lusin area funcion and he inrinsic gλ-funcion. The p( )-Carleson measure characerizaion for he dual space of H p( ) ( ), he variable exponen Campanao space L,p( ),s ( ), in erms of he inrinsic funcion is also presened. 2 Mahemaics Subjec Classificaion: Primary: 42B25, Secondary: 42B3, 42B35, 46E3 Keywords and phrases: Hardy space, variable exponen, inrinsic square funcion, Carlson measure, aom. Inroducion Variable exponen Lebesgue spaces are a generalizaion of he classical L p ( ) spaces, in which he consan exponen p is replaced by an exponen funcion p( ) : (, ), namely, hey consis of all funcions f such ha f(x) p(x) dx <. These spaces were inroduced by Birnbaum-Orlicz [3] and Orlicz [34], and widely used in he sudy of harmonic analysis as well as parial differenial equaions; see, for example, [, 2, 6, 7, 8,, 2, 3, 4, 2, 3, 43, 48, 5]. For a sysemaic research abou he variable exponen Lebesgue space, we refer he reader o [8, 3] Recenly, Nakai and Sawano [32] exended he heory of variable Lebesgue spaces via sudying he Hardy spaces wih variable exponens on, and Sawano in [35] furher gave more applicaions of hese variable exponen Hardy spaces. Independenly, Cruz-Uribe and Wang in [9] also invesigaed he variable exponen Hardy space wih some weaker condiions han hose used in [32], which also exends he heory of variable exponen Lebesgue spaces. Recall ha he classical Hardy spaces H p ( ) wih p (, ] on he Euclidean space and heir duals are well sudied (see, for example, [, 38]) and have been playing an imporan and fundamenal role in various fields of analysis such as harmonic analysis and parial differenial equaions; see, for example, [4, 3]. Communicaed by Rosihan M. Ali, Dao. Received: January 28, 24; Revised: March 26, 24. Corresponding auhor

2 2 Ciqiang Zhuo, Dachun Yang and Yiyu Liang On he oher hand, he sudy of he inrinsic square funcion on funcion spaces, including Hardy spaces, has recenly araced many aenions. To be precise, Wilson [44] originally inroduced inrinsic square funcions, which can be hough of as grand maximal square funcions of C. Fefferman and E. M. Sein from [], o sele a conjecure proposed by R. Fefferman and E. M. Sein on he boundedness of he Lusin area funcion S(f) from he weighed Lebesgue space L 2 M(v) (Rn ) o he weighed Lebesgue space L 2 v( ), where v L loc (Rn ) and M denoes he Hardy-Lilewood maximal funcion. The boundedness of hese inrinsic square funcions on he weighed Lebesgue spaces L p ω( ), when p (, ) and ω belongs o Muckenhoup weighs A p ( ), was proved by Wilson [45]. The inrinsic square funcions dominae all square funcions of he form S(f) (and he classical ones as well), bu are no essenially bigger han any one of hem. Similar o he Fefferman-Sein and he Hardy-Lilewood maximal funcions, heir generic naures make hem poinwise equivalen o each oher and exremely easy o work wih. Moreover, he inrinsic Lusin area funcion has he disinc advanage of being poinwise comparable a differen cone openings, which is a propery long known no o hold rue for he classical Lusin area funcion; see Wilson [44, 45, 46, 47] and also Lerner [24, 25]. Laer, Huang and Liu in [9] obain he inrinsic square funcion characerizaions of he weighed Hardy space Hω(R n ) under he addiional assumpion ha f L ω( ), which was furher generalized o he weighed Hardy space Hω(R p n ) wih p (n/(n + α), ) and α (, ) by Wang and Liu in [42], under anoher addiional assumpion. Very recenly, Liang and Yang in [28] esablished he s-order inrinsic square funcion characerizaions of he Musielak-Orlicz Hardy space H ϕ ( ), which was inroduced by Ky [23] and generalized boh he Orlicz-Hardy space (see, for example, [2, 4]) and he weighed Hardy space (see, for example, [6, 36]), in erms of he inrinsic Lusin area funcion, he inrinsic g-funcion and he inrinsic gλ -funcion wih he bes known range λ (2 + 2(α + s)/n, ). More applicaions of such inrinsic square funcions were also given by Wilson [46, 47] and Lerner [24, 25]. Moivaed by [28], in his aricle, we esablish inrinsic square funcion characerizaions of he variable exponen Hardy space H p( ) ( ) inroduced by Nakai and Sawano in [32], including he inrinsic Lilewood-Paley g-funcion, he inrinsic Lusin area funcion and he inrinsic gλ -funcion by firs obaining characerizaions of H p( ) ( ) via he Lilewood-Paley g-funcion, he Lusin area funcion and he gλ -funcion. We also esablish he p( )-Carleson measure characerizaion for he dual space of H p( ) ( ), he variable exponen Campanao space L,p( ),s ( ) in [32], in erms of he inrinsic square funcion. To sae he resuls, we begin wih some noaion. In wha follows, for a measurable funcion p( ) : (, ) and a measurable se E of, le p (E) := ess inf p(x) and p +(E) := ess sup p(x). x E For simpliciy, we le p := p ( ), p + := p + ( ) and p := min{p, }. Denoe by P( ) he collecion of all measurable funcions p( ) : (, ) saisfying < p p + <. For p( ) P( ), he space is defined o be he se of all measurable funcions such ha { [ ] p(x) f(x) f := inf λ (, ) : dx } <. λ Remark.. I was poined ou in [32, p. 367] (see also [8, Theorem 2.7]) ha he follows hold rue: (i) f, and f = if and only if f(x) = for almos every x ; (ii) λf = λ f for any λ C; x E (iii) f + g l f l + g l for all l (, p ];

3 Inrinsic Square Funcion Characerizaions 3 (iv) for all measurable funcions f wih f, [ f(x) / f ] p(x) dx =. A funcion p( ) P( ) is said o saisfy he locally log-hölder coninuous condiion if here exiss a posiive consan C such ha, for all x, y and x y /2, (.) p(x) p(y) C log(/ x y ), and p( ) is said o saisfy he decay condiion if here exis posiive consans C and p such ha, for all x, (.2) p(x) p C log(e + x ). In he whole aricle, we denoe by S( ) he space of all Schwarz funcions and by S ( ) is opological dual space. Le S ( ) denoe he space of all Schwarz funcions ϕ saisfying ϕ(x)x β dx = for all muli-indices β Z n + := ({,,...}) n and S (R n ) is opological dual space. For N N := {, 2,... }, le (.3) F N ( ) := ψ S(Rn ) : ( + x ) N D β ψ(x) x, β Z n +, β N sup where, for β := (β,..., β n ) Z n +, β := β + + β n and D β := ( x ) β ( x n ) βn. Then, for all f S ( ), he grand maximal funcion fn,+ of f is defined by seing, for all x Rn, f N,+(x) := sup { f ψ (x) : (, ) and ψ F N ( )}, where, for all (, ) and ξ, ψ (ξ) := n ψ(ξ/). For any measurable se E and r (, ), le L r (E) be he se of all measurable funcions f such ha f Lr (E) := { E f(x) r dx } /r <. For r (, ), denoe by L r loc ( ) he se of all r-locally inegrable funcions on. Recall ha he Hardy-Lilewood maximal operaor M is defined by seing, for all f L loc (Rn ) and x, M(f)(x) := sup B x B B f(y) dy, where he supremum is aken over all balls B of conaining x. Now we recall he noion of he Hardy space wih variable exponen, H p( ) ( ), inroduced by Nakai and Sawano in [32]. For simpliciy, we also call H p( ) ( ) he variable exponen Hardy space. Definiion.2. Le p( ) P( ) saisfy (.) and (.2), and ( ) n (.4) N + n +, N. p The Hardy space wih variable exponen p( ), denoed by H p( ) ( ), is defined o be he se of all f S ( ) such ha f N,+ Lp( ) ( ) wih he quasi-norm f H p( ) ( ) := f N,+. Remark.3. (i) Independenly, Cruz-Uribe and Wang in [9] inroduced he variable exponen Hardy space, denoed by H p( ), in he following way: Le p( ) P( ) saisfy ha here exis p (, p ) and a posiive consan C, only depending on n, p( ) and p, such ha (.5) M(f) L p( )/p (Rn ) C f L p( )/p ( ).

4 4 Ciqiang Zhuo, Dachun Yang and Yiyu Liang If N (n/p + n +, ), hen he variable exponen Hardy space H p( ) is defined o be he se of all f S ( ) such ha f N,+ Lp( ) ( ). In [9, Theorem 3.], i was shown ha he space H p( ) is independen of he choice of N (n/p + n +, ). (ii) We poin ou ha, in [32, Theorem 3.3], i was proved ha he space H p( ) ( ) is independen of N as long as N is sufficienly large. Alhough he range of N is no presened explicily in [32, Theorem 3.3], by he proof of [32, Theorem 3.3], we see ha N as in (.4) does he work. Le φ S( ) be a radial real-valued funcion saisfying (.6) supp φ {ξ : /2 ξ 2} and (.7) φ(ξ) C if 3/5 ξ 5/3, where C denoes a posiive consan independen of ξ and, for all φ S( ), φ denoes is Fourier ransform. Obviously, φ S ( ). Then, for all f S (R n ), he Lilewood-Paley g-funcion, he Lusin area funcion and he gλ -funcion wih λ (, ) of f are, respecively, defined by seing, for all x, { g(f)(x) := f φ (x) 2 d } /2, { } /2 2 dy d S(f)(x) := φ f(y) and { (.8) gλ(f)(x) := {y : y x <} ( ) λn φ f(y) + x y n+ 2 dy d n+ For all f S ( ) and φ S( ) saisfying (.6) and (.7), we le, for all (, ), j Z, a (, ) and x, } 2. (φ φ f(x + y) f) a (x) := sup y ( + y /) a and (φ φ j f(x + y) j f) a (x) := sup y ( + 2 j y ) a. Then, for all f S ( ), a (, ) and x, define { g a, (f)(x) := } /2 [(φ f) a (x)] 2 d and σ a, (f)(x) := The following conclusion is he firs main resul of his aricle. [ (φ j f) a (x) ] 2 Theorem.4. Le p( ) P( ) saisfy (.) and (.2). Then f H p( ) ( ) if and only if f S (R n ) and S(f) ; moreover, here exiss a posiive consan C, independen of f, such ha C S(f) f H p( ) ( ) C S(f) L p( ) (R ). n The same is rue if S(f) is replaced, respecively, by g(f), g a, (f) and σ a, (f) wih a (n/ min{p, 2}, ). j Z /2.

5 Inrinsic Square Funcion Characerizaions 5 Corollary.5. Le p( ) P( ) and λ ( + 2/min{2, p }, ). Then f H p( ) ( ) if and only if f S ( ) and g λ (f) Lp( ) ( ); moreover, here exiss a posiive consan C, independen of f, such ha C g λ (f) f H p( ) ( ) C g λ (f). Remark.6. (i) We poin ou ha he conclusion of Theorem.4 is undersood in he following sense: if f H p( ) ( ), hen f S (R n ) and here exiss a posiive consan C such ha, for all f H p( ) ( ), S(f) C f H p( ) (R ); conversely, if f S (R n ) n and S(f), hen here exiss a unique exension f S ( ) such ha, for all h S ( ), f, h = f, h and f H p( ) ( ) C S(f) wih C being a posiive consan independen of f. In his sense, we idenify f wih f. (ii) Recall ha, Hou e al. [8] characerized he Musielak-Orlicz Hardy space H ϕ ( ), which was inroduced by Ky [23], via he Lusin area funcion, and Liang e al. [26] esablished he Lilewood-Paley g-funcion and he g λ -funcion characerizaions of Hϕ ( ). Observe ha, when (.9) ϕ(x, ) := p(x) for all x and [, ), hen H ϕ ( ) = H p( ) ( ). However, a general Musielak-Orlicz funcion ϕ saisfying all he assumpions in [23] (and hence [8, 26]) may no have he form as in (.9). On he oher hand, i was proved in [49, Remark 2.23(iii)] ha here exiss an exponen funcion p( ) saisfying (.) and (.2), bu p( ) is no a uniformly Muckenhoup weigh, which was required in [23] (and hence [8, 26]). Thus, he Musielak-Orlicz Hardy space H ϕ ( ) in [23] (and hence in [8, 26]) and he variable exponen Hardy space H p( ) ( ) in [32] (and hence in he presen aricle) can no cover each oher. Moreover, Liang e al. [26, Theorem 4.8] esablished he gλ -funcion characerizaion of he Musielak-Orlicz Hardy space H ϕ ( ) wih he bes known range for λ. In paricular, in he case of he classical Hardy space H p ( ), λ (2/min{p, 2}, ); see, for example, [5, p. 22, Corollary 7.4] and [37, p. 9, Theorem 2]. However, i is sill unclear wheher he gλ -funcion, when λ (2/min{p, 2}, + 2/min{p, 2}], can characerize H p( ) ( ) or no, since he mehod used in [26, Theorem 4.8] srongly depends on he properies of uniformly Muckenhoup weighs, which are no saisfied by p( ). Indeed, a key fac ha used in he proof of [26, Theorem 4.8], which may no hold in he presen seing, is ha, if ϕ is a Musielak-Orlicz funcion as in [26], hen here exiss a posiive consan C such ha, for all λ (, ), α (, ) and measurable se E, ϕ(x, λ) dx Cα nq ϕ(x, λ) dx, U(E;α) where U(E; α) := {x : M(χ E )(x) > α} and q [, ) is he uniformly Muckenhoup weigh index of ϕ. To see his, following [32, Example.3], for all x R, le { p(x) := max e 3 x, min ( 6/5, max{/2, 3/2 x 2 } )}. Then p( ) saisfies (.) and (.2). Now, le E := (, 2), hen, for all x R, M(χ E )(x) = χ E (x) + E + 2 x 3/2 χ R\E(x). I is easy o see ha, for all λ (, ), E λp(x) dx = λ /2 and U(E;/) λ p(x) dx = λ p(x) dx > E λ p(x) 2 dx + λ p(x) dx = λ /2 + λ 6/5. 2

6 6 Ciqiang Zhuo, Dachun Yang and Yiyu Liang Thus, we find ha lim λ U(E;/) λp(x) dx E λp(x) dx =, which implies ha here does no exis a posiive consan C, independen of λ, such ha, λ p(x) dx C λ p(x) dx. U(E;/) Thus, he mehod used in he proof of [26, Theorem 4.8] is no suiable for he presen seing. For any s Z +, C s ( ) denoes he se of all funcions having coninuous classical derivaives up o order no more han s. For α (, ] and s Z +, le C α,s ( ) be he family of funcions φ C s ( ) such ha supp φ {x : x }, φ(x)x γ dx = for all γ Z n + and γ s, and, for all x, x 2 and ν Z n + wih ν = s, (.) D ν φ(x ) D ν φ(x 2 ) x x 2 α. For all f L loc (Rn ) and (y, ) + + := (, ), le A α,s (f)(y, ) := E sup f φ (y). φ C α,s( ) Then, he inrinsic g-funcion, he inrinsic Lusin area inegral and he inrinsic gλ -funcion of f are, respecively, defined by seing, for all x and λ (, ), { g α,s (f)(x) := } /2 [A α,s (f)(x, )] 2 d, and { S α,s (f)(x) := { gλ,α,s(f)(x) := {y : y x <} } /2 [A α,s (f)(y, )] 2 dy d n+ ( ) } λn /2 2 dy d [A α,s(f)(y, )] + x y n+. We also recall anoher kind of similar-looking square funcions, defined via convoluions wih kernels ha have unbounded suppors. For α (, ], s Z + and ɛ (, ), le C (α,ɛ),s ( ) be he family of funcions φ C s ( ) such ha, for all x, γ Z n + and γ s, D γ φ(x) ( + x ) n ɛ, φ(x)x γ dx = and, for all x, x 2, ν Z n + and ν = s, (.) D ν φ(x ) D ν φ(x 2 ) x x 2 α [ ( + x ) n ɛ + ( + x 2 ) n ɛ]. Remark ha, in wha follows, he parameer ɛ usually has o be chosen o be large enough. For all f saisfying (.2) f( ) ( + ) n ɛ L ( ) and (y, ) + +, le (.3) Ã (α,ɛ),s (f)(y, ) := sup f φ (y). φ C (α,ɛ),s ( )

7 Inrinsic Square Funcion Characerizaions 7 Then, for all x and λ (, ), we le and { ] } 2 /2 d g (α,ɛ),s (f)(x) := [Ã(α,ɛ),s (f)(x, ), { S (α,ɛ),s (f)(x) := { g λ,(α,ɛ),s (f)(x) := {y : y x <} ] 2 dy d [Ã(α,ɛ),s (f)(y, ) n+ } /2 ( ) } λn /2 2 dy d [Ã(α,ɛ),s(f)(y, )] + x y n+. These inrinsic square funcions, when s =, were original inroduced by Wilson [44], which were furher generalized o s Z + by Liang and Yang [28]. In wha follows, for any r Z +, we use P r ( ) o denoe he se of all polynomials on wih order no more han r. We now recall he noion of he Campanao space wih variable exponen, which was inroduced by Nakai and Sawano in [32]. Definiion.7. Le p( ) P( ), s be a nonnegaive ineger and q [, ). Then he Campanao space L q,p( ),s ( ) is defined o be he se of all f L q loc (Rn ) such ha f Lq,p( ),s ( ) := sup Q Q χ Q [ ] f(x) P s Q Qf(x) q q dx <, Q where he supremum is aken over all cubes Q of and PQ s g denoes he unique polynomial P P s ( ) such ha, for all h P s ( ), [f(x) P (x)]h(x) dx =. Q Now we sae he second main resul of his aricle. Recall ha f S ( ) is said o vanish weakly a infiniy, if, for every φ S( ), f φ in S ( ) as ; see, for example, [5, p. 5]. Theorem.8. Le p( ) P( ) saisfy (.), (.2) and p + (, ]. Assume ha α (, ], s Z + and p (n/n + α + s, ]. Then f H p( ) ( ) if and only if f (L,p( ),s ( )), he dual space of L,p( ),s ( ), f vanishes weakly a infiniy and g α,s (f) ; moreover, i holds rue ha C g α,s(f) f H p( ) ( ) C g α,s (f) wih C being a posiive consan independen of f. The same is rue if g α,s (f) is replaced by g (α,ɛ),s (f) wih ɛ (α + s, ). Observe ha, for all x, S α,s (f)(x) and g α,s (f)(x) as well as S (α,ɛ),s (f)(x) and g (α,ɛ),s (f)(x) are poinwise comparable (see [28, Proposiion 2.4]), which, ogeher wih Theorem.8, immediaely implies he following Corollary.9. Corollary.9. Le p( ) P( ) saisfy (.), (.2) and p + (, ]. Assume ha α (, ], s Z + and p (n/(n + α + s), ]. Then f H p( ) ( ) if and only if f (L,p( ),s ( )), f vanishes weakly a infiniy and S α,s (f) ; moreover, i holds rue ha C S α,s(f) f H p( ) ( ) C S α,s (f) wih C being a posiive consan independen of f. The same is rue if S α,s (f) is replaced by S (α,ɛ),s (f) wih ɛ (α + s, ).

8 8 Ciqiang Zhuo, Dachun Yang and Yiyu Liang Theorem.. Le p( ) P( ) saisfy (.), (.2) and p + (, ]. Assume ha α (, ], s Z +, p (n/(n + α + s), ] and λ (3 + 2(α + s)/n, ). Then f H p( ) ( ) if and only if f (L,p( ),s ( )), f vanishes weakly a infiniy and gλ,α,s (f) Lp( ) ( ); moreover, i holds rue ha C g λ,α,s(f) f H p( ) ( ) C gλ,α,s(f) wih C being a posiive consan independen of f. The same is rue if gλ,α,s (f) is replaced by g λ,(α,ɛ),s (f) wih ɛ (α + s, ). Remark.. (i) We poin ou ha here exiss a posiive consan C such ha, for all φ C α,s ( ), Cφ C (α,ɛ),s ( ) and hence φ L,p( ),s ( ); see Lemma 2.8 below. Thus, he inrinsic square funcions are well defined for funcionals in (L,p( ),s ( )). Observe ha, if φ S( ), hen φ L,p( ),s ( ); see also Lemma 2.8 below. Therefore, if f (L,p( ),s ( )), hen f S ( ) and f vanishing weakly a infiniy makes sense. (ii) Recall ha Liang and Yang [28] characerized he Musielak-Orlicz Hardy space H ϕ ( ) in erms of he inrinsic square funcions original inroduced by Wilson [44]. Moreover, Liang and Yang [28] esablished he inrinsic gλ -funcions g λ,α,s and g λ,(α,ɛ),s wih he bes known range λ (2 + 2(α + s)/n, ) via some argumen similar o ha used in he proof of [26, Theorem 4.8]. However, i is sill unclear wheher he inrinsic gλ -funcions g λ,α,s and g λ,(α,ɛ),s, when λ (2 + 2(α + s)/n, 3 + 2(α + s)/n], can characerize Hp( ) ( ) or no. Based on he same reason as in Remark.6(ii), we see ha he mehod used in he proof of [28, Theorem.8] is no available for he presen seing. (iii) Le p (, ]. When (.4) p(x) := p for all x, hen H p( ) ( ) = H p ( ). In his case, Theorem.8 and Corollary.9 coincide wih he corresponding resuls of he classical Hardy space H p ( ); see [28, Theorem.6] and [28, Corollary.7]. (iv) We also poin ou ha he mehod used in his aricle does no work for he variable exponen Hardy space invesigaed by Cruz-Uribe and Wang in [9], since i srongly depends on he locally log-hölder coninuiy condiion (.) and he decay condiion (.2) of p( ). Thus, i is sill unknown wheher he variable exponen Hardy space in [9] has any inrinsic square funcion characerizaions or no. Definiion.2. Le p( ) P( ). A measure dµ on + + is called a p( )-Carleson measure if Q /2 { /2 dµ p( ) := sup dµ(x, ) } <, Q χ Q Q where he supremum is aken over all cubes Q and Q denoes he en over Q, namely, Q := {(y, ) + + : B(x, ) Q}. Theorem.3. Le p( ) P( ) saisfy (.) and (.2). Assume ha p + (, ], s Z +, p (n/(n + s + ), ] and φ S( ) is a radial funcion saisfying (.6) and (.7). (i) If b L,p( ),s ( ), hen dµ(x, ) := φ b(x) 2 dxd for all (x, ) + + is a p( )-Carleson measure on + + ; moreover, here exiss a posiive consan C, independen of b, such ha dµ p( ) C b L,p( ),s ( ).

9 Inrinsic Square Funcion Characerizaions 9 (ii) If b L 2 loc (Rn ) and dµ(x, ) := φ b(x) 2 dxd for all (x, ) + + is a p( )-Carleson measure on + +, hen b L,p( ),s ( ) and, moreover, here exiss a posiive consan C, independen of b, such ha b L,p( ),s ( ) C dµ p( ). In wha follows, for α (, ], s Z +, ɛ (, ) and b L,p( ),s ( ), he measure µ b on + + is defined by seing, for all (x, ) + +, (.5) dµ b (x, ) := [Ã(α,ɛ),s(b)(x, )] 2 dxd, where à (α,ɛ),s (b) is as in (.3) wih f replaced by b. Theorem.4. Le α (, ], s Z +, ɛ (α+s, ), p( ) P( ) saisfy (.), (.2), p + (, ] and p (n/(n + α + s), ]. (i) If b L,p( ),s ( ), hen dµ b as in (.5) is a p( )-Carleson measure on + + ; moreover, here exiss a posiive consan C, independen of b, such ha dµ b p( ) C b L,p( ),s ( ). (ii) If b L 2 loc (Rn ) and dµ b as in (.5) is a p( )-Carleson measure on + +, hen i follows ha b L,p( ),s ( ); moreover, here exiss a posiive consan C, independen of b, such ha b L,p( ),s ( ) C dµ b p( ). Remark.5. (i) Fefferman and Sein [] shed some ligh on he igh connecion beween BMO-funcions and Carleson measures, which is he case of Theorem.3 when s = and p(x) := for all x. (ii) When p( ) is as in (.4) wih p (, ], Theorem.3 is already known (see [29, Theorem 4.2]). (iii) When p( ) is as in (.4) wih p (, ], Theorem.4 was obained in [28, Theorem.] wih p (n/(n + α + s), ]. Thus, he range of p in Theorem.4 is reasonable and he bes known possible, even in he case ha p( ) being as in (.4) wih p (, ]. This aricle is organized as follows. Secion 2 is devoed o he proofs of Theorems.4,.8,.,.3 and.4. To prove Theorem.4, we esablish an equivalen characerizaion of H p( ) ( ) via he discree Lilewood-Paley g- funcion (see Proposiion 2.3 below) by using he nonangenial maximal funcion characerizaion of H p( ) ( ) obained in his aricle and he Lilewood-Paley decomposiion of H p( ) ( ) which was proved in [32]. In he proof of Theorem.4, we also borrow some ideas from he proofs of [4, Theorem 2.8] (see also [27, Theorem 3.2]). The key ools used o prove Theorem.8 are he Lilewood-Paley g-funcion characerizaion of H p( ) ( ) in Theorem.4, he aomic decomposiion of H p( ) ( ) esablished in [32] (see also Lemma 2. below), he dual space of H p( ) ( ), L,p( ),s ( ), given in [32] and he fac ha he inrinsic square funcions are poinwise comparable proved in [28]. As an applicaion of Theorems.4 and.8, we give he proof of Theorem. via showing ha, for all x, he inrinsic square funcions S (α,ɛ),s (f)(x) and g λ,(α,ɛ),s (f)(x) are poinwise comparable under he assumpion λ (3 + 2(α + s)/n, ). The proof of Theorem.3 is similar o ha of [29, Theorem 4.2], which depends on aomic decomposiion of he en space wih variable exponen, he fac ha he dual space of H p( ) ( ) is L,p( ),s ( ) (see [32, Theorem 7.5]) and some properies of L,p( ),s ( ). To complee he proof of Theorem.3, we firs inroduce he en space wih variable exponen and obain is

10 Ciqiang Zhuo, Dachun Yang and Yiyu Liang aomic decomposiion in Theorem 2.6 below. Then we give an equivalen norm of L,p( ),s ( ) via esablishing a John-Nirenberg inequaliy for funcions in L,p( ),s ( ). A he end of Secion 2, we give he proof of Theorem.4 by using Theorem.3 and some ideas from he proof of [28, Theorem.]. Finally, we make some convenions on noaion. Throughou he paper, we denoe by C a posiive consan which is independen of he main parameers, bu i may vary from line o line. The symbol A B means A CB. If A B and B A, hen we wrie A B. If E is a subse of, we denoe by χ E is characerisic funcion. For any x and r (, ), le B(x, r) := {y : x y < r} be he ball. For β := (β,..., β n ) Z n +, le β! := β! β n!. For α R, we use α o denoe he maximal ineger no more han α. For a measurable funcion f, we use f o denoe is conjugae funcion. 2 Proofs of main resuls In wha follows, for all f S ( ) and N N, he nonangenial maximal funcion fn defined by seing, for all x, (2.) fn(x) := sup sup f ψ (y), ψ F N ( ) (, ) y x < where F N ( ) is as in (.3). The following proposiion is an equivalen characerizaion of H p( ) ( ). of f is Proposiion 2.. Le p( ) P( ) saisfy (.) and (.2), and N be as in (.4). Then f H p( ) ( ) if and only if f S ( ) and f N Lp( ) ( ); moreover, here exiss a posiive consan C such ha, for all f H p( ) ( ), C f H p( ) ( ) f N C f H p( ) ( ). Proof. Le f S ( ) and fn Lp( ) ( ). Observing ha, for all x, fn,+ (x) f N (x), we hen conclude ha f H p( ) ( ) = fn,+ fn and hence f H p( ) ( ). This finishes he proof of he sufficiency of Proposiion 2.. To prove he necessiy, we need o show ha, for all f H p( ) ( ), fn f H p( ) (R ). n To his end, for all Φ F N ( ), x, (, ) and y wih y x <, le, for all z, ψ(z) := Φ(z + (y x)/). Then we see ha β Z n +, β N sup z ( + z ) N D β ψ(z) = β Z n +, β N sup z which implies ha 2 N ψ F N ( ). From his, we deduce ha f Φ (y) = f ψ (x) 2 N f N,+(x), and hence f N (x) f N,+ (x) for all x Rn, which furher implies ha f N f N,+ f H p( ) ( ). This finishes he proof of he necessiy par and hence Proposiion 2.. ( + z y z N ) D β Φ(z) 2 N, Corollary 2.2. Le p( ) be as in Proposiion 2. and f H p( ) ( ). Then f vanishes weakly a infiniy.

11 Inrinsic Square Funcion Characerizaions Proof. Observe ha, for any f H p( ) ( ) wih f H p( ) ( ), φ S( ), x, (, ) and y B(x, ), f φ (x) fn (y), where f N is as in (2.) wih N as in (.4). By his and Remark.(iv), we see ha min{ f φ (x) p+, f φ (x) p } inf min{[f N(y)] p+, [fn (y)] p } y B(x,) min{[fn (y)] p+, [fn(y)] p } dy B(x, ) B(x,) B(x,) [ [f N (y)] p(y) dy B(x, ) f N (y) f N ] p(y) fn p(y) dy B(x, ) B(x, ) max{ f N p, f N p+ }, as, which implies ha f vanishes weakly a infiniy. This finishes he proof of Corollary 2.2. In wha follows, denoe by P poly ( ) he se of all polynomials on. For f S ( ) and φ S( ) saisfying (.6) and (.7), le σ(f)(x) := φ j f(x) 2 j Z /2 and { } Hσ p( ) ( ) := f S (R n ) : f p( ) H σ ( ) := σ(f) <. Proposiion 2.3. Le p( ) P( ) saisfy (.) and (.2). Then H p( ) ( ) = Hσ p( ) ( ) in he following sense: if f H p( ) ( ), hen f Hσ p( ) ( ) and here exiss a posiive consan C such ha, for all f H p( ) ( ), f p( ) H σ ( ) C f H p( ) (R ); conversely, if f H p( ) n σ ( ), hen here exiss a unique exension f S ( ) such ha, for all h S ( ), f, h = f, h and f H p( ) ( ) C f p( ) H wih C being a posiive consan independen of f. σ ( ) Proof. Le f H p( ) ( ). Then f S ( ) S (R n ) and, by [32, Theorem 5.7] (see also [35, Theorem 3.]), we see ha f p( ) H σ ( ) f H p( ) ( ) and hence f Hσ p( ) ( ). Conversely, le f Hσ p( ) ( ). Then f S (R n ). From [7, Proposiion ], we deduce ha here exiss f S ( ) such ha f f P poly ( ). By [32, Theorem 5.7] and he fac ha φ j f = φ j f for all j Z and φ as in definiion of σ(f), we know ha f H p( ) ( ) σ( f) σ(f) f H p( ) σ ( ), which implies ha f H p( ) ( ). Suppose ha here exiss anoher exension of f, for example, g H p( ) ( ). Then g S ( ) and g = f in S (R n ), which, ogeher wih [7, Proposiion ], implies g f P poly ( ). From his, g f H p( ) ( ) and Corollary 2.2, we deduce ha g = f since nonzero polynomials p( ) fail o vanish weakly a infiniy. Therefore, f is he unique exension of f H σ ( ), which complees he proof of Proposiion 2.3.

12 2 Ciqiang Zhuo, Dachun Yang and Yiyu Liang The following esimae is a special case of [27, Lemma 3.5], which is furher raced back o [4, (2.29)] and he argumen used in he proof of [4, Theorem 2.6] (see also [27, Theorem 3.2]), Lemma 2.4. Le f S ( ), N N and Φ S( ) saisfy (.6) and (.7). Then, for all [, 2], a (, N ], l Z and x, i holds rue ha [(Φ 2 l f) a(x)] r C (r) k= 2 knr 2 (k+l)n (Φ k+l) f(y) r ( + 2 l dy, x y ) ar where r is an arbirary fixed posiive number and C (r) a posiive consan independen of Φ, f, l,, bu may depend on r. We poin ou ha Lemma 2.4 plays an imporan role in he proof of Theorem.4. The following vecor-valued inequaliy on he boundedness of he Hardy-Lilewood maximal operaor M on he variable Lebesgue space was obained in [6, Corollary 2.]. Lemma 2.5. Le r (, ). Assume ha p( ) : [, ) is a measurable funcion saisfying (.), (.2) and < p p + <, hen here exiss a posiive consan C such ha, for all sequences {f j } j= of measurable funcions, /r /r (Mf j ) r C f j r. j= j= Proof of Theorem.4. We firs prove ha, for all f S ( ), (2.2) g(f) S(f) g a, (f) σ(f) σ a, (f) L p( ) (R ). n To prove (2.2), we firs show ha, for all f S ( ), (2.3) g(f) g a, (f) and σ(f) σ a, (f). For similariy, we only give he proof for he firs equivalence. By definiions, we easily see ha g(f) g a, (f). Conversely, we show ha g a, (f) g(f). Since a (n/min{p, 2}, ), i follows ha here exiss r (, min{p, 2}) such ha a (n/r, ). By Lemma 2.4 and he Minkowski inegral inequaliy, we find ha g a, (f)(x) = j Z j Z 2 2 [(φ 2 j f) a(x)] 2 d /2 [ 2 knr 2 (k+j)n k= [ 2 knr 2 (k+j)n j Z k= ] 2 (φ k+j) f(y) r r ( + 2 j dy x y ) ar 2 [ (φ k+j) f(y) 2 d ] r 2 ( + 2 j x y ) ar dy which, ogeher wih he Minkowski series inequaliy and Remark.(iii), implies ha d ] 2 r /2 /2, (2.4) g a, (f) r

13 Inrinsic Square Funcion Characerizaions 3 k= k= k= 2 k(nr n) 2 k(nr n) 2 k(nr n) y 2 i j j Z 2 j 2n r j Z [ 2 j Z 2 j 2n r 2 j 2n r [ [ [ [ ( i= 2 (φ k+j) f(y) 2 d ] r 2 ( + 2 j y ) ar dy 2 (φ k+j) f(y) 2 d ] r 2 ( + 2 j y ) ar dy 2 iar (φ k+j ) f(y) 2 d ] r 2 dy ) 2 r 2 r, ] 2 r ] 2 r r 2 p( ) L r ( ) r 2 where N N is sufficienly large and x y 2 i j means ha x y < 2 j if i =, or 2 i j x y < 2 i j if i N. Applying he Minkowski inequaliy and Lemma 2.5, we conclude ha g a, (f) r 2 knr+kn k= i= 2 knr+kn k= i= 2 iar+in 2 iar+in which complees he proof of (2.3). Nex we prove ha [ ( [ 2 M j Z [ 2 j Z (φ k+j ) f 2 d ] r 2 (φ k+j ) f 2 d ] 2 (2.5) S(f) g a, (f). 2 r )] 2 r 2 r L p( ) r ( ) g(f) r, I suffices o show ha g a, (f) S(f), since he inverse inequaliy holds rue rivially. From [27, (3.9)], we deduce ha g a, (f) [ = j Z i= k= y 2 i j 2 knr+2(k+j)n ( 2 z <2 (k+j) ) r (φ k+j ) f(y + z) 2 dzd 2 dy 2 r, where N N is sufficienly large and y 2 i j is he same as in (2.4). Then, by an argumen similar o ha used in he proof of (2.3), we conclude ha g a, (f) S(f), which complees he proof of (2.5).

14 4 Ciqiang Zhuo, Dachun Yang and Yiyu Liang By argumens similar o hose used in he proofs of (2.3), (2.5) and [4, Theorem 2.8], we conclude ha (2.6) σ(f) g a, (f) σ a, (f). Now, from (2.3), (2.5) and (2.6), we deduce ha (2.2) holds rue, which, ogeher wih Proposiion 2.3, implies ha f H p( ) ( ) if and only if f S ( ) and S(f) ; moreover, f H p( ) ( ) S(f). This finishes he proof of Theorem.4. Proof of Corollary.5. Assume f S (R n ) and gλ (f) Lp( ) ( ). I is easy o see ha, for all λ (, ) and x, S(f)(x) gλ (f)(x), which, ogeher wih Theorem.4, implies ha f H p( ) ( ) and f H p( ) ( ) S(f) gλ (f) L p( ) (R ). n Conversely, le f H p( ) ( ). Then f S ( ) S (R n ). By he fac ha λ ( + 2/min{2, p }, ), we see ha here exiss a (n/min{2, p }, ) such ha λ ( + 2a/n, ). Then, by his, we furher find ha, for all x, { ( ) } λn /2 gλ(f)(x) = φ f(y) 2 dy d R + x y n+ n { { [(φ f) a (x)] 2 [(φ f) a (x)] 2 d From his and Theorem.4, we deduce ha which complees he proof of Corollary.5. ( + ) } 2a λn /2 x y dy d n+ } /2 g a, (f)(x). g λ(f) g a, (f) f H p( ) ( ), To prove Theorem.8, we need more preparaions. The following echnical lemma is essenially conained in [32]. Lemma 2.6. Le p( ) P( ) saisfy (.) and (.2). Then here exiss a posiive consan C such ha, for all cubes Q Q 2, ( ) /p+ Q (2.7) χ Q C χ Q2 Q 2 Lp( )(Rn) and χ Q2 C ( ) /p Q2 χ Q Q Lp( )(Rn). Proof. For similariy, we only show (2.7). Le z Q. If l(q 2 ), hen, by [32, Lemma 2.2()] and is proof, we see ha χ Q χ Q2 ( ) Q Q 2 p(z ) ( Q Q 2 ) p +. If l(q ), hen by [32, Lemma 2.2(2)], we find ha χ Q χ Q2 ( ) Q Q 2 p ( Q Q 2 ) p +,

15 Inrinsic Square Funcion Characerizaions 5 where p is as in (.2). If l(q ) < < l(q 2 ), hen by [32, Lemma 2.2], we know ha χ Q χ Q2 Q /p(z) Q 2 /p ( ) Q p +, Q 2 which complees he proof of (2.7) and hence Lemma 2.6. The following Lemma 2.7 comes from [39, p.38]. Lemma 2.7. Le g L loc (Rn ), s Z + and Q be a cube in. Then here exiss a posiive consan C, independen of g and Q, such ha sup PQg(x) s C g(x) dx. x Q Q Lemma 2.8. Le α (, ], s Z + and ɛ (α + s, ). Assume ha p( ) P( ) saisfies (.), (.2) and p (n/(n + α + s), ]. If f C (α,ɛ),s ( ) or S( ), hen f L,p( ),s ( ). Proof. For similariy, we only give he proof for C (α,ɛ),s ( ). For any f C (α,ɛ),s ( ), x and cube Q := Q(x, r) wih (x, r) + +, le p Q (x) := β s Q D β f(x ) (x x ) β P s ( ). β! Then, from Lemma 2.7 and Taylor s remainder heorem, we deduce ha, for any x Q, here exiss ξ(x) Q such ha (2.8) f(x) PQf(x) s dx f(x) p Q (x) dx + PQ(p s Q f)(x) dx Q Q Q f(x) p Q (x) dx Q D β f(ξ(x)) D β f(x ) (x x ) β Q β! dx. β =s Now, if x + r, namely, Q Q(, n), hen, by Lemma 2.6, (2.8), (.) and he fac ha p (n/(n + α + s), ], we see ha (2.9) f(x) P χ Q Qf(x) s dx Q sup D β f(x) D β f(y) x, y,x y x y α ξ(x) x α x x s dx χ Q Q β =s Q(, n) /p Q +(α+s)/n /p. χ Q(, n) If x + r > and x 2r, hen r > /3 and Q Q(, n( x + r)). From Lemma 2.7 and f(x) ( + x ) n ɛ for all x, we deduce ha (2.) f(x) P χ Q Qf(x) s dx Q

16 6 Ciqiang Zhuo, Dachun Yang and Yiyu Liang [ f(x) dx sup ( + y ) n+ε f(y) ] χ Q Q y χ Q [ ] /p Q(, n( x + r)). Q χ Q(, n( x +r)) Q dx ( + x ) n+ε If x + r > and x > 2r, hen, for all x Q, i holds ha x x. By his, (2.8), Lemma 2.6 and (.), we find ha (2.) f(x) P χ Q Qf(x) s dx Q ξ(x) x α ( + x ) n ɛ x x s dx χ Q Q α+s Q + n ( x + r) n ɛ χ Q ( ) Q +(α+s)/n Q(, n( x p + r)) ( x + r) n+ɛ Q χ Q(, n( x +r)) Combining (2.9), (2.) and (2.), we see ha f L,p( ),s ( ), which complees he proof of Lemma 2.8. Remark 2.9. We poin ou ha, from he proof of Lemma 2.8, we know ha C (α,ε),s ( ) and S( ) are coninuously embedding ino L,p( ),s ( ), which, in he case of s = and p(x) := for all x, was proved in [33, Proposiion 2.]. Indeed, by he proof of Lemma 2.8, we see ha, for all f C (α,ε),s ( ) or S( ), f L,p( ),s ( ) sup ( + x ) n+ε f(x) x { [ ] } + sup ( + x ) n+ε + D β f(x) D β f(y) ( + y ) n+ε x y α ; x, y,x y β =s moreover, if f C (α,ε),s ( ), hen f L,p( ),s ( ) ; if f S( ), hen f L,p( ),s ( ) sup x β Z n +, β s+ ( + x ) n+ε D β f(x). In his sense, C (α,ε),s ( ) and S( ) are coninuously embedding ino L,p( ),s ( ). Now we recall he aomic Hardy space wih variable exponen inroduced by Nakai and Sawano [32]. Le p( ) P( ), s (n/p n, ) Z + and q [, ] saisfy ha q [p +, ). Recall ha a measurable funcion a on is called a (p( ), q, s)-aom if i saisfies he following hree condiions: (i) supp a Q for some Q ; (ii) a Q /q χ Q ; (iii) a(x)x β dx = for any β Z n + and β s..

17 Inrinsic Square Funcion Characerizaions 7 The aomic Hardy space wih variable p( ), denoed by H p( ),q aom ( ), is defined o be he se of all f S ( ) ha can be represened as a sum of muliples of (p( ), q, s)-aoms, namely, f = j λ ja j in S ( ), where, for each j, λ j is a nonnegaive number and a j is a (p( ), q, s)-aom suppored in some cube Q j wih he propery ( λj χ Qj (x) j χ Qj ) p p(x) p dx < wih p := min{p, }. The norm of f H p( ),q aom ( ) is defined by f p( ),q H := inf aom (Rn ) A({λ j} j, {Q j } j ) : f = λ j a j in S ( ), j where he infimum is aken over all decomposiions of f as above and A({λ j } j, {Q j } j ) := inf λ (, ) : j p(x) [ ] p p λ j χ Qj (x) dx λ χ Qj. The following conclusion is jus [32, Lemma 4.]. Lemma 2.. Le p( ) P( ) saisfy (.) and (.2). Then here exis β (, ) and a posiive consan C such ha, if q (, ) saisfies /q (, log 2 β /(n + )), hen, for all sequences {λ j } j of nonnegaive numbers, measurable funcions {b j } j and cubes {Q j } j saisfying supp b j Q j and b j L q (Q j) for each j, ( ) p λ j b j Q j /q p CA({λ b j j Lq (Q j) χ Qj j } j, {Q j } j ). Le q [, ] and s Z +. Denoe by L q,s comp( ) he se of all funcions f L ( ) wih compac and { L q,s comp( ) := f L q comp( ) : } f(x)x α dx =, α s. As poin ou in [32, p. 377], L q,s comp( ) is dense in H p( ),q aom ( ). The conclusions of he following Lemmas 2. and 2.2 were, respecively, jus [32, Theorems 4.6] and [32, Theorem 7.5], which play key roles in he proof of Theorem.4. Lemma 2.. Le q [, ] and p( ) P( ) saisfy (.), (.2) and p + (, q). Assume ha q is as in Lemma 2.. Then H p( ) ( ) = H p( ),q aom ( ) wih equivalen quasi-norms. Lemma 2.2. Le p( ) P( ) saisfy (.), (.2), p + (, ], q (p +, ) and s (n/p n, ) Z +. Then he dual space of H p( ),q aom ( ), denoed by (H p( ),q aom ( )), is L q,p( ),s( ) in he following sense: for any b L q,p( ),s( ), he linear funcional (2.2) l b (f) := b(x)f(x) dx,

18 8 Ciqiang Zhuo, Dachun Yang and Yiyu Liang iniial defined for all f L q,s comp( ), has a bounded exension o H p( ),q aom ( ); conversely, if l is a bounded linear funcional on H p( ),q aom ( ), hen l has he form as in (2.2) wih a unique b L q,p( ),s( ). Moreover, b Lq,p( ),s ( ) l b p( ),q (H, aom (Rn )) where he implici posiive consans are independen of b. The following Lemma 2.3 is jus from [28, Theorem 2.6], which, in he case when s =, was firs proved by Wilson [44, Theorem 2]. Lemma 2.3. Le α (, ], s Z + and ɛ (max{α, s}, ). Then here exiss a posiive consan C such ha, for all f saisfying (.2) and x, C g α,s(f)(x) g (α,ɛ),s (f)(x) Cg α,s (f)(x). The following Lemma 2.4 is a special case of [28, Proposiion 3.2]. Lemma 2.4. Le α (, ], s Z + and q (, ). Then here exiss a posiive consan C such ha, for all measurable funcions f, [g α,s (f)(x)] q dx C f(x) q dx. Now we come o give a proof of Theorem.8. Proof of Theorem.8. For ɛ (α + s, ), by Lemma 2.3, we see ha g α,s (f) and g (α,ɛ),s (f) are poinwise comparable. Thus, o prove Theorem.8, i suffices o show ha he conclusion of Theorem.8 holds rue for he inrinsic square funcion g α,s (f). Le f (L,p( ),s ( )) vanish weakly a infiniy and g α,s (f). Then, by Lemma 2.8, we find ha f S ( ) S ( ). Noice ha, for all x, g(f)(x) g (α,ɛ),s (f)(x) g α,s (f)(x) (see Lemma 2.3), i follows ha g(f). From his and Theorem.4, we deduce ha here exiss a disribuion f S ( ) such ha f = f in S ( ), f H p( ) ( ) and f H p( ) ( ) g(f), which, ogeher wih Corollary 2.2 and he fac ha f vanishes weakly a infiniy, implies ha f = f in S ( ) and hence f H p( ) ( ) f H p( ) ( ) g(f) g α,s (f). This finishes he proof of he sufficiency of Theorem.4. I remains o prove he necessiy. Le f H p( ) ( ). Then, by Corollary 2.2, we see ha f vanishes weakly a infiniy and, by Lemmas 2. and 2.2, we have f (L,p( ),s ( )). If q (, ) is as in Lemma 2., hen, by Lemma 2., we know ha here exis a sequence {λ j } j of nonnegaive numbers and a sequence {a j } j of (p( ), q, s)-aoms, wih supp a j Q j for all j, such ha f = j λ ja j in S ( ) and also in H p( ) ( ) and, moreover (2.3) A({λ j } j, {Q j } j ) f H p( ) ( ). Thus, by Lemma 2.8, we find ha, for all φ C (α,ε),s ( ), f φ = j λ ja j φ poinwise and hence, for all x, g α,s (f)(x) j λ jg α,s (a j )(x). Now, for a (p( ), q, s)-aom a wih supp a Q := Q(x, r), we esimae g α,s (a). By Lemma 2.4, we find ha (2.4) Q /q g α,s (a) L q (2 nq) a L q ( ), Q

19 Inrinsic Square Funcion Characerizaions 9 here and hereafer, 2 nq denoes he cube wih he cener same as Q bu wih he side lengh 2 n imes Q. On he oher hand, for all x / 2 nq, by he vanishing momen condiion of a and (.), ogeher wih Taylor s remainder heorem, we see ha (2.5) a φ (x) = n ( ) x y a(y) φ D β ( ) φ( x x ) β x y dy R β! n β s Rn a(y) y x α+s ( r ) n+α+s n+α+s dy. χ Q Noice ha supp φ {x : x }. If x / 2 nq and φ a(x), hen, here exiss a y Q such ha x y / and hence x y x x x y > x x /2. From his and (2.5), we deduce ha [ ] 2 /2 d g α,s (a)(x) = sup a φ (x) φ C α,s( ) { r n+α+s χ Q ( r χ Q x x which implies ha (2.6) g α,s (f) λ j g α,s (a j )χ 2 nqj j =: I + I 2. x x 2 2(n+α+s) d } /2 ) n+α+s [M(χ Q)(x)] n+α+s n j χ Q + [M(χ Qj )] n+α+s n λ j χ Qj, For I, by aking b j := g α,s (a j )χ 2 nqj for each j in Lemma 2., (2.4) and Lemma 2.6, we conclude ha I λ j b j Q j q (2.7) b j j Lq (2 nq j) χ 2 nqj ( ) λ j b j Q j p /p q A({λ b j j L q (2 nq j) χ 2 nqj j } j, {Q j } j ). For I 2, leing θ := (n + α + s)/n, by Lemma 2.5 and p (n/(n + α + s), ), we find ha θ λ j [M(χ Qj )] θ θ I 2 λ j χ Qj χ j Qj χ j Qj L θp( ) ( ) ( ) p λ j χ p Qj A({λ χ j Qj j } j, {Q j } j ).

20 2 Ciqiang Zhuo, Dachun Yang and Yiyu Liang From his, ogeher wih (2.3), (2.6) and (2.7), we deduce ha which complees he proof of Theorem.8. g α,s (f) f H p( ) ( ), For s Z +, α (, ] and ɛ (, ), le C (α,ɛ),s (y, ), wih y and (, ), be he family of funcions ψ C s ( ) such ha, for all γ Z n +, γ s and x, D γ ψ(x) n γ ( + y x /) n ɛ, ψ(x)x γ dx = and, for all x, x 2, ν Z n + and ν = s, [ ( D ν ψ(x ) D ν ψ(x 2 ) x x 2 α n+γ+α + y x ) n ɛ ( + + y x ) ] n ɛ 2. The proof of Theorem. needs he following Lemma 2.5, whose proof is rivial, he deails being omied. Lemma 2.5. Le s Z +, α (, ], ɛ (, ) and f be a measurable funcion saisfying (.2). (i) For any y and (, ), i holds rue ha à (α,ɛ),s (f)(y, ) = sup ψ C (α,ɛ),s (y,) ψ(x)f(x) dx. (ii) If, 2 (, ), < 2, y and ψ C (α,ɛ),s (y, ), hen ( 2 ) n+s+α ψ C (α,ɛ),s (y, 2 ). Proof of Theorem.. If f (L,p( ),s ( )), g λ,(α,ɛ),s (f) Lp( ) ( ) and f vanishes weakly a infiniy, hen, by Lemma 2.8, we see ha f S ( ) and, by he fac ha, for all x, g λ(f)(x) g λ,α,s(f)(x) g λ,(α,ɛ),s (f)(x) and Theorem.4, we furher know ha f H p( ) ( ) and f H p( ) ( ) g λ(f) g λ,α,s(f) g λ,(α,ɛ),s (f). This finishes he proof of he sufficiency of Theorem.. Nex we prove he necessiy of Theorem.. Le f H p( ) ( ). Then, as in he proof of Theorem.8, we see ha f (L,p( ),s ( )) and f vanishes weakly a infiniy. For all x, we have (2.8) [ g λ,(α,ɛ),s (f)(x)]2 ( = + k= y x < [ S (α,ɛ),s (f)(x)] 2 + [ S (α,ɛ),s (f)(x)] x y 2 k y x <2 k ) λn 2 dy d [Ã(α,ɛ),s(f)(y, )] n+ dy d n+ 2 kλn k= 2 kλn 2 kn k= [Ã(α,ɛ),s(f)(y, )] y x <2 k y x < 2 dy d n+ [Ã(α,ɛ),s(f)(y, 2 k 2 dy d )] n+.

21 Inrinsic Square Funcion Characerizaions 2 By Lemma 2.5, we find ha, for all k N and (y, ) + Ã (α,ɛ),s (f)(y, 2 k ) = sup ψ C (α,ɛ),s (y,2 k ) 2 k(n+s+α) +, ψ(x)f(x) dx sup ψ(x)f(x) dx = 2k(n+s+α) Ã (α,ɛ),s (f)(y, ), ψ C (α,ɛ),s (y,) which, ogeher wih (2.8) and λ (3 + 2(s + α)/n, ), implies ha [ g λ,(α,ɛ),s (f)(x)]2 2 kλn 2 k(3n+2s) [ S (α,ɛ),s (f)(x)] 2 [ S (α,ɛ),s (f)(x)] 2. k= From his, ogeher wih Theorem.9, we deduce ha g λ,(α,ɛ),s (f) Lp( ) ( ) and gλ,α,s(f) g λ,(α,ɛ),s (f) S (α,ɛ),s (f) f H p( ) (R ), n which complees he proof of Theorem.. To prove Theorem.3, we firs inroduce he en space wih variable exponen. measurable funcions g on + + and x, define { A(g)(x) := {y : y x <} } /2 2 dy d g(y, ). n+ For all Recall ha a measurable funcion g is said o belong o he en space T p 2 (Rn+ + ) wih p (, ), if g T p 2 (Rn+ + ) := A(g) L p ( ) <. Le p( ) P( ) saisfy (.) and (.2). In wha follows, we denoe by T p( ) 2 (+ + ) he space of all measurable funcions g on + + such ha A(g) and, for any g T p( ) 2 (+ + ), is quasi-norm is defined by { ( ) p(x) A(g)(x) g p( ) T 2 (+ + ) := A(g) := inf λ (, ) : dx }. λ Le p (, ). A funcion a on + + is called a (p( ), p)-aom if here exiss a cube Q such ha supp a Q and a T p 2 (Rn+ + ) Q /p χ Q. Furhermore, if a is a (p( ), p)-aom for all p (, ), we hen call a a (p( ), )-aom. For funcions in he space T p( ) 2 (+ + ), we have he following aomic decomposiion. Theorem 2.6. Le p( ) P( ) saisfy (.) and (.2). Then, for any f T p( ) 2 (+ + ), here exis {λ j } j C and a sequence {a j } j of (p( ), )-aoms such ha, for almos every (x, ) + +, f(x, ) = j λ ja j (x, ). Moreover, here exiss a posiive consan C such ha, for all f T p( ) 2 (+ + ), A ({λ j } j, {Q j } j ) C f p( ) T ), where (2.9) 2 (+ + A ({λ j } j, {Q j } j ) := inf λ (, ) : and, for each j, Q j appears in he suppor of a j. j Q j [ λ j λ χ Qj ] p(x) dx

22 22 Ciqiang Zhuo, Dachun Yang and Yiyu Liang Remark 2.7. Assume ha p + (, ]. Then, by [32, Remark 4.4], we know ha, for any sequences {λ j } j of nonnegaive numbers and cubes {Q j } j, j λ j A ({λ j } j, {Q j } j ). The proof of Theorem 2.6 is similar o ha of [8, Theorem 3.2] (see also [22, Theorem 3.]). To his end, we need some known facs as follows (see, for example, [22, Theorem 3.]). Le F be a closed subse of and O := \F =: F. Assume ha O <. For any fixed γ (, ), x is said o have he global γ-densiy wih respec o F if, for all (, ), B(x, ) F / B(x, ) γ. Denoe by F γ he se of all such x and le O γ := (F γ ). Then O γ = {x : M(χ O )(x) > γ} is open, O Oγ and here exiss a posiive consan C (γ), depending on γ, such ha Oγ C (γ) O. For any ν (, ) and x, le Γ ν (x) := {(y, ) + + : x y < ν} be he cone of aperure ν wih verex x and Γ(x) := Γ (x). Denoe by R ν F he union of all cones wih verices in F, namely, R ν F := x F Γ ν (x). The following Lemma 2.8 is jus [22, Lemma 3.]. Lemma 2.8. Le ν, η (, ). Then here exis posiive consans γ (, ) and C such ha, for any closed subse F of whose complemen has finie measure, and any nonnegaive measurable funcion H on + +, H(y, ) n dyd C R ν(fγ ) F { Γ η H(y, ) dyd where F γ denoes he se of poins in wih he global γ-densiy wih respec o F. Proof of Theorem 2.6. Assume ha f T p( ) 2 (+ + ). For any k Z, we le O k := { x : A(f)(x) > 2 k} and F k := Ok p( ). Since f T2 (+ + ), for each k, O k is an open se of and O k <. Le γ (, ) be as in Lemma 2.8 wih η = = ν. In wha follows, we denoe (F k ) γ and (O k ) γ simply by Fk and O k. By he proof of [8, Theorem 3.2], we know ha supp f ( k ZÔ k E), where E + + saisfies ha dy d E =. For each k Z, considering he Whiney decomposiion of he open se of Ok, we obain a se I k of indices and a family {Q k,j } j Ik of closed cubes wih disjoin ineriors such ha (i) j Ik Q k,j = Ok and, if i j, hen Q k,j Q k,i =, where E denoes he inerior of he se E; (ii) nl(q k,j ) dis(q k,j, (Ok ) ) 4 nl(q k,j ), where l(q k,j ) denoes he side-lengh of Q k,j and dis(q k,j, (Ok ) ) := inf{ z w : z Q k,j, w (Ok ) }. Now, for each j I k, le R k,j be he cube wih he same cener as Q k,j and wih he radius n/2-imes l(q k,j ). Se A k,j := R k,j (Q k,j (, )) (Ô k \Ô k+ ), a k,j := 2 k χ Rk,j fχ A k,j and λ k,j := 2 k χ Rk,j L p( ) (R ). Noice ha (Q n k,j (, )) (Ô k \Ô k+ ) R k,j. From his and supp f ( k Z Ôk E), we deduce ha f = k Z j I k λ k,j a k,j almos everywhere on + +. } dx,

23 Inrinsic Square Funcion Characerizaions 23 Nex we firs show ha, for each k Z and j I k, a k,j is a (p( ), )-aom suppor in R k,j. Le p (, ) and h T p 2 (Rn+ + ) wih h T p. Since A 2 (Rn+ + ) k,j (Ôk+ ) = R (Fk+ ), by Lemma 2.8 and he Hölder inequaliy, we have dy d a k,j, h := a k,j (y, )χ Ak,j (y, )h(y, ) + + dy d a k,j (y, )h(y, ) dx A(a F k+ Γ(x) n+ k,j )(x)a(h)(x) dx F k+ { } /p 2 k χ Rk,j [A(f)(x)] p dx h T p 2 (3R k,j ) F (Rn+ + ) k+ R k,j /p χ Rk,j, which, ogeher wih (T p 2 (Rn+ + )) = T p 2 (Rn+ + ) (see [5]), where (T p 2 (Rn+ + )) denoes he dual space of T p 2 (Rn+ + ), implies ha a k,j T p 2 (Rn+ + ) R k,j /p χ Rk,j. Thus, a k,j is a (p( ), p)-aom suppor in R k,j up o a harmless consan for all p (, ) and hence a (p( ), )-aom up o a harmless consan. Finally, we prove ha A ({λ j } j, {Q j } j ) f T p( ) 2 (+ + ). By he fac ha χ R k,j M(χ r Q k,j ) for any r (, p ), we know ha A ({λ k,j }, {R k,j }) p λ χ k,j p Rk,j ( = 2 k ) p χ χ k Z j I Rk,j p Rk,j k k Z j I k /p [ ] p M(2 kr χ r r Q k,j )(x), k Z j I k p which, ogeher wih Lemma 2.5 and he Whiney decomposiion of Ok, implies ha /p A ( ({λ k,j }, {R k,j }) 2 k ) p { } /p ( χ Qk,j 2 k ) p χ O. k k Z j I k k Z From he fac ha χ O k M(χ r O k ) wih r (, p ) and Lemma 2.5 again, we furher deduce ha A ({λ k,j }, {R k,j }) { ( M(2 kr χ r O k ) ) } /p { p /r ( 2 k χ Ok) k Z L p( ) ( k Z ) p }/p { } /p ( 2 k ) p χ Ok \O k+ A(f) L k Z p( ) ( ) f p( ) T 2 (+ ), + which complees he proof of Theorem 2.6.

24 24 Ciqiang Zhuo, Dachun Yang and Yiyu Liang To prove Theorem.3, we also need following echnical lemmas. Lemma 2.9. Le Q := Q(x, δ), ε (n(/p ), ), p( ) P( ) saisfy (.) and (.2), and s (n/p n, ) Z +. Then here exiss a posiive consan C such ha, for all f L,p( ),s ( ), Rn δ ε f(x) PQ s f(x) δ n+ε + x x n+δ dx C χ Q Lp( ) (Rn ) f Q L,p( ),s ( ). To prove Lemma 2.9, we need he following Lemma 2.2 which was proved in [32, Lemma 6.5]. Lemma 2.2. Le p( ) P( ) and q [, ]. Assume ha p( ) saisfies (.) and (.2), and s (n/p n, ) Z +. Then here exiss a posiive consan C such ha, for all Q Q, j Z and f L q,p( ),s ( ), { f(x) P s 2 j Q Q f(x) } /q q dx C2 jn( p ) χ Q f 2 j Q Q Lq,p( ),s ( ), where 2 j Q denoes he cube wih he same cener as Q bu 2 j imes side-lengh of Q. Proof of Lemma 2.9. For any k Z, le Q k := 2 k Q, namely, Q k has he same cener wih Q bu wih 2 k imes side-lengh of Q. Then we have Rn δ ε f(x) PQ s I := f(x) δ n+ε dx + x x n+δ ( ) δ ε f(x) PQ s = + f(x) Q k= Q k+ \Q k δ n+ε dx + x x n+δ f(x) P s Q Qf(x) dx + (2 k δ) n ε δ ε f(x) PQf(x) Q k= Q s dx k+ χ Q L p( ) ( ) 2 k(n+ε) [ f Q L,p( ),s ( )+ f(x) P s Q Qk f(x) + PQ s k f(x) PQf(x) s ] dx. Q k k= By Lemmas 2.7 and 2.2, we find ha, for all x Q k, P s Q k f(x) P s Qf(x) = P s Q k (f P s Qf)(x) Q k 2 k( n p n) χ Q f Q L,p( ),s ( ), which, ogeher wih ε (n(/p ), ), implies ha I χ Q Q f L,p( ),s ( ) + χ Q f Q L,p( ),s ( ). This finishes he proof of Lemma 2.9. k= 2 k(ε+n n Q k f(x) P s Qf(x) dx p ) χ Q f Q L,p( ),s ( ) Nex we esablish a John-Nirenberg inequaliy for funcions in L,p( ),s ( ).

25 Inrinsic Square Funcion Characerizaions 25 Lemma 2.2. Le p( ) P( ) saisfy (.) and (.2), f L,p( ),s ( ) wih s (n/p n, ) Z +. Assume ha p + (, ]. Then here exis posiive consans c and c 2, independen of f, such ha, for all cubes Q and λ (, ), { } {x Q : f(x) PQf(x) s c 2 Q λ > λ} c exp Q. f L,p( ),s ( ) χ Q Proof. Le f L,p( ),s ( ) and a cube Q. Wihou loss of generaliy, we may assume ha f L,p( ),s ( ) χ Q = Q. Oherwise, we replace f by f Q /[ f L,p( ),s ( ) χ Q ]. Thus, o show he conclusion of Lemma 2.2, i suffices o show ha (2.2) {x Q : f(x) P s Qf(x) > λ} c exp { c 2 λ} Q. For any λ (, ) and cube R Q, le I(λ, R) := {x R : f(x) PR s f(x) > λ} and (2.2) F(λ, Q) := sup R Q I(λ, R). R Then i is easy o see ha F(λ, Q). From Lemma 2.6, f L,p( ),s ( ) χ Q = Q and p + (, ], we deduce ha here exis a posiive consan c such ha, for any cube R Q, f(x) P s R Rf(x) dx χ R f R L,p( ),s ( ) c. R Applying he Calderón-Zygmund decomposiion of f PR s f a heigh σ (c, ) on he cube R, here exiss a family {R k } k of cubes of R such ha f(x) PR s f(x) σ for almos every x R\( k R k ), R k R j = if k j and, for all k, σ < R k f(x) PR s f(x) dx/ R k 2 n σ. From his, we deduce ha (2.22) R k f(x) P σ Rf(x) s dx f(x) P R k σ Rf(x) s dx c R σ R. k k If λ (σ, ), hen, for almos every x R\( k R k ), f(x) PR s f(x) σ < λ and hence (2.23) I(λ, R) k {x R k : f(x) P s Rf(x) > λ} k I(λ η, R k ) + k {x R k : P s R k f(x) P s Rf(x) > η} =: I + I 2, where η (, λ) is deermined laer. For I, by (2.2) and (2.22), we have (2.24) I k F(λ η, Q) R k C F(λ η, Q) R. σ For I 2, by Lemma 2.7, we find ha here exiss a posiive consan C such ha, for any x R k, PR s k f(x) PRf(x) s = PR s k (f PRf)(x) s C f(x) P R k Rf(x) s dx 2 n C σ. R k Now, le σ := 2c and η = 2 n C σ. Then, when λ (η, ), I 2 =, which, ogeher wih (2.23) and (2.24), implies ha I(λ, R) F(λ η, Q) R /2 for all R Q. Thus, i follows ha F(λ, Q) F(λ η, Q)/2. If m N saisfies mη < λ (m + )η, hen F(λ, Q) 2 F(λ η, Q) F(λ mη, Q). 2m