Aales Academiæ Scietiarum Feicæ Series A. I. Mathematica Volume 9, 994, 35 46 CARLESON MEASURE, ATOMIC ECOMPOSITION AN FREE INTERPOLATION FROM BLOCH SPACE Jie Xiao Pekig Uiversity, epartmet of Mathematics Beijig 0087, P.R. Chia Abstract. Several characterizatios, Carleso measures ad atomic decompositio for the Bloch space B are give. For their applicatios, free iterpolatios from B are also discussed.. Itroductio Let = { z : z < } be the uit disk i the fiite complex plae C ad dm α z) = z ) α dmz) the two-dimesioal Lebesgue measure with weight z ) α, α >. eote by A ad H the sets of fuctios aalytic ad boudedly aalytic o, respectively. For f A we say f B if.) f B = f0) + z ) f z) < ; z also f A α if.) f,α = fz) dmα z) <. B ad A ) are the so-called Bloch space ad the Bergma space weighted by z α, 6], 5]. It is well kow that the dual space of A 0 is idetified with B uder the followig ier product: f, g = π lim fz)gz) dmz) t t.3) = f)z) z ) g π z) dmz) + f0)g0) for f A 0 ad g B, where t 0, ), t = { z : z < t } ad f)z) = fz) f0) ] /z; see 3]. 99 Mathematics Subject Classificatio: Primary 30C55, 3A0, 46G0. Research ported by the Natioal Sciece Foudatio of Chia.
36 Jie Xiao I ] we discussed the atomic decompositio ad the free iterpolatio o the Bergma space A 0. Sice A 0 ) = B, it is very atural to cosider similar problems o the Bloch space. As far as we kow, these questios have ot bee thoroughly dealt with yet 8], 3]), which is what we try to do i this paper. First, i Sectio, we give several characterizatios of B as well as relatios betwee B ad Carleso measure. Next, i Sectio 3, we obtai a atomic decompositio of B by meas of the pseudohyperbolic metric. Fially, i Sectio 4, we study the free iterpolatios by fuctios from B by meas of the direct costructio ad the operator theory. We wish to express our deepest gratitude to Professor X.C. She for helpful suggestios, especially durig his illess. Also, we are grateful to Professors R. Aulaskari ad O. Martio for their kid help. Besides, we would like to thak the referee for his/her commets, ad the secretary of epartmet Mathematics of the Joesuu Uiversity for her dedicated typig.. Bloch space ad Carleso measure There are may works o the Bloch space, ], e.g. ], 7], 0]. Here we will give several iterestig characterizatios, some of which are ew. For z ad w i, let ϕ w z) = w z)/ wz), w, z) = ϕw z) ad dw, z) = log{ + w, z)]/ w, z)] }. Here, ) ad d, ) are called the pseudohyperbolic ad hyperbolic distaces, respectively. Also, deote the measure of set E, relative to dm α z), by m α E) = E dm αz) = E z ) α dmz). The we have the followig result. Theorem.. Let f A. The the followig statemets are equivalet: i) f B; ii) w,z fw) fz) /dw, z) < ; iii) there is a costat C > 0 such that w exp C f ϕw )z) fw) ] dm α z) <. Proof. We will show this fact accordig to i) = ii) = iii) = i). Firstly, i) = ii). Let f B, ad g w λ) = f ϕ w )λ) fw), λ, w. The g w 0) = 0 ad g w B = z z ) f z) f B <. Further, gw λ) = λ Settig z = ϕ w λ) we obtai i.e., ii) holds. 0 g w ζ) dζ f B log + λ λ. fz) fw) f + z, w) B log z, w) = f Bdz, w),
Carleso measure, atomic decompositio ad free iterpolatio 37 Secodly, ii) = iii). Suppose that 0 < f B = fw) fz) /dw, z) < ; w,z the for t 0, { z : z, gw z) > t } { t ) ] / t ) ]} z : z, z > exp f exp B f +. B Moreover, whe 0 < C < α + )/ f B], exp C g w z) ] dm α z) = C C 0 0 exp Ct) m α { z : z, gw z) > t } ) dt exp Ct) 4π α + exp α + )t ) f dt B = 4πC f B α + ) α + ) C f B ]. Thirdly, iii) = i). Let f B = exp C f ϕ w )z) fw) ] dm α z) < w for some costat C > 0. The f ϕw )z) fw),α f B C <. Sice g w has a Taylor series a z which coverges uiformly o t 0 < t < ), a simple calculatio gives a = g w 0) = α + )α + ) + α + )t ] g t ) α+ w z)z dm α z). By lettig t we get i.e., So, f B. g w 0) α + )α + ) t gw z) dmα z), ) w f w) α + )α + ) ] f C B.
38 Jie Xiao Remark. This theorem tells us that B is Lipschiz s class, relative to the hyperbolic metric d, ). However, we kow that B ca be idetified with the Zygmud class see ], 5]). Hece our result is much clearer tha the oe i ]. I what follows we characterize coectio betwee the Bloch space ad Carleso measure. For w let w, r) = { z : z, w, z) < r }, r 0, ). w, r) is called the pseudohyperbolic disk. It is more coveiet to use w, r) ot Carleso square) for discussig Borel measure o the Bergma space A α; see 6], ]. Similarly, we have the followig theorem. Theorem.. Let p 0, ) ad r 0, ), ad let µ be a oegative Borel measure o. The the followig statemets are equivalet: ) i) fz) fw) p w +α ] /p f p dµz) < ; B wz 4+α w 0 f B ii) w iii) w ) µ w, r) ] ) < ; m α w, r) ) w +α dµz) wz 4+α ] <. Proof. ii) iii) has bee derived i 3], so we oly eed to claim i) ii). O the oe had, if ii) is true, it follows by Theorem. that f ϕ w )z) fw) p dm α z)] /p C f B for f B, where C > 0 is a costat idepedet of f. Further, by 5], ] ad 6] it yields aother costat C 0 depedig o the coditio ii) such that ) fz) fw) p w +α dµz) wz 4+α C 0 = C 0 fz) fw) p ] /p w ) +α wz 4+α dm αz) ] /p f ϕ w )z) fw) p dm α z)] /p C 0 C f B. O the other had, let i) hold. Takig f 0 z) = / w 0 z) ] for w 0 = r + ) + w) / r + ) w), r + ) w, r 0, ), we get
Carleso measure, atomic decompositio ad free iterpolatio 39 f 0 B = w 0 / w 0 ), f0 z) f 0 w 0 ) = w0 z w 0 / w 0 z w 0 ) ad w 0 z = w ) r + ) z w,r) λ r w r + ) + r + r) w ). r) w λ wλ) Also, there are two costats C > 0 ad C > 0 depedig oly o α ad r such that see 4]) We also have > λ f p B 0 f B Therefore C w ) +α mα w, r) ) C w ) +α. f 0 p B w0 ) p w 0 ) fz) fλ) p λ +α dµz) λz 4+α f0 z) f 0 w 0 ) p w0 ) +α w 0 z w,r) ] /p ) w +α dµz) 4+α dµz) ) p w0 z, w0 w 0 ) ] p ] /p ] /p r) 4+α) + r + r ) 4+α r) 4+α) 4 4+α w0, w) z, w) ] ] /p p w,r) w dµz) ) +α ) ] r)4α+8+p)/p µ w, r) /p C /p 4α+8+p)/p ). m α w, r) ) ] µ w, r) ) <. w m α w, r) The measure µ satisfyig oe of the three statemets i Theorem. is said to be α-carleso measure. The followig fact is iterestig. Theorem.3. Let f A. The the followig statemets are equivalet: i) f B; ii) f z) log / z ) dmz) is 0-Carleso measure; iii) f z) z ) dmz) is 0-Carleso measure.
40 Jie Xiao Proof. We will give the whole claim i accordace with the order i) = ii) = iii) = i). First of all, i) = ii). Uder f B, we cosider the itegral below: I = w ) wz 4 f z) log z ) dmz) = { z > 4 } + { z 4 } Sice log/ z ) C ) z whe z > 4, f z) ) w ) z { z > 4 } { } dmz) C { z > 4 } C f B w ) ) { } dmz). wz 4 dmz) wz 4 dmz) πc f B, where C > 0 is a absolute costat. At the same time 6 ) f ) w { } dmz) { z 4 } 5 B log ) dmz) { z 4 } wz 4 z 6 ) 4 4 5 3 4 f B log ) dmz) = C f B z, { z 4 } where C > 0 is a absolute costat. Cosequetly I πc + C ) f B. So, from Theorem. iii) we see that f z) ) log/ z ) dmz) is 0-Carleso measure. Next ii) = iii). This is obvious, sice z ) ) 4 log/ z ) for all z. Fially iii) = i). Assumig that f z) ) z dmz) is 0-Carleso measure, we have I = w f z) ) z ) w wz 4 dmz) <, ad obviously > I f z) ) z dmz). Moreover, f w) ) 3 = f λ) ) ) λ π wλ) dmλ). 4 Hece ) w f w) 3 f λ) ) λ ) w π wλ 4 dmλ) I <, i.e., f B. Supposig g z, w) = log wz)/w z) the Gree s fuctio o ), we just have
Carleso measure, atomic decompositio ad free iterpolatio 4 Corollary.4. Let f A. The f B if ad oly if.) f z) gz, w) dmz) <. w Proof. This fact is readily derived from the equivalece betwee f ϕ w B ad f B, ad Theorem.3 ii). Nevertheless, the result ca also be show by Theorem. ad.3. To begi with, we let ad follows: 3.) = 3. Atomic decompositio { {c } : {c } C, {c } = stad for the usual sequece spaces as } c <, 3.) = { {c } : {c } C, {c } = c < }. Both are Baach spaces. Also, pose that {z } is a sequece of poits o. A sequece of poits {z } is called δ-weakly separated if δ = if m z m, z ) > 0 ad η-uiformly separated if η = if m z m, z ) > 0. Clearly a η- uiformly separated sequece must be δ-weakly separated. A sequece of poits {z } is said to be ε-dese if = z, ε), where z, ε) = { z : z, z, z) < ε } ad ε 0, ). Lueckig 6] ad Xiao ] proved the quasi-atomic decompositio theorem of A α as follows. Lemma 3.. Let {z } be a sequece of poits o, α > ad f A α. If {z } is δ-weakly separated, there is a costat C > 0 depedig oly o δ ad α so that 3.3) f,α C z ) +α fz ). Furthermore, there are a ε 0 > 0 ad a costat C > 0 depedig oly o δ ad α so that 3.4) f,α C z ) +α fz ) if {z } is also ε-dese with 0 < ε ε 0.
4 Jie Xiao After the above lemma, we ca state a atomic decompositio theorem o the Bloch space. Theorem 3.. Let {z } be a sequece of poits o. If {z } is δ-weakly separated, the fuctio of the form 3.5) is i B for ay {c } 3.5) fz) = z ). c z z Moreover, there is a ε 0 > 0 such that every f B has the form 3.5) for some {c } if {z } is also ε-dese with 0 < ε ε 0. Proof. Let {z } be δ-weakly separated. The T, defied as follows, is a bouded liear operator from A 0 to, 3.6) Tf = { Tf) } = { z ) fz ) }, i that 3.3) holds uder {z } beig δ-weakly separated. Thus T, the adjoit operator of T give by 3.7), is a bouded liear operator from = ) ) to B = A 0 ) ), 3.7) Tf, y = f,t y, f A 0, y, where the left, is just the usual ier product betwee ad. To compute T, we take {, m = y = e, e ) m = 0, m, so Tf, e = Tf) = z ) fz ) = z ) f, Kz, where K z z) = / z z) is the reproducig kerel for A 0. Hece ad T y = T e = z ) Kz z) c z ) z z) for y = {c }, i.e., the fuctio i the form 3.5) is i B. Ideed, it is easy to derive T y B by meas of the direct computatio. Now we tur to showig the secod part of Theorem 3.. I fact, it is oly ecessary to claim T to be surjective. However, T is oto if ad oly if T is bouded below. By Lemma 3., there exists a ε 0 > 0 such that T is bouded below if {z } is ε-dese with 0 < ε ε 0. That is to say, there is a ε 0 > 0 such that every f B has the form 3.5) for some {c } as {z } is ε-dese with 0 < ε ε 0. Therefore the proof is completed.
Carleso measure, atomic decompositio ad free iterpolatio 43 4. Free iterpolatio As is well-kow, a give sequece of poits {z } o is called a H - iterpolatig sequece if for ay {c } there exists f H satisfyig fz ) = c for all. Carleso stated i 4] that {z } is a H -iterpolatig sequece if ad oly if {z } is η-uiformly separated. Here we wat to exted this fact to the Bloch space. Yet, it is ufortuate that the η-uiformly separated property is oly a sufficiet coditio for B. A sequece of poits {z } is said to be a B-iterpolatig sequece if there is f B such that fz ) = c for all ad ay {c }. Theorem 4.. Let {z } be a sequece of poits o. If {z } is a B- iterpolatig sequece, {z } is δ-weakly separated. Coversely, if {z } is δ- weakly separated ad 4.) or 4.) is true, the {z } is a B-iterpolatig sequece where zm ) z ) 4.) z z m <, m zm ) 4.) m z z m <. Proof. Firstly, if {z } is a B-iterpolatig sequece, the T B, where T f = { fz ) }. Sice B is a Baach space, relative to B, it follows from the ope mappig theorem that there is a uiform costat C > 0 ad f B so that f B C with fz ) = w for all ad {w }. Pickig w m = 0, m ; w m =, m =, there exist f B, f B C satisfyig f z ) = ; f z m ) = 0, m. Theorem. yields f z ) f z m ) C, m, dz, z m ) ad so if m dz, z m ) /C > 0, i.e., δ = if m z m, z ) e /C )/e /C + ) > 0. Coversely, let {z } be δ-weakly separated. If 4.) is true, {z } is η- uiformly separated ad hece = T H T B sice H is a proper subspace of B. Furthermore, if 4.) holds, we cosider the liear operator T, give by T {c } ) = c z )/ z z) ), {c }. Clearly, T is bouded from to B by Theorem 3.), while T T I){c } = zm ) c m z m z m {c } zm ). m z z m
44 Jie Xiao So, T T I) <, where I is the idetify operator, i.e., T T has a iverse, deoted by T T ). Further, T has a right iverse T T T ), that is to say, T T T T ) ) = I, ad thus T B. So, {z } is a B-iterpolatig sequece. Note that T H T B. I geeral, it is ecessary to take ito cosideratio the geeric free iterpolatio problem from B. That is, for which {w } C there is f B satisfyig { fz ) } = T f = {w }. For this we obtai the followig fact. Theorem 4.. Let {z } be a δ-weakly separated sequece of poits o. If {fz )} = T f = {w } is solvable i B for {w } C, the followig assertios i) ad ii) hold: i) there are a costat C > 0 ad a fuctio βz) such that 4.3) z for γ > ; ii) there is a costat C > 0 such that 4.4) z z, z ) ] γ exp C w βz) ] < z, z ) ] γ exp C w hz) ] < for γ >, where hz) = { w z, z ) ] γ}/ z, z ) ] γ. Coversely, if i) or ii) holds for γ =, the { fz ) } = T f = {w } is solvable i B. Proof. First we cosider the case i). If { fz ) } = T f = {w } is solvable i B, the Theorem. yields exp C g z w) ] dm α w) < z for C < α + )/ f B f B > 0 is aturally assumed), where g z w) = f ϕ z )w) fz). The above statemet meas that expc g z ) is i A α. Cosequetly, by Lemma 3., 4.5) z exp C g z z ) ] z ) α+ C expc g z,α <, z where { z } = {ϕ z z )}, C > 0 is a costat idepedet of g z, ad { z } is also a δ-weakly separated sequece of poits o sice {z } is such a sequece. Thus 4.5) meas that i) holds for γ = α + > ad βz) = fz). Now let us cosider ii).
Carleso measure, atomic decompositio ad free iterpolatio 45 Because {z } is δ-weakly separated, we get z, z ) ] α+ < by Lemma 3.. By 4.5) we further have {: w fz) >t} z, z ) ] α+ C exp C t) for t 0, where C ad C are costats with C < α + )/ f B, f B > 0, ad f is the iterpolatig fuctio for T f = {w } i B. Thus, for γ = α + > hz) fz) z, z ) ] γ w fz) z, z ) ] γ = z, z ) ] γ z, z ) ] γ C { + log ad, cosequetly, z 0 0 C { z, z ) ] γ {: w fz) >t} z, z ) ] } γ dt { mi z, z ) ] } γ, C exp C t) dt z, z ) ] γ C exp w hz) ] z, z ) ] γ C exp w fz) ] C ] exp fz) hz) z {{ z, z ) ] γ exp C w fz) ]} /{ z, z ) ] γ z ) exp { } / } C z, z ) ] γ = C e { z } z, z ) ] γ exp C w fz) ]} / <. That is to say, ii) is true for γ = α + >. Next we show the cotrary assertio. If i) or ii) holds for γ =, the it follows from 4.4) that z z, z ) ] <. } /
46 Jie Xiao This, together with {z } beig δ-weakly separated, shows that {z } is η- uiformly separated. Also, it follows by 9] that there is f BMOA ) to make T f = { fz ) } = {w } for {w } C which is satisfied with 4.3) or 4.4) for γ =. Sice BMOA ) B, there exists f B such that T f = { fz ) } = {w } uder the previous assumptio. Thus the theorem is proved. Refereces ] Aderso, J.M.: Bloch fuctios: The basic theory. - I: Operators ad fuctio theory, edited by S.C. Power ad. Reidel, ordrecht, 985, 7. ] Aderso, J.M., J. Cluie, ad Ch. Pommereke: O Bloch fuctios ad ormal fuctios. - J. Reie Agew. Math. 70, 974, 37. 3] Axler, S.: Bergma spaces ad their operators. - Lecture otes at the Idiaa Uiversity Fuctio Theoretic Operator Theory Coferece, 985. 4] Carleso, L.: A iterpolatio problem for bouded aalytic fuctios. - Amer. J. Math. 80, 958, 9 930. 5] Garett, J.B.: Bouded aalytic fuctios. - Academic Press, New York, 98. 6] Lueckig,.H.: Represetatio ad duality i weighted spaces of aalytic fuctios. - Idiaa Uiv. Math. J. 34, 985, 39 336. 7] Pommereke, Ch.: O Bloch fuctios. - J. Lodo Math. Soc. ), 970, 689 695. 8] Rochberg, R.: Iterpolatio by fuctios i Bergma spaces. - Michiga Math. J. 9, 98, 9 36. 9] Sudberg, C.: Values of BMOA fuctios o iterpolatig sequeces. - Michiga Math. J. 3, 984, 30. 0] Xiao, J.: Equivalece betwee Bloch space ad BMOA space. - J. Math. Res. Expositio 0:, 990, 87 88. ] Xiao, J.: Atomic decompositio ad sequece iterpolatio o Bergma space A. - Approx. Theory Appl. 8:, 99, 40 49. ] Xiao, J.: Carleso measure ad atomic decompositio o Bergma space A p ϕ) < p < ). - J. Liaoig Normal Uiv. Natural Sciece) 5:3, 99, 88 94. 3] Xiao, J.: Iterpolatig sequeces for A ϕ)-fuctios. - Sci. Chia Ser. A 35:8, 99, 907 96. 4] Xiao, J.: Compact Toeplitz operators o Bergma spaces A p ϕ) p < ). - Acta Math. Sci. B) 3:, 993, 56 64. 5] Xiao, J.: ual space, Carleso measure ad sequece iterpolatio o A p ϕ) 0 < p < ). - Acta Math. Siica N.S.) to appear). Received 3 October 99