AN INTEGRAL REPRESENTATION FOR BESOV AND LIPSCHITZ SPACES 1 1 KEHEZHU 0 2 ABSTRACT. ItiswellknownthatfunctionsintheanalyticBesovspace n B1ontheunitdiskDadmitsanintegralrepresentation a J z−w f(z)= dµ(w), 5 ZD 1−zw 1 whereµisacomplexBorelmeasurewith|µ|(D) < ∞. Wegeneralize this result to all Besov spaces B with 0 1. We also obtainaversionforBergmanandFock t C spaces. . h t a m [ 1. INTRODUCTION 1 LetDdenotetheunitdiskinthecomplexplaneC,H(D)denotethespace v of all analytic functions in D, and dA denote the normalized area measure 5 9 on D. For 0 < p < ∞ we consider the analytic Besov space B consisting p 9 offunctionsf ∈ H(D)with theproperty that(1−|z|2)kf(k)(z) belongsto 2 . Lp(D,dλ),where 1 0 dA(z) dλ(z) = 1 (1−|z|2)2 1 : istheMo¨biusinvariantareameasureonDandk isanypositiveintegersuch v i thatpk > 1. ThespaceB is independentoftheintegerk used. X p Itiswellknownthatananalyticfunctionf inDbelongstoB ifandonly r 1 a if there exists a complex Borel measure µ on the D such that |µ|(D) < ∞ and z −w f(z) = dµ(w), z ∈ D. (1) Z 1−zw D See [1, 2, 10]. The purpose of this paper is to generalize the above result toseveralotherspaces,includingBesovspaces,Lipschitzspaces,Bergman spaces, and Fock spaces. We state our main results as Theorems A and B below. 2000MathematicsSubjectClassification. 30H20,30H25,32A36. Key words and phrases. Bergmanspaces, Besov spaces, Fock spaces, Carleson measures,atomicdecomposition,Berezintransform. 1 2 KEHEZHU Theorem A. Suppose0 < p ≤ 1, 0 < r < 1, and f is analyticin D. Then f ∈ B ifandonlyifitadmitsa representation p z −w f(z) = dµ(w), z ∈ D, ZD 1−zw where µ is a complex Borel measure on D such that the localized function z 7→ |µ|(D(z,r))belongstoLp(D,dλ),where z −w D(z,r) = w ∈ D : < r (cid:26) (cid:12)1−zw(cid:12) (cid:27) (cid:12) (cid:12) (cid:12) (cid:12) isthepseudo-hyperbolicdiskat z withra(cid:12)diusr. (cid:12) Recall that for any real number t, the analytic Lipschitz space Λ on the t unit disk consists of functions f ∈ H(D) such that (1 − |z|2)k−tf(k)(z) is bounded,wherek isanynonnegativeintegergreater thant. Theorem B. Suppose t > 1, 0 < r < 1, and f is analytic in D. Then f ∈ Λ if andonlyif t z −w f(z) = dµ(w), z ∈ D, ZD 1−zw forsomecomplexBorel measureµwith thepropertythat |µ|(D(z,r)) sup < ∞. z∈D (1−|z|2)t In addition to Besov and Lipschitz spaces in dimension 1, where the in- tegralrepresentationlooksparticularlynice,wewillalsoconsiderBergman andFock spaces inhigherdimensions. 2. PRELIMINARIES ON MEASURES Suppose µ is a complex Borel measure on D and r ∈ (0,1). We can definetwofunctionson Das follows. µ(D(z,r)) µ (z) = µ(D(z,r)), µ (z) = . r r (1−|z|2)2 Itis wellknownthatthearea ofthepseubdo-hyperbolicdiskD(z,r) is 1−|z|2 2 πr2 , (cid:18)1−r2|z|2(cid:19) which is comparable to (1 − |z|2)2 whenever r is fixed. That is why we think of µ as an averaging function for the measure µ. We will call µ a r r localizedfunction forµ. Thebehaviorofµ and µ isoften independentof r r theparticbularradiusr beingused. b INTEGRALREPRESENTATION 3 Another averaging function for µ is the so-called Berezin transform of µ. We need the assumption |µ|(D) < ∞ in order to define the Berezin transform: (1−|z|2)2 µ(z) = dµ(w), z ∈ D. Z |1−zw|4 D See[10]forbasiecinformationabout theseaveragingoperations. Wewillneedtodecomposetheunitdiskintoroughlyequal-sizedpartsin thepseudo-hyperbolicmetric. Morespecifically,wewillneedthefollowing result. Lemma 1. For any 0 < r < 1 there exists a sequence {z } in D and a n sequenceofBorel subsets{D } ofD with thefollowingproperties: n (a) D = D ∪D ∪···∪D ∪···. 1 2 n (b) ThesetsD aremutuallydisjoint. n (c) D(z ,r/4) ⊂ D ⊂ D(z ,r)forevery n. n n n Proof. Thisiswell known. See [10]forexample. (cid:3) Any sequence {z } satisfying the three conditions above will be called n an r-latticeinthepseudo-hyperbolicmetric. Lemma 2. Suppose µ is a positive Borel measure on D and 0 < p ≤ ∞. If r and s are two radii in (0,1) and {z } is an r-lattice in the pseudo- n hyperbolicmetric. Then thefollowingconditionsareequivalent. (a) Thefunctionµ (z) belongsto Lp(D,dλ). s (b) Thesequenceµ (z ) belongstolp. r n b If1/2 < p ≤ ∞,then theaboveconditionsarealsoequivalentto b (c) Thefunctionµ(z) belongstoLp(D,dλ). Proof. Thisisalso weellknown. See[10]forexample. (cid:3) As a consequence of the above lemma on the averaging function µ we r obtainseveralequivalentconditionsforthelocalized functionµ . r b Corollary 3. Suppose 0 < p ≤ ∞ and µ is a positive Borel measure on D. If r and s are two radii in (0,1) and {z } is an r-lattice in the pseudo- n hyperbolicmetric. Then thefollowingconditionsareequivalent. (a) Thesequence{µ (z )}belongsto lp. r n (b) Thefunctionµ (z) belongsto Lp(D,dλ). s Ifp 6= ∞,theaboveconditionsarealsoequivalentto (c) Thefunctionµ (z) belongsto Lp(D,dA ), where s 2(p−1) dA (z) = (1−|z|2)2(p−1)dA(z). b2(p−1) 4 KEHEZHU Proof. ConsiderthepositiveBorel measure dν(z) = (1−|z|2)2dµ(z). Itiswellknownthat(1−|z|2)2iscomparableto(1−|z |2)2forzinD(z ,t), n n where t is any fixed radius in (0,1). See [10] for example. Thus ν (z) is t comparableto µ (z), and ν (z) is also comparableto (1−|z|2)2µ (z). The t t t desiredresult thenfollowsfrom Lemma2. b (cid:3) b b In view of the equivalence of conditions (a) and (c) in Lemma 2, it is temptingto conjecture that condition (c) in Corollary 3 aboveis equivalent to µ ∈ Lp(D,dA ) whenever p > 1/4. It turns out that this is not 2(p−1) true. This already fails at p = 1. In fact, if p = 1, the condition µ ∈ Lp(eD,dA ) means 2(p−1) e +∞ > µ(z)dA(z) Z D e dµ(w) = (1−|z|2)2dA(z) ZD ZD |1−zw|4 (1−|z|2)2dA(z) = dµ(w) . ZD ZD |1−zw|4 ThistogetherwithLemma3.10in[10]showsthat,forp = 1, thecondition µ ∈ Lp(D,dA )is thesameas 2(p−1) e 1 log dµ(w) < ∞. ZD 1−|w|2 On the other hand, it is easy to see that condition (a) in Corollary 3, for p = 1, simply means µ(D) < ∞, which is obviously different from the integralconditionabove. TheBerezin transformwillnotreallybeusedintherestofthepaper,but itisalways interestingand insightfultocomparethebehaviorofµ and µ. r Another notion critical to the integral representation of Lipschitz spaces isthatofCarleson measures. b e Let t > 0. We say that a positive Borel measure µ on the unit disk D is t-Carleson if µ(D(z,r)) sup < ∞ z∈D (1−|z|2)t for some r ∈ (0,1). It is well known that if the above condition holds for somer ∈ (0,1),thenitholdsforeveryr ∈ (0,1). Thusbeingt-Carleson is independentoftheradiusr used inthedefinition. INTEGRALREPRESENTATION 5 If t > 1, then every t-Carleson measure is finite. In fact, in this case, we useLemma1to get ∞ ∞ µ(D) = µ(D ) ≤ µ(D(z ,r)) n n Xn=1 Xn=1 ∞ ∞ ≤ C (1−|z |2)t ≤ C′ (1−|z|2)t−2dA(z) n Z Xn=1 Xn=1 Dn = C′ (1−|z|2)t−2dA(z) < ∞. ZD It is clear from the above argument that not every t-Carleson measure is finitewhen t ≤ 1. Ift > 1,thenµist-Carelsonifandonlyifforsome(orevery)0 0 suchthat p |f(z)|pdµ(z) ≤ C |f(z)|p(1−|z|2)t−2dA(z) Z Z D D for all f ∈ H(D). Because of this, such measures are also called Carleson measures for Bergman spaces. When t > 1, it is also known that µ is t- CarlesonifandonlyifthereisaconstantC > 0suchthatµ(S ) ≤ Cht for h all“Carleson squares”S ofsidelengthh. See [11, 8]. h We warn the reader that there is a fine distinction between 1-Carleson measures defined above and the classical Carleson measures (for Hardy spaces). This can be seen by considering an arbitrary Bloch function f intheunitdisk. In fact, forsuchafunctionifwedefinethemeasureµ by dµ(z) = (1−|z|2)|f′(z)|2dA(z), thenµ is1-Carleson, because µ(D(z,r)) = (1−|w|2)|f′(w)|2dA(w) Z D(z,r) 1 ∼ (1−|w|2)2|f′(w)|2dA(w) 1−|z|2 Z D(z,r) ≤ C(1−|z|2). But µ is not a classical Carleson measure, because being so would mean thatf isinBMOA.See[5]. ItiscertainlywellknownthatBMOAisstrictly contained in the Bloch space. A classical Carleson measure is 1-Carleson, butnottheotherway around. Similarly, for 0 < t < 1, there is a subtle difference between measures satisfying the condition µ(S ) ≤ Cht and those satisfying the condition h µ(D(z,r)) ≤ C(1−|z|2)t. 6 KEHEZHU 3. BESOV SPACES IN THE UNIT DISK We prove Theorem A in this section. For this purpose we need to make useofweightedBergman spaces. Thus foranyα > −1 welet dA (z) = (α+1)(1−|z|2)αdA(z) α denotetheweightedarea measureonD. Thespaces Ap = H(D)∩Lp(D,dA ), 0 < p < ∞, α α arecalled weightedBergman spaces withstandard weights. We will see that Theorem A can be thought of as an extension of the following atomic decomposition for weighted Bergman spaces, which can befoundin[10]forexample. Theorem 4. Suppose0 −1, and b > max(1,1/p)+(α+1)/p. (2) There exists some positive number δ such that for any r-lattice {z } with n r < δ the weighted Bergman space Ap consists exactly of functions of the α form ∞ (1−|z |2)(pb−2−α)/p f(z) = c n , (3) n (1−zz )b Xn=1 n where{c } ∈ lp. n Atomicdecompositionfor Bergman spaces was first obtained in [4]. We willfollowtheproofoftheabovetheorem as found in[10]. We beginwith an explictconstructionforameasurein(1)when f is apolynomial. Lemma 5. If f is a polynomial and 0 < p ≤ ∞, there exists a complex Borel measure µ such that the localized function |µ| (z) is in Lp(D,dλ) r and z −w f(z) = dµ(w) ZD 1−zw forallz ∈ D. Proof. Iff isanonzero constantfunction,weusethemeasure |w| dµ(w) = c (1−|w|2)N dA(w), w wherecisanappropriateconstantandN issufficientlylarge(dependingon p). Iff(w) = wn withn ≥ 1, weusethemeasure dµ(w) = cwn−1(1−|w|2)N dA(w), wherecisanappropriateconstantandN issufficientlylarge(dependingon p). Thisfollowsfrom theTaylorexpansionofthefunction1/(1−zw) and polarcoordinates. (cid:3) INTEGRALREPRESENTATION 7 We now proceed to the proof of Theorem A. First assume that f ∈ B p for some 0 1. Let b = k + 1 and α = pk − 2. Then b satisfies the condition in (2). In fact, since0 < p ≤ 1, wehave 1 α+1 1 pk −1 max 1, + = + = k < b. (cid:18) p(cid:19) p p p Also, f ∈ B if and only if its k-th order derivative f(k) is in Ap and the p α exponentin thenumeratorof(3)is pb−2−α = 1. p Itfollowsfrom Theorem4that wecan find an r-lattice{z }in thepseudo- n hyperbolicmetricandasequence{c } ∈ lp suchthat n ∞ 1−|z |2 f(k)(z) = c n , z ∈ D. n (1−zz )k+1 Xn=1 n There is considerable freedom in the choice of the r-lattice in Theorem 4. So we may assume that z 6= 0 for each n and |z | → 1 as n → ∞. We n n thenconsiderthefunction ∞ z −z g(z) = c′ n , z ∈ D, n1−zz Xn=1 n where c c′ = n , n ≥ 1. n k!zk−1 n Clearly, the sequence {c′ } is still in lp ⊂ l1. This is where we use the n assumption 0 < p ≤ 1 in a critical way to ensure that the infinite series defining g actually converges. Differentiating term by term shows that the functiong satisfiesg(k) = f(k). Thusthere isapolynomialP(z) suchthat ∞ z −z f(z) = P(z)+ c′ n . n1−zz Xn=1 n By Lemma 5, there is a measure ν such that the localized function |ν| (z) r isinLp(D,dλ)and z −w P(z) = dν(w), z ∈ D. ZD 1−zw Ifwedefine ∞ µ = ν + c′ δ , n zn Xn=1 8 KEHEZHU where δ denotes the unit point mass at z , we obtain the desired repre- zn n sentation for f with the localized function |µ| (z) belonging to Lp(D,dλ). r ThisprovesonedirectionofTheoremA. ToprovetheotherdirectionofTheoremA,letusassumethatµisacom- plex Borel measure such that the function |µ| (z) is in Lp(D,dλ). Let r be r asufficientlysmallradiusand{z }beanr-latticeinthepseudo-hyperbolic n metric. By Corollary 3 thesequence|µ| (z ) is inlp. Nowif r n z −w f(z) = dµ(w), z ∈ D, Z 1−zw D then (1−|w|2)wk−1 f(k)(z) = k! dµ(w), z ∈ D. Z (1−zw)k+1 D We use Lemma 1 to decompose D into the disjoint union of {D } and n rewrite ∞ (1−|w|2)wk−1 f(k)(z) = k! dµ(w), Z (1−zw )k+1 Xn=1 Dn n sothat ∞ 1−|w|2 |f(k)(z)| ≤ k! d|µ|(w). Z |1−zw|k+1 Xn=1 Dn Foreach n ≥ 1 andz ∈ D thereissomepointw (z) ∈ D such that n n 1−|w (z)|2 1−|w|2 n = sup . |1−zw (z)|k+1 |1−zw|k+1 n w∈Dn Thus, ∞ 1−|w (z)|2 |f(k)(z)| ≤ c n n|1−zw (z)|k+1 Xn=1 n for all z ∈ D, where c = k!|µ|(D ) is a sequence in lp as |µ|(D ) ≤ n n n |µ|(D(z ,r)). ByLemma4.30of[10],thereexistsaconstantC > 0(inde- n pendentofnand z) suchthat 1−|w (z)|2 1−|z |2 n n ≤ C |1−zw (z)|k+1 |1−zz |k+1 n n foralln ≥ 1 and allz ∈ D. Therefore, ∞ 1−|z |2 |f(k)(z)| ≤ C c n n|1−zz |k+1 Xn=1 n forallz ∈ D. Since 0 < p ≤ 1,weapplyHo¨lder’sinequalitytoget (1−|z |2)p |f(k)(z)|p ≤ Cp |c |p n . n |1−zz |p(k+1) Xn=1 n INTEGRALREPRESENTATION 9 Integrate term by term and apply Proposition 1.4.10 of [7]. We obtain an- otherconstantC > 0 (independentoff)suchthat ∞ |f(k)(z)|p(1−|z|2)pk−2dA(z) ≤ C |c |p < ∞, n ZD Xn=1 whichshowsthat f ∈ B and completestheproofofTheoremA. p It is clear that Theorem A cannot possibly be true when p > 1. This is because any function f represented by the integral in Theorem A must be bounded,whilethereareunboundedfunctionsinB when p > 1. p 4. BERGMAN TYPE SPACES ON THE UNIT BALL In this section we show how to extend Theorem A to Bergman type spaces on the unit ball B in Cn. The main reference for this section is n [8]. Whenα > −1, all backgroundinformationcan alsobefoundin [11]. Foranyreal parameterα weconsidertheweighted volumemeasure dv (z) = (1−|z|2)αdv(z), α wheredv is theLebesguevolumemeasureonB . n Forrealαand0 −1 and Rf is thestandard radial derivativedefined by ∂f ∂f Rf(z) = z +···+z . 1 n ∂z ∂z 1 n It is well known that the space Ap is independent of the integer k used in α thedefinition. Various names exist in theliteraturefor thespaces Ap: Bergman spaces, α Besov spaces, and Sobolev spaces, among others. We follow [8] and call them Bergman spaces here. When α > −1, Ap are indeed the weighted α Bergman spaceswithstandardweights. Forα = −(n+1),Ap becomethe α so-calleddiagonalBesov spaces. If p is fixed, all the spaces Ap are isomorphic as Banach spaces for 1 ≤ α p < ∞ and as complete metric spaces for 0 < p < 1. The isometry can be realized by certain fractional radial differential operators. Because of this, itisoftenenoughforusjusttoconsiderthecaseα = 0andobtaintheother cases byfractional differentiationorfractional integration. Ontheunitballthereexistsauniquefamilyofinvolutiveautomorphisms ϕ (z) thatare highdimensionalanalogsoftheMo¨biusmaps a a−z ϕ (z) = a 1−az 10 KEHEZHU on the unit disk. See [7] and [11]. The pseudo-hyperbolic metric on B n is still the metric defined by d(z,w) = |ϕ (w)|. For any complex Borel z measure µ on B the localized function µ and the averaging function µ n r r aredefined in exactlythesameway as before. Wecan nowextendTheoremA toall thespaces Ap as follows. b α Theorem 6. Supposeα isreal, 0 max 1, + . (cid:18) p(cid:19) p Then afunctionf ∈ H(B ) belongstoAp ifand onlyif n α (1−|w|2)(pb−n−1−α)/p f(z) = dµ(w) (4) Z (1−hz,wi)b Bn forsomecomplexBorelmeasureµonB withthelocalizedfunction|µ| (z) n r belongingtoLp(D,dλ),where dv(z) dλ(z) = (1−|z|2)n+1 istheMo¨biusinvariantvolumemeasureon B . n Proof. That every function f ∈ Ap has the desired integral representation α in (4), with µ being atomic, follows from Theorem 32 in [8], which is the atomicdecompositionforthesespaces. On the other hand, if f is a function represented by (4), we follow the secondhalfoftheproofofTheorem A toobtainthefollowingestimate, ∞ (1−|z |2)(pb−n−1−α)/p |RNf(z)| ≤ C c k , z ∈ B , k |1−hz,z i|b+N n X k k=1 where C is some positiveconstant, {c } ∈ lp, and N is a sufficiently large k positive integer. Using the arguments on pages 92-93 of [10] (the proof for atomic decompositionof Bergman spaces) we can then show that RNf belongsto theBergman spaceAp , whichmeans thatf ∈ Ap. (cid:3) Np+α α One particular case is worth mentioning. If p = 1 and µ is a positive Borel measure on B , then we use Fubini’s theorem, the fact that n

An integral representation for Besov and Lipschitz spaces

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