WEIGHTED COMPOSITION OPERATORS BETWEEN DIFFERENT WEIGHTED BERGMAN SPACES AND DIFFERENT HARDY SPACES

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1 Illinois Journl of Mthemtics Volume 51, Number 2, Summer 2007, Pges S WEIGHTE COMPOSITION OPERATORS BETWEEN IFFERENT WEIGHTE BERGMAN SPACES AN IFFERENT HARY SPACES ZELJKO CU CKOVIĆ AN RUHAN ZHAO Abstrct. We chrcterize bounded nd compct weighted composition opertors cting between weighted Bergmn spces nd between Hrdy spces. Our results use certin integrl trnsforms tht generlize the Berezin trnsform. We lso estimte the essentil norms of these opertors. As pplictions, we chrcterize bounded nd compct pointwise multipliction opertors between weighted Bergmn spces nd estimte their essentil norms. 1. Introduction Let be the open unit disk in the complex plne. Let ϕ : be n nlytic self-mp of nd let u be n nlytic function on. The weighted composition opertor uc ϕ is defined on the spce of nlytic functions on by (uc ϕ )f(z) = u(z)(f ϕ)(z). We re interested in weighted composition opertors restricted to Hrdy spces nd weighted Bergmn spces. In our previous work [CZ] we chrcterized bounded nd compct weighted composition opertors mpping every weighted Bergmn spce into itself. The min tool ws the generlized Berezin trnsform. We needed generl Poisson trnsform to find chrcteriztion of boundedness nd compctness of these opertors from Hrdy spce into itself. In this pper we continue this line of investigtion nd study weighted composition opertors from one weighted Bergmn spce into nother weighted Bergmn spce. We study the sme question bout boundedness nd compctness of these opertors cting between different Hrdy spces. We lso obtin estimtes for the essentil norms of uc ϕ on these spces. Let da(z) = (1/π)dxdy be the normlized Lebesgue mesure on nd da α (z) = (1 + α)(1 z 2 ) α da(z) be the weighted Lebesgue mesure, where 1 < α <. For 0 < p < nd 1 < α <, the weighted Bergmn spce Received Jnury 27, 2005; received in finl form August 3, Mthemtics Subject Clssifiction. 47B c 2007 University of Illinois

2 480 ZELJKO CU CKOVI Ć AN RUHAN ZHAO L p,α consists of those functions f nlytic on tht stisfy f p = f(z) p da L p,α α (z) <. For 0 < p <, the Hrdy spce H p consists of functions f nlytic on tht stisfy 2π f p H = sup f(re iθ ) p dθ <. p 0<r<1 0 We would like to mention other relevnt work in this direction. Composition opertors between different Hrdy spces nd Bergmn spces were studied by mny uthors, for exmple, Goebeler [G], Gorkin nd McCluer [GM], Hmmond nd McCluer [HM], Hunziker nd Jrchow [HJ], Jrchow [J], Smith [Sm] nd Smith nd Yng [SY]. Boundedness nd compctness of weighted composition opertors between Hrdy spces were studied by Contrers nd Hernndez-iz [CH] using Crleson mesures. Our pproch uses the generlized Berezin trnsform nd relted integrl opertors to chrcterize bounded nd compct weighted composition opertors mpping L p,α into L q,β nd H p into H q. The generlized Berezin trnsform lso ppers in [Li] in chrcteriztions of bounded nd compct composition opertors cting on the Bergmn spces on strictly pseudoconvex domins in C n. As one would expect, our results re different for the p q cse nd the q < p cse. Our results lso provide n nswer to question posed by Contrers nd Hernndez-iz [CH] regrding one of the cses mentioned bove in the Hrdy spce setting. Our first result concerns bounded weighted composition opertors mpping into L q,β for p q. For unweighted spces, tht would men mpping lrger Bergmn spce into smller one. Our results will be expressed in L p,α terms of the integrl opertor ( ) 1 2 (2+α)q/p I ϕ,α,β (u)() = 1 āϕ(w) 2 u(w) q da β (w). Theorem 1. Let u be n nlytic function on nd ϕ be n nlytic self-mp of. Let 0 < p q <, nd α, β > 1. Then the weighted composition opertor uc ϕ is bounded from L p,α into L q,β if nd only if (1) sup I ϕ,α,β (u)() <. We hve the following estimtes for the essentil norm of uc ϕ. Theorem 2. Let u be n nlytic function on nd ϕ be n nlytic self-mp of. Let 1 < p q <, nd α, β > 1. Let uc ϕ be bounded from L p,α into L q,β. Then there is n bsolute constnt C 1 such tht lim sup 1 I ϕ,α,β (u)() uc ϕ q e C lim sup I ϕ,α,β (u)(). 1

3 WEIGHTE COMPOSITION OPERATORS 481 The following corollry is now immedite. Corollry 1. Let u be n nlytic function on nd ϕ be n nlytic self-mp of. Let 1 < p q <, nd α, β > 1. Let uc ϕ be bounded from L p,α into L q,β. Then the weighted composition opertor uc ϕ is compct from L p,α into L q,β if nd only if lim sup I ϕ,α,β (u)() = 0. 1 Let σ z (w) = (z w)/(1 zw) be Möbius trnsformtion on. efinition. Let ϕ be n nlytic self-mp of the unit disk. Let 1 < α, β <. The weighted ϕ-berezin trnsform of mesurble function h is defined s follows. B ϕ,α,β (h)(z) = σ z(ϕ(w)) 2+α h(w) da β (w) = (1 z 2 ) 2+α h(w) 1 zϕ(w) 4+2α da β(w). We lso write I ϕ,α = I ϕ,α,α, B ϕ,α = B ϕ,α,α nd B ϕ = B ϕ,0. If ϕ(z) = z, B ϕ,α is just the usul weighted Berezin trnsform B α. For the cse q < p, we hve the following chrcteriztion of the boundedness of uc ϕ. Theorem 3. Let ϕ be n nlytic self-mp of the unit disk nd u be n nlytic function on. Let 1 q < p <, nd let 1 < α, β <. Then the following sttements re equivlent: (i) uc ϕ is bounded from L p,α to L q,β ; (ii) uc ϕ is compct from L p,α to L q,β ; (iii) B ϕ,α,β ( u q ) L p/(p q),α. For the weighted composition opertors between Hrdy spces, we obtin nlogous results using the relted integrl opertor ( ) 1 2 q/p I ϕ, 1 (u)() = 1 āϕ(w) 2 u(w) q dσ(w), where, is the unit circle nd dσ is the normlized rc length mesure on. Theorem 4. Let u be n nlytic function on nd ϕ be n nlytic self-mp of. Let 0 < p q <. Then the weighted composition opertor uc ϕ is bounded from H p into H q if nd only if sup I ϕ, 1 (u)() <.

4 482 ZELJKO CU CKOVI Ć AN RUHAN ZHAO We lso hve the following estimtes for the essentil norm of uc ϕ. Theorem 5. Let u be n nlytic function on nd ϕ be n nlytic self-mp of. Let 1 < p q <. Let uc ϕ be bounded from H p into H q. Then there is n bsolute constnt C 1 such tht lim sup 1 I ϕ, 1 (u)() uc ϕ q e C lim sup I ϕ, 1 (u)(). 1 In prticulr, uc ϕ is compct from H p into H q if nd only if lim sup I ϕ, 1 (u)() = 0. 1 Theorem 6. Let ϕ be n nlytic self-mp of the unit disk nd u be n nlytic function on. Let 1 q < p <. Let uc ϕ be bounded from H p into H q. Then uc ϕ is compct from H p into H q if nd only if ϕ(z) < 1.e on. Theorems 1 6 re going to be proved in the Sections 2 5, respectively. Throughout the pper, C represents constnt which my vry from line to line. 2. Boundedness between L p,α nd L q,β for p q In this section we prove Theorem 1. Our min tool is the Crleson mesure on the weighted Bergmn spce. Let µ be positive Borel mesure on. Let X be Bnch spce of nlytic functions on. Let q > 0. We sy tht µ is n (X, q)-crleson mesure if there is constnt C > 0 such tht, for ny f X, f(z) q dµ(z) C f q X. Let I be n rc in the unit circle. Let S(I) be the Crleson squre defined by S(I) = {z : 1 I z < 1, z/ z I}. The following result is well-known. Theorem A. Let µ be positive Borel mesure on. Let 0 < p q < nd 1 < α <. Then the following sttements re equivlent: (i) There is constnt C 1 > 0 such tht, for ny f L q, f(z) q dµ(z) C 1 f q L. p,α (ii) There is constnt C 2 > 0 such tht, for ny rc I, µ(s(i)) C 2 I (2+α)q/p.

5 WEIGHTE COMPOSITION OPERATORS 483 (iii) There is constnt C 3 > 0 such tht, for every, σ (z) (2+α)q/p dµ(z) C 3. The result ws proved by severl uthors. The equivlence of (i) nd (ii) cn be found in [H] nd [L1], nd proof of the equivlence of (ii) nd (iii) cn be found in [ASX]. Notice tht the best constnts C 1, C 2 nd C 3 in this theorem re in fct comprble, which mens tht there is positive constnt M, independent of µ, such tht 1 M C 1 C 2 MC 1, 1 M C 1 C 3 MC 1. To check this fct, one my refer to the proof of Theorem in [Zhu1, p ] nd [ASX]. We define µ(s(i)) µ = sup I I. (2+α)q/p Then µ nd the bove constnts re comprble. Now we re redy to prove Theorem 1. Proof of Theorem 1. By definition, uc ϕ is bounded from L p,α nd only if for ny f L p,α tht is, (2) Letting w = ϕ(z) we get, (uc ϕ )f q L q,β C f q L p,α, u(z) q f(ϕ(z)) q da β (z) C f q L p,α. f(w) q dµ u (w) C f q L p,α, into L q,β where µ u = ν u ϕ 1 nd dν u (z) = u(z) q da β (z). But (2) mens tht dµ u is n (L p,α, q)-crleson mesure. By Theorem A, this is equivlent to σ (w) (2+α)q/p dµ u (w) <. sup Chnging the vrible bck to z we get (1). The proof is complete. Using the corresponding results on (H p, q)-crleson mesures for 0 < p q < nlogous to Theorem A (see [] nd [ASX]), the proof of Theorem 4 follows similrly. if

6 484 ZELJKO CU CKOVI Ć AN RUHAN ZHAO We need the following two lemms. Lemm Essentil norm estimtes Let 0 < r < 1. Let µ be positive Borel mesure on. Let Nr = sup σ (z) (2+α)q/p dµ(z). r If µ is n (L p,α, q)-crleson mesure for 0 < p q <, then so is µ r = µ \r, where r = {z : z < r}. Moreover, µ r MN r, where M is n bsolute constnt. The proof is the sme s the proof of Lemm 1 nd Lemm 2 in [CZ], nd thus is omitted here. For f(z) = k=0 kz k nlytic on, let K n f(z) = n k=0 kz k nd R n = I K n, where If = f is the identity mp. Hence R n f(z) = k=n+1 kz k. Then we hve: Lemm 2. then If uc ϕ is bounded from L p,α uc ϕ e lim inf n uc ϕr n. into L q,β for 0 < p q <, Proof. Since (R n + K n )f = f nd K n is compct, we hve for ech n, uc ϕ e uc ϕ R n + uc ϕ K n e uc ϕ R n e uc ϕ R n. Therefore uc ϕ e lim inf n uc ϕ R n. The proof of Theorem 2 is similr to the proof of Theorem 2 in [CZ], nd uses some estimtes from [Sh]. We sketch the proof here. Proof of Theorem 2. First we prove the upper estimte. By Lemm 2, uc ϕ e lim inf uc ϕr n lim inf n n sup f L p,α 1 (uc ϕ R n )f L q,β. However, for ny fixed 0 < r < 1, (3) (uc ϕ R n )f q = u(z) q (R L q,β n f)(ϕ(z)) q da β (z) = R n f(w) q dµ u (w) = R n f(w) q dµ u (w) + \ r R n f(w) q dµ u (w) r = I 1 + I 2, where µ u is the pull-bck mesure induced by ϕ defined in Section 2. Since uc ϕ is bounded from L p,α into L q,β, µ u is n (L p,α, q)-crleson mesure.

7 WEIGHTE COMPOSITION OPERATORS 485 From the proof of Proposition 3 in [CZ] we see tht, for given ε > 0, nd n big enough, Thus Hence, for fixed r, I 2 ε q f q L p,α R n f(w) ε f L p,α. µ u( r ) ε q f q L p,α sup f L p,α 1 I 2 0 s n. u q. L q,β On the other hnd, if we set µ u,r = µ u \r, then, by Theorem A nd Lemm 1, I 1 = R n f(w) q dµ u,r (w) K µ u,r R n f q L p,α \ r KCMN r f q L p,α, where K, C nd M re constnts independent of u nd r, nd Nr is defined s in Lemm 1. Here we hve lso used the inequlity R n f q C f q L p,α L, p,α which cn be esily proved by the tringle inequlity, nd the inequlity K n f q C f q, obtined in [Zhu3] (see Proposition 1 nd Corollry 4 there). Tking the supremum in (3) over nlytic functions f in the L p,α L p,α unit bll of L p,α, nd letting n, we get lim inf n sup f L p,α (uc ϕ R n )f q 1 L p,α Thus uc ϕ q e KCMN r. Letting r 1 we get lim inf n KCMN r = KCMN r. uc ϕ q e KCM lim Nr = KCM lim sup σ r 1 (w) (2+α)q/p dµ u (w) 1 = KCM lim sup σ (ϕ(z)) (2+α)q/p u(z) q da β (z) 1 = KCM lim sup I ϕ,α,β (u)(), 1 which gives us the desired upper bound. Now let us prove the lower estimte. Consider the normlized kernel function k (z) = σ (z) = (1 2 )/(1 āz) 2. Let f = k (2+α)/p. Then f L p,α = 1, nd f 0 uniformly on compct subsets of s 1. Fix compct opertor K from L p,α into L q,β. Then Kf L q,β 0 s 1.

8 486 ZELJKO CU CKOVI Ć AN RUHAN ZHAO Therefore, Thus uc ϕ K lim sup (uc ϕ K)f L q,β 1 lim sup 1 ( (uc ϕ )f L q,β = lim sup (uc ϕ )f. L q,β 1 uc ϕ q e lim sup (uc ϕ )f q L q,β 1 Kf L q,β ) = lim sup I ϕ,α,β (u)(). 1 The proof of Theorem 5 is similr to tht of Theorem 2, using modified versions of Lemm 1 (with α = 1) nd Lemm 2 for Hrdy spces. 4. Boundedness between L p,α nd L q,β for q < p In this section we prove Theorem 3. We need chrcteriztion of the (L p,α, q)-crleson mesure. The ide of the proof follows tht of Theorem 4.4 in [CKY]. efinition. For positive mesure µ on, 1 < α <, nd fixed number r (0, 1), define µ((z, r)) µ r,α (z) = (z, r), B α(µ)(z) = σ z(w) 2+α dµ(w), 1+α/2 where (z, r) = {w σ z (w) < r} is the pseudohyperbolic disk with center z nd rdius r nd (z, r) denotes the Lebesgue re mesure of (z, r). Recll tht for mesurble function h on, B α (h)(z) = σ z(w) 2+α h(w) da α (w). We need severl lemms. Lemm 3. Given 0 < r < 1, there exists constnt C = C r > 0 such tht C r g(z) g(w) da(w) (z, r) (z,r) for ll g subhrmonic on, nd ll z. The proof is the sme s tht of Proposition in [Zhu1, p. 62]. Lemm 4. Let 1 < α <. Let µ be positive mesure on. Given 0 < r < 1, there exists constnt C = C r > 0 such tht g(w) dµ(w) C g(w) µ r,α (w) da α (w). for ll g subhrmonic on.

9 WEIGHTE COMPOSITION OPERATORS 487 Proof. Since (1 z 2 ) 2 (1 w 2 ) 2 (z, r) (w, r) s z (w, r) (see for exmple [Zhu1, p. 61]), we hve dµ(z) µ((w, r)) C C(1 w 2 ) α µ r,α (w), (z, r) (w, r) (w,r) for some constnt C nd for ll w. Therefore, from Lemm 3 nd Fubini s Theorem, g(z) dµ(z) C = C The proof is complete. C 1 (z, r) g(w) (w,r) (z,r) g(w) da(w) dµ(z) dµ(z) (z, r) da(w) g(w) µ r,α (w)(1 w 2 ) α da(w). Lemm 5. Let 1 < α <. Let µ be positive mesure on. Given 0 < r < 1, there exists constnt C = C r > 0 such tht B α (µ)(z) CB α ( µ r,α )(z) for ny z. Proof. Setting g(w) = σ z(w) 2+α in Lemm 4, we get B α (µ)(z) = σ z(w) 2+α dµ(w) C σ z(w) 2+α µ r,α (w) da α (w) The proof is complete. = CB α ( µ r,α )(z). Lemm 6. ny p > 1. Let 1 < α <. Then B α is bounded opertor on L p,α for Proof. Let h(z) = (1 z 2 ) 1/(pq). By Lemm in [Zhu1, p. 53], it cn be esily checked tht σ w(z) 2+α h(z) q da α (z) Ch(w) q, nd σ w(z) 2+α h(w) p da α (w) Ch(z) p. Thus by Schur s Theorem (see, for exmple, Theorem in [Zhu1, p. 42]), B α is bounded on L p,α. Lemm 7. Let 1 < α <. Let µ be positive mesure on. Given 0 < r < 1, there exists constnt C = C r > 0 such tht µ r,α (z) CB α (µ)(z) for ny z.

10 488 ZELJKO CU CKOVI Ć AN RUHAN ZHAO Proof. Since (1 z 2 ) 2 (1 w 2 ) 2 (z, r) s w (z, r), we hve B α (µ)(z) = σ z(w) 2+α (1 σ z (w) 2 ) 2+α dµ(w) = (1 w 2 ) 2+α dµ(w) (1 r 2 ) 2+α dµ(w) (1 w 2 ) 2+α C µ((z, r)) r (z, r) 1+α/2 = C r µ r,α (z). The proof is complete. (z,r) Theorem 7. Let µ be positive mesure on. Let 0 < q < p < nd 1 < α <. Then the following sttements re equivlent: (i) µ is n (L p,α, q)-crleson mesure. (ii) For fixed r (0, 1), µ r,α L p/(p q),α. (iii) B α (µ) L p/(p q),α. Proof. The equivlence of (i) nd (ii) is given by Luecking [L2] [L4], for the cse α = 0. For 1 < α <, the result cn be similrly proved s in [L4]. We just need to prove (ii) nd (iii) re equivlent. However, (iii) (ii) is direct consequence of Lemm 7. To prove (ii) (iii), let µ r,α L p/(p q),α. By Lemm 6, B α ( µ r,α ) L p/(p q),α. By Lemm 5 we get tht B α (µ) L p/(p q),α. The proof is complete. Proof of Theorem 3. Let dν u (z) = u(z) q da β (z) nd µ u = ν u ϕ 1 be the pull-bck mesure of ν u. Then uc ϕ is bounded from L p,α to L q,β if nd only if for ny f L p,α, or u(z) q f(ϕ(z)) q da β (z) C f q p,α, f(w) q dµ u (w) C f q p,α. Thus µ u is n (L p,α, q)-crleson mesure. By Theorem 7, this is equivlent to B α (µ u ) L p/(p q),α. Thus (i) nd (iii) re equivlent since B α (µ u ) = B ϕ,α,β ( u q ). The equivlence of (i) nd (ii) follows from generl result of Bnch spce theory. It is known tht, for 1 q < p <, every bounded opertor from l p to l q is compct (see, for exmple [LT, p. 31, Theorem I.2.7]). Since the Bergmn spce L p,α is isomorphic to l p (see, [W, p. 89, Theorem 11]), we get the impliction (i) (ii) directly from the bove result. On the other hnd, it is obvious tht (ii) implies (i). If α = β = 0, then B ϕ,α,β (h) = B ϕ (h). consequence. Thus we hve the following

11 WEIGHTE COMPOSITION OPERATORS 489 Corollry 2. Let 1 q < p <. Then uc ϕ is compct from L p to L q if nd only if B ϕ ( u q ) L p/(p q). As byproduct, we show the boundedness of B ϕ,α on L p,α here. Proposition 1. For ny nlytic self-mp ϕ on nd p > 1, B ϕ,α is bounded opertor on L p,α. Proof. Let h L p,α. Let dν h = h(z) da α (z) nd µ h = ν h ϕ 1. Let q = p 1 nd (1/p ) + (1/p) = 1. Noticing tht qp = p, we hve, for ny f L p,α, h(z) f(ϕ(z)) q da α (z) ( ) 1/p ( h p da α (z) f(ϕ(z)) p da α (z) = h p,α f ϕ q p,α C h p,α f q p,α. ) 1/p The lst inequlity is true since the composition opertor C ϕ is lwys bounded on L p,α. Hence µ h is n (L p,α, q)-crleson mesure, nd by Theorem 7, B α (µ h ) L p,α. Noticing tht B ϕ,α (h) B ϕ,α ( h ) = B α (µ h ), we get tht B ϕ,α (h) L p,α. A stndrd ppliction of the Closed Grph Theorem shows tht B ϕ,α is bounded on L p,α. 5. Compctness between H p nd H q for q < p We first prove the following result, which is of independent interest. Theorem 8. Let ϕ be n nlytic self-mp of the unit disk, nd u be n nlytic function on. Let 1 < p <. Let uc ϕ be bounded from H p into H 1. Then uc ϕ is compct from H p into H 1 if nd only if ϕ(z) < 1.e. on. Proof. It is well-known tht the sequence {z n } is n H p -wekly null sequence. Thus the compctness of uc ϕ from H p to H 1 implies tht uc ϕ z n H 1 0 s n, i.e., lim n 2π 0 u(e iθ ) ϕ(e iθ ) n dθ = 0. Becuse uc ϕ is bounded from H p to H 1, it is cler tht u = uc ϕ 1 H 1. Hence the convergence condition bove mens tht {ξ : ϕ(ξ) = 1} hs mesure 0. Conversely, suppose ϕ(z) < 1.e. on. Let {f n } H p be n rbitrry wekly null sequence. This implies tht {f n } converges to 0 uniformly on compct subsets of. Since uc ϕ is bounded from H p to H 1, it tkes wekly null sequence in H p into wekly null sequence in H 1. Hence u(f n ϕ) 0

12 490 ZELJKO CU CKOVI Ć AN RUHAN ZHAO wekly in H 1. Since ϕ(z) < 1.e. on, it follows tht u(f n ϕ) 0.e. on. This mens tht u(f n ϕ) 0 in mesure. By the unford-pettis Theorem (see [S]), we hve uc ϕ f n H 1 0. Hence uc ϕ is completely continuous nd, by the reflexivity of H p, uc ϕ is compct. For proving Theorem 6, we first give the following criterion for boundedness of uc ϕ from H p to H q. Proposition 2. Let ϕ be n nlytic self-mp of the unit disk nd u be n nlytic function on. Let 1 q < p <. Then uc ϕ is bounded from H p to H q if nd only if ( 2π ) p/(p q) dµ u (w) 1 w 2 dθ <, 0 Γ(θ) where µ u = ν u ϕ 1 nd dν u (z) = u(z) q dσ(z) with dσ(z) the normlized mesure of, nd Γ(θ) is the Stolz ngle t θ, which is defined for rel θ s the convex hull of the set {e iθ } {z : z < 1/2}. Proof. The opertor uc ϕ being bounded from H p to H q mens tht, for ny f H p, u(z)f(ϕ(z)) q dσ(z) C f q H p. With the chnge of vrible w = ϕ(z) we get f(w) q dµ u (w) C f q H p, which mens tht dµ u is n (H p, q)-crleson mesure. By result of I. V. Videnskii [V] nd. Luecking [L3], this is equivlent to ( 2π ) p/(p q) dµ u (w) 1 w 2 dθ <. The result is proved. 0 Γ(θ) Remrk. From this result we cn esily see tht if u hs no zeros in, then uc ϕ is bounded from H p to H q if nd only if u q C ϕ is bounded from H p/q to H 1. Proof of Theorem 6. The necessity follows in the sme wy s in the previous theorem. To prove the sufficiency, we follow the rgument of Goebeler [G, Theorem 4]. Let us first ssume tht u is n outer function. Suppose {f n } is sequence in the unit bll of H p. For ech n, we write f n = B n F n, where B n is inner, F n is outer. Clerly, both sequences {B n } nd {F n } re contined in the unit bll of H p. The locl boundedness of these sequences shows tht they re

13 WEIGHTE COMPOSITION OPERATORS 491 norml fmilies; we cn use Montel s Theorem to extrct subsequences {B nj } nd {F nj } tht converge uniformly on compct subsets of. Put G j = Fn q j. Then G j is in the unit bll of H p/q. Now recll tht uc ϕ is bounded from H p to H q. From the remrk fter Proposition 2, this is equivlent to sying tht u q C ϕ is bounded from H p/q to H 1. Since ϕ(z) < 1.e. on by ssumption, Theorem 8 pplies nd it follows tht u q C ϕ is compct from H p/q to H 1. Therefore there is subsequence {G jk } of {G j } such tht the sequence {u q (G jk ϕ)} converges in the norm of H 1. Also, the fct tht ϕ(z) < 1.e. on implies {u q (G jk ϕ)} converges lmost everywhere on. Vitli s Convergence Theorem implies lim sup u q G jk ϕ dσ = 0, σ(e) 0 k E where σ denotes the normlized Lebesgue mesure on. As in Goebeler s proof, this implies lim sup u q f njk ϕ q dσ lim sup u q G jk ϕ dσ = 0. σ(e) 0 k σ(e) 0 k E E Agin, since ϕ(z) < 1.e. on, {u q (f njk ϕ)} converges lmost everywhere on. Using Vitli s Theorem gin, we conclude tht u(f njk ϕ) converges in H q. Hence uc ϕ is compct from H p to H q. In generl, if u is not outer, we cn fctor u = B u F u, where B u is inner nd F u is outer. It is cler tht uc ϕ is compct from H p to H q if nd only if F u C ϕ is compct from H p to H q. By the proof bove, this is equivlent to sying tht ϕ < 1.e. on. 6. Pointwise multipliction opertors In this section we show how our results led to the corresponding results bout boundedness, compctness nd essentil norm estimtes for the pointwise multipliction opertors between weighted Bergmn spces. In this setting, the results re expressed in terms of much simpler expressions thn the integrl opertors I ϕ,α,β used for the weighted composition opertors. Some of these results were given by the second uthor in [Zh]. We need the following lemms. Lemm 8. Let 0 < q <, 1 < β < nd 1 < s <. Then there is constnt C > 0, depending only on β nd s, such tht u() q (1 2 ) β+2 s C σ (w) s u(w) q da β (w).

14 492 ZELJKO CU CKOVI Ć AN RUHAN ZHAO Proof. By subhrmonicity we know tht for n nlytic function g in nd for ny fixed r, 0 < r < 1 (for exmple, one my choose r = 1/4), g(0) q 1 r 2 r g(ζ) q da(ζ). Replcing g by u σ, we hve u() q 1 r 2 u(σ (ζ)) q da(ζ) r = 1 r 2 u(w) q σ (w) 2 da(w) (,r) 16 r 2 (1 2 ) 2 (,r) u(w) q da(w). It is known tht for w (, r), 1 w (cf. [Zhu1, p. 61]). So there is constnt C, depending only on β nd s, such tht u() q (1 ) β+2 s 16(1 2 ) β s r 2 (,r) u(w) q da(w) 16C r 2 u(w) q (1 w 2 ) β s da(w) (,r) 16C r 2 (1 r 2 ) s u(w) q (1 w 2 ) β s (1 σ (w) 2 ) s da(w) (,r) = C σ (w) s u(w) q da β (w), where C = 16C /((1 + β)r 2 (1 r 2 ) s ). The proof is complete. Lemm 9., Let 0 < q <, 1 < β < nd 1 < s <. Then for every σ (w) s u(w) q da β (w) 1 + β s 1 sup u(w) q (1 w 2 ) β+2 s. w

15 WEIGHTE COMPOSITION OPERATORS 493 Proof. Since s > 1, σ (w) s u(w) q da β (w) (1 + β) sup u(w) q (1 w 2 ) β+2 s σ (w) s (1 w 2 ) s 2 da(w) w = (1 + β) sup u(w) q (1 w 2 ) β+2 s w (1 σ (w) 2 ) s (1 w 2 ) 2 da(w) = (1 + β) sup u(w) q (1 w 2 ) β+2 s (1 z 2 ) s 2 da(z) w = 1 + β s 1 sup u(w) q (1 w 2 ) β+2 s. w The proof is complete. Let M u denote the pointwise multipliction opertors. Then M u (f) = uf, nd M u is the weighted composition opertor uc ϕ with ϕ = id, the identity mp of. We hve the following result. Theorem 9. Let u be n nlytic function on. Let 0 < p q <, nd α, β > 1. Then the pointwise multipliction opertor M u is bounded from L p,α into L q,β if nd only if (4) sup u() (1 2 ) γ <, where γ = (β + 2)/q (α + 2)/p. Proof. By Theorem 1, we know tht M u is bounded from L p,α into L q,β nd only if sup I id,α,β (u)() = σ (w) (2+α)q/p u(w) q da β (w) <. The result clerly follows from Lemm 8 nd Lemm 9 (with s = (α + 2)q/p). Remrk. Let α > 0, nd let the Bloch type spce B α be the spce of nlytic functions f on such tht sup z f (z) (1 z 2 ) α <. It is known tht, s α > 1, f B α if nd only if sup z f(z) (1 z 2 ) α 1 < (see [Zhu2]). Therefore, if γ = (β + 2)/q (α + 2)/p > 0, condition (4) mens tht u B 1+γ. If γ = 0 or γ < 0, condition (4) is clerly the sme s u H, or u 0, respectively. Theorem 9 ws first proved by the second uthor in [Zh, Theorem 1 (i), (ii) nd (iii)]. if

16 494 ZELJKO CU CKOVI Ć AN RUHAN ZHAO The following result is new. Theorem 10. Let u be n nlytic function on. Let 1 < p q <, nd α, β > 1. Let M u be bounded from L p,α into L q,β. Then there is n bsolute constnt C 1 such tht lim sup 1 u() (1 2 ) γ M u e C lim sup u() (1 2 ) γ, 1 where γ = (β + 2)/q (α + 2)/p. Consequently, M u is compct from L p,α if nd only if L q,β (5) lim sup u() (1 2 ) γ = 0. 1 Proof. By Theorem 2, into M u e lim sup(i id,α,β (u)()) 1/q. 1 Applying Lemm 8 with s = (α + 2)q/p, we obtin constnt C such tht C(I id,α,β (u)()) 1/q u() (1 2 ) γ, which gives the lower estimte. We now obtin n upper estimte. Agin by Theorem 2 we know M u q e C lim sup(i id,α,β (u)()) 1 = C lim sup σ (w) (α+2)q/p u(w) q da β (w). 1 For ny fixed 0 < r < 1, we write the bove integrl s I 1 + I 2, where I 1 = σ (w) (α+2)q/p u(w) q da β (w), \ r nd I 2 = σ (w) (α+2)q/p u(w) q da β (w). r

17 WEIGHTE COMPOSITION OPERATORS 495 Then nd I 1 (1 + β) sup u(w) q (1 w 2 ) β+2 (α+2)q/p w \ r (1 σ (w) \r 2 ) (α+2)q/p (1 w 2 ) 2 da(w) (1 + β) sup u(w) q (1 w 2 ) β+2 (α+2)q/p w \ r (1 z 2 ) (α+2)q/p 2 da(z) C sup ( u(w) (1 w 2 ) γ ) q, w \ r I 2 (1 + β) sup u(w) q (1 w 2 ) β+2 (α+2)q/p w (1 σ (w) r 2 ) (α+2)q/p (1 w 2 ) 2 da(w) (1 + β) sup u(w) q (1 w 2 ) β+2 (α+2)q/p w (1 z 2 ) (α+2)q/p 2 da(z) (,r) C(1 2 ) (α+2)q/p 2 sup ( u(w) (1 w 2 ) γ ) q (, r) w C(1 2 ) (α+2)q/p sup ( u(w) (1 w 2 ) γ ) q, w where (, r) is the normlized re mesure of (, r). Here we used the fct tht (1 z 2 ) 2 (1 2 ) 2 (, r) for fixed r nd for ny z (, r). Since M u is bounded from L p,α into L q,β, by Theorem 9, we know tht sup w ( u(w) (1 w 2 ) γ ) q <. Notice tht (α + 2)q/p > 1 nd hence I 2 0 s 1. Therefore M u e lim sup I 1/q 1 C sup u(w) (1 w 2 ) γ 1 w \ r for ny fixed 0 < r < 1, which implies M u e lim sup w 1 u(w) (1 w 2 ) γ. The proof is complete. Remrk. Let α > 0, nd let B0 α be the spce of nlytic functions f on such tht lim z 1 f (z) (1 z 2 ) α = 0. It is known tht, s α > 1, f B0 α if nd only if lim z 1 f(z) (1 z 2 ) α 1 = 0 (see [Zhu2]). Therefore, s γ = (β + 2)/q (α + 2)/p > 0, condition (5) mens tht u B 1+γ 0. As γ 0, condition (5) implies u 0. Thus, s γ 0, M u is compct from L p,α into L q,β if nd only if u 0.

18 496 ZELJKO CU CKOVI Ć AN RUHAN ZHAO For the cse 0 < q < p <, Attele [A] chrcterized nlytic multipliers from L p into L q, nd the second uthor [Zh] extended this result to the weighted cses. We show here how these results follow from our Theorem 3. Theorem 11. Let u be n nlytic function on. Let 1 q < p <, nd α, β > 1. Then the following sttements re equivlent: (i) M u is bounded from L p,α to L q,β ; to L q,β ;, where 1/s = 1/q 1/p nd δ/s = β/q α/p. (ii) M u is compct from L p,α (iii) u L s,δ Proof. By Theorem 3, (i) nd (ii) re equivlent, nd both re equivlent to the condition (6) B id,α,β ( u q ) L p/(p q),α. Suppose (6) holds. From Lemm 8 it follows B id,α,β ( u q )() = σ (w) 2+α u(w) q da β (w) C 1 u() q (1 2 ) β α. We conclude tht u() q (1 2 ) β α L p/(p q),α, which is the sme s u L s,δ. Conversely, if u L s,δ, then by Hölder s inequlity we esily get M u is bounded from L p,α to L q,β. The proof is complete. References [A] K. R. M. Attele, Anlytic multipliers of Bergmn spces, Michign Mth. J. 31 (1984), MR (86g:46039) [ASX] R. Aulskri,. A. Stegeng, nd J. Xio, Some subclsses of BMOA nd their chrcteriztion in terms of Crleson mesures, Rocky Mountin J. Mth. 26 (1996), MR (97k:30045) [CKY] B. R. Choe, H. Koo, nd H. Yi, Positive Toeplitz opertors between the hrmonic Bergmn spces, Potentil Anl. 17 (2002), MR (2003d:47037) [CH] M.. Contrers nd A. G. Hernández-íz, Weighted composition opertors between different Hrdy spces, Integrl Equtions Opertor Theory 46 (2003), MR (2004c:47048) [CZ] Z. Cu cković nd R. Zho, Weighted composition opertors on the Bergmn spce, J. London Mth. Soc. (2) 70 (2004), MR (2005f:47064) [S] N. unford nd J. Schwrtz, Liner Opertors, Prt 1, Pure nd Applied Mthemtics, Vol. 7, Interscience Publishers, New York, MR (22 #8302) [] P. L. uren, Theory of H p spces, Pure nd Applied Mthemtics, Vol. 38, Acdemic Press, New York, MR (42 #3552) [G] T. E. Goebeler, Jr., Composition opertors cting between Hrdy spces, Integrl Equtions Opertor Theory 41 (2001), MR (2002g:47046) [GM] P. Gorkin nd B.. McCluer, Essentil norms of composition opertors, Integrl Equtions Opertor Theory 48 (2004), MR (2004j:47051) [HM] C. Hmmond nd B.. McCluer, Isoltion nd component structure in spces of composition opertors, Integrl Equtions Opertor Theory 53 (2005), MR (2006h:47041)

19 WEIGHTE COMPOSITION OPERATORS 497 [H] W. W. Hstings, A Crleson mesure theorem for Bergmn spces, Proc. Amer. Mth. Soc. 52 (1975), MR (51 #11082) [HJ] H. Hunziker nd H. Jrchow, Composition opertors which improve integrbility, Mth. Nchr. 152 (1991), MR (93d:47061) [J] H. Jrchow, Compctness properties of composition opertors, Rend. Circ. Mt. Plermo (2) Suppl. (1998), 91 97, Interntionl Workshop on Opertor Theory (Ceflù, 1997). MR (2000e:47047) [Li] S.-Y. Li, Trce idel criteri for composition opertors on Bergmn spces, Amer. J. Mth. 117 (1995), MR (96g:47023) [LT] J. Lindenstruss nd L. Tzfriri, Clssicl Bnch spces, Springer-Verlg, Berlin, 1973, Lecture Notes in Mthemtics, Vol MR (54 #3344) [L1]. H. Luecking, Forwrd nd reverse Crleson inequlities for functions in Bergmn spces nd their derivtives, Amer. J. Mth. 107 (1985), MR (86g:30002) [L2], Multipliers of Bergmn spces into Lebesgue spces, Proc. Edinburgh Mth. Soc. (2) 29 (1986), MR (87e:46034) [L3], Embedding derivtives of Hrdy spces into Lebesgue spces, Proc. London Mth. Soc. (3) 63 (1991), MR (92k:42030) [L4], Embedding theorems for spces of nlytic functions vi Khinchine s inequlity, Michign Mth. J. 40 (1993), MR (94e:46046) [Sh] J. H. Shpiro, The essentil norm of composition opertor, Ann. of Mth. (2) 125 (1987), MR (88c:47058) [Sm] W. Smith, Composition opertors between Bergmn nd Hrdy spces, Trns. Amer. Mth. Soc. 348 (1996), MR (96i:47056) [SY] W. Smith nd L. Yng, Composition opertors tht improve integrbility on weighted Bergmn spces, Proc. Amer. Mth. Soc. 126 (1998), MR (98d:47070) [V] I. V. Videnskiĭ, An nlogue of Crleson mesures, okl. Akd. Nuk SSSR 298 (1988), MR (89j:30047) [W] P. Wojtszczyk, Bnch spces for nlysts, Cmbridge Studies in Advnced Mthemtics, vol. 25, Cmbridge University Press, Cmbridge, MR (93d:46001) [Zh] R. Zho, Pointwise multipliers from weighted Bergmn spces nd Hrdy spces to weighted Bergmn spces, Ann. Acd. Sci. Fenn. Mth. 29 (2004), MR (2004m:30059) [Zhu1] K. H. Zhu, Opertor theory in function spces, Monogrphs nd Textbooks in Pure nd Applied Mthemtics, vol. 139, Mrcel ekker Inc., New York, MR (92c:47031) [Zhu2], Bloch type spces of nlytic functions, Rocky Mountin J. Mth. 23 (1993), MR (95d:46020) [Zhu3], ulity of Bloch spces nd norm convergence of Tylor series, Michign Mth. J. 38 (1991), MR (92h:30004)

20 498 ZELJKO CU CKOVI Ć AN RUHAN ZHAO Zeljko Cu cković, eprtment of Mthemtics, University of Toledo, Toledo, OH , USA E-mil ddress: Ruhn Zho, eprtment of Mthemtics, University of Toledo, Toledo, OH , USA Current ddress: eprtment of Mthemtics, SUNY Brockport, Brockport, NY 14618, USA E-mil ddress: