Weighted Composition Operator from Bers-Type Space to Bloch-Type Space on the Unit Ball

Transcription

1 BULLETIN of the MALAYSIAN MATHEMATICAL SCIENCES SOCIETY Bull. Malays. Math. Sci. Soc , Weighted Compositio Operator from Bers-Type Space to Bloch-Type Space o the Uit Ball 1 YU-XIA LIANG, 2 ZE-HUA ZHOU AND 3 XING-TANG DONG 1,2,3 Departmet of Mathematics, Tiaji Uiversity, Tiaji , P.R. Chia 1 liagyx1986@126.com, 2 zehuazhou2003@yahoo.com.c, 3 dogxigtag@163.com Abstract. I this paper, we characterize the boudedess ad compactess of weighted compositio operator from Bers-type space to Bloch-type space o the uit ball of C. 1. Itroductio 2010 Mathematics Subject Classificatio: Primary: 47B38; Secodary: 32A37, 32A38, 32H02, 47B33 Keywords ad phrases: Weighted compositio operator, Bers-type space, Bloch-type space, boudedess, compactess. Let HB be the class of all holomorphic fuctios o B ad SB the collectio of all the holomorphic self-maps of B, where B is the uit ball i the -dimesioal complex space C. The Bloch space B see, e.g. [22] is defied as the space of holomorphic fuctios such that f B = sup { 1 z 2 R f z : z B } <. For each α > 0, we defie a weighted-type spaces Hα see, e.g. [11] as follows: { } Hα = f HB : sup 1 r 2 α M f,r < 0<r<1, where M f,r = sup z =r f z. It is easy to see that f H α if ad oly if sup z B 1 z 2 α f z <, so we defie the orm f H α = sup 1 z 2 α f z ad Hα with this orm is a Baach space. It is sometimes called Bers-type space which is a special case of the weighted-type space Hµ see, e.g. [4]. Whe α = 0, the space Hα is just H see, e.g. [8, 9, 18], which is defied by { } H = Commuicated by Rosiha M. Ali, Dato. Received: October 26, 2011; Revised: Jauary 21, f HB : f = sup f z <.

2 834 Y. X. Liag, Z. H. Zhou ad X. T. Dog A positive cotiuous fuctio µ o [0,1 is called ormal [12], if there exist three costats a,b 0 < a < b <, ad δ 0 δ < 1, such that for r [δ,1 µr 1 r a 0, µr 1 r b as r 1. I the rest of this paper we always assume that µ is ormal o [0,1, ad from ow o if we say that a fuctio µ : B [0, is ormal we will also assume that it is radial o B, that is, µz = µ z,z B. Now f HB is said to belog to Bloch-type space B µ see, e.g. [10, 14], if f µ,1 = sup µz f z <, where f z = f / z 1 z,..., f / z z is the complex gradiet of f. It is clear that B µ is a Baach space with orm f Bµ = f 0 + f µ,1. For f HB, we deote f µ,2 = sup z B µz R f z ad f µ,3 = sup z B Q µ f z, where R f z = f z, z = j=1 f z j z, Q µ f z = sup z j u C \{0} { } G z µ u,u = 1 µ 2 z µ 2 z σµ z 2 u µ2 z σµ z 2 z,u 2 z 2 G µ 0 u,u = u 2 µ 2 0 ad 1 σ µ t = 1 t µ0 + dτ 1 τ 1/2 µτ 0 f z,ū, G z µ u,u z 0, 0 t < 1. It was proved that f µ,1, f µ,2 ad f µ,3 are equivalet for f B µ i [1] ad [21]. Let ϕ SB ad ψ HB. The weighted compositio operator T ψ,ϕ is defied by T ψ,ϕ f = ψ f ϕ, f HB. We ca regard this operator as a geeralizatio of a multiplicatio operator M ψ ad a compositio operator C ϕ see, e.g. [1, 2, 5, 6, 15, 25, 26]. That is whe ϕz z we obtai T ψ,ϕ f z = M ψ f z = ψz f z ad whe ψz 1 we obtai T ψ,ϕ f z = C ϕ f z = f ϕz. Recetly, Stević characterized the boudedess ad compactess of the weighted compositio operators betwee mixed-orm spaces ad H α spaces i the uit ball i [7]. Moreover, Zhag ad coauthor discussed the coditios for which the weighted compositio operator is bouded or compact from Bergma space to µ-bloch space i [19] ad [21]. Zhou ad Che discussed weighted compositio operators from Fp, q, s to Bloch type spaces o the uit ball i [20]. For some recet related results, see also [13, 16, 17, 23, 24] ad the refereces therei. Now i this article, we give some ecessary ad sufficiet coditios for the weighted compositio operator T ψ,ϕ to be bouded ad compact from weighted-type spaces H α to Bloch-type space B µ o the uit ball of C. Throughout the remaider of this paper, C will deote a positive costat, the exact value of which may vary from oe appearace to the ext. The otatio A B meas that there is a positive costat C such that B/C A CB. The symbol N stads for the set of positive itegers.

3 Weighted Compositio Operator from Bers-Type Space to Bloch-Type Space o the Uit Ball Some lemmas To begi the discussio, let us state a couple of lemmas which will be used i the proof of the mai results. The followig lemma was proved i [3]. Lemma 2.1. Let α > 0 ad m be a positive iteger. The for every f HB it holds sup 1 r 2 α M f,r f 0 + sup 1 r 2 α+m M R m f,r. 0<r<1 0<r<1 Lemma 2.2. Let α > 0. The for every f HB it holds f H R f z C α 1 z 2 α+1. Proof. Usig Lemma 2.1 with m = 1 we obtai f H α = sup 1 r 2 α M f,r C sup 1 r 2 α+1 M R f,r C1 z 2 α+1 R f z. 0<r<1 0<r<1 From which the desired estimate follows. From Lemma 2.2 we ca easily obtai f B α+1 ad f B α+1 C f H α for f Hα. For z B, u C, deote H z u,u = 1 z 2 u 2 + z,u 2 1 z 2 2. It is well-kow that H z u,u is the Bergma metric of B see, e.g. [22]. Lemma 2.3. Let α > 0, vr = 1 r 2 α+1 ad ϕ SB. The G v ϕz Jϕzz,Jϕzz CH ϕzjϕzz,jϕzz 1 ϕz 2 2α for all z B, where Jϕz deotes the Jacobia matrix of ϕz ad T ϕ 1 ϕ Jϕzz = z k,..., z k z k. z k k=1 Proof. If ϕz = 0, the desired result is obvious. If ϕz 0, for the defiitio of σ v, we have 1 r σ v r = 1 + dt 0 1 t 1/2 1 t 2 α+1 1 r2 1/2, 0 r < 1. vr Thus, G v ϕz Jϕz,Jϕzz [ 1 v 2 ] ϕz = v 2 ϕz σv 2 ϕz Jϕzz v2 ϕz ϕz,jϕzz 2 σv 2 ϕz ϕz 2 [ 1 v 2 ] ϕz = Jϕzz 2 v 2 ϕz σv 2 ϕz,jϕzz 2 ϕz ϕz 2 + ϕz,jϕzz 2 ϕz 2 [ ] C v 2 1 ϕz 2 Jϕzz 2 ϕz,jϕzz 2 ϕz ϕz 2 + ϕz,jϕzz 2 ϕz 2 C [ = 1 ϕz 2 v 2 Jϕzz 2 + ϕz,jϕzz 2] ϕz k=1

4 836 Y. X. Liag, Z. H. Zhou ad X. T. Dog = C1 ϕz 2 2 v 2 H ϕz ϕz Jϕzz,Jϕzz = C H ϕzjϕzz,jϕzz 1 ϕz 2 2α. From which the desired result follows. Lemma 2.4. Assume that f HB ad ϕ SB. The Proof. R f ϕz = R f ϕz = f ϕz,jϕzz. = i=1 f ϕ z i = z i f ϕ j=1 w j i=1 i i=1z j=1 z i ϕ j z i f ϕ ϕ j w j z i = f ϕz,jϕzz. By Motel theorem ad the defiitio of compact operator, the followig lemma follows. The iterested reader ca also see the Lemma 2.1 i [5]. Hece we omit it. Lemma 2.5. Assume that 0 < α <, µ is a ormal fuctio o [0,1, ϕ SB ad ψ HB. The T ψ,ϕ : H α B µ is compact if ad oly if for ay bouded sequece { f k } k N H α which coverges to zero uiformly o compact subsets of B as k, we have T ψ,ϕ f k Bµ 0 as k. 3. The boudedess ad compactess of T ψ,ϕ : H α B µ. I this sectio we characterize the boudedess ad compactess of the operator T ψ,ϕ : H α B µ. Theorem 3.1. Suppose that 0 < α <, µ is a ormal fuctio o [0,1,ϕ SB ad ψ HB. The T ψ,ϕ : H α B µ is bouded if ad oly if µz Rψz 3.1 M 1 := sup 1 ϕz 2 α < ad µz ψz { 3.2 M 2 := sup Hϕz 1 ϕz 2 α Jϕzz,Jϕzz } 1/2 <. Proof. Assume that 3.1 ad 3.2 hold. The for ay f H α, if Jϕzz 0, for z B. By Lemma 2.4, Lemma 2.3 ad Lemma 2.2 we have µz RT ψ,ϕ f z µz Rψz f ϕz + µz ψz R f ϕz 3.3 µz Rψz f H α 1 ϕz 2 α + µz ψz f ϕz,jϕzz M 1 f H α + Cµz ψz {H ϕzjϕzz,jϕzz} 1/2 f ϕz,jϕzz 1 ϕz 2 α G v ϕz Jϕzz,Jϕzz M 1 f H α +CM 2 f B1 r 2 α+1 C f H α.

5 Weighted Compositio Operator from Bers-Type Space to Bloch-Type Space o the Uit Ball 837 Whe z B ad Jϕzz = 0, from 3.1 we ca easily obtai that 3.4 µz RT ψ,ϕ f z M 1 f H α. Combiig 3.3 with 3.4 it follows that T ψ,ϕ f Bµ = sup µz RT ψ,ϕ f z C f H α. From which the boudedess of T ψ,ϕ : Hα B µ follows. For the coverse directio, we suppose that T ψ,ϕ : Hα B µ is bouded. First, we assume that w B ad ϕw = r w e 1, where r w = ϕw ad e 1 is the vector 1,0,0,...,0. If 1 rw η η 2 η 1, where Jϕww = η 1,...,η T. We cosider the fuctio { } 1 r 2 α f w z = w z1 r w 1 r w z r w z 1 The sup1 z 2 α 1 z 1 2 α f w z sup 1 z 1 α { } 1 r 2 α w 4 α. 1 r w It shows that f w Hα ad f w H α C. Note that f w ϕw = 0 ad 1 f w ϕw = 1 rw 2,0,...,0. α+1 It follows from Lemma 2.4 that 3.5 T ψ,ϕ f w Bµ µw Rψ f w ϕw µw ψw R f w ϕw µw Rψw f w ϕw = µw ψw f w ϕw,jϕww = µw ψw η 1 1 r 2 w α+1. By the defiitio of H ϕw Jϕww,Jϕww ad 3.5, it follows that 3.6 µw ψw 1 ϕw 2 α {H ϕwjϕww,jϕww} 1/2 = µw ψw {1 ϕw 2 Jϕww 2 + ϕw,jϕww 2 } 1/2 1 ϕw 2 α+1 = µw ψw {1 r2 w η η 2 + η 1 2 } 1/2 1 ϕw 2 α+1 2µw ψw η1 1 rw 2 α+1 C T ψ,ϕ f w Bµ. This shows that whe 1 r 2 w η η 2 η 1, the result 3.2 holds. O the other had, if 1 r 2 w η η 2 > η 1. For j = 2,...,, let θ j = argη j ad a j = e iθ j, whe η j 0 or a j = 0 whe η j = 0. Takig f w z = a 2z a z 1 r w z 1 α+1.

6 838 Y. X. Liag, Z. H. Zhou ad X. T. Dog It is easy to check that sup1 z 2 α f w z sup1 z 2 α 1 1 z 1 α 1 r w C, which implies that f w Hα ad f w H α C. Notice that f w ϕw = 0 ad a 2 f w ϕw = 0, 1 rw 2 α+1,..., a 1 rw 2 α+1. Similar to the proof of 3.5, we obtai that 3.7 It follows from 3.7 that 3.8 µw ψw η η 1 r 2 w α+1 C T ψ,ϕ f w Bµ. µw ψw 1 ϕw 2 α {H ϕwjϕww,jϕww} 1/2 = µw ψw {1 ϕw 2 Jϕww 2 + ϕw,jϕww 2 } 1/2 1 ϕw 2 α+1 = µw ψw {1 r2 w η η 2 + η 1 2 } 1/2 1 ϕw 2 α+1 µw ψw {21 r2 w η η 2 } 1/2 1 ϕw 2 α+1 C µw ψw 21 r 2 w η η 1 r 2 w α+1 C T ψ,ϕ f w Bµ. Therefore, whe 1 r 2 w η η 2 > η 1, we ca also obtai 3.2. Combiig the two cases we kow that 3.2 holds. For the geeral situatio, we ca use the uitary trasform U w to make ϕw = r w e 1 U w. I order to prove 3.2, we first give a propositio. Propositio 3.1. Suppose that 0 < α <, µ is a ormal fuctio o [0,1, ϕ SB ad ψ HB. Let ϕz = U w ϕz, ad g = f Uw 1 for ay f Hα. The a H ϕz J ϕzz,j ϕzz = H ϕz Jϕzz,Jϕzz; b g H α = f H α ; c T ψ, ϕ g Bµ = T ψ,ϕ f Bµ. Proof. a Note that J ϕzz = U w Jϕzz ad ϕz 2 = ϕz 2, we have 1 ϕz 2 J ϕzz 2 + ϕz,j ϕzz 2 H ϕz J ϕzz,j ϕzz = 1 ϕz 2 2 = 1 ϕz 2 Jϕzz 2 + ϕz,jϕzz 2 1 ϕz 2 2 = H ϕz Jϕzz,Jϕzz.

7 Weighted Compositio Operator from Bers-Type Space to Bloch-Type Space o the Uit Ball 839 b g H α = sup 1 z 2 α gz = sup 1 z 2 α f U 1 w z = sup 1 z 2 α f z = f H α. I the last equality, we use the liear coordiate traslatio w = Uw 1 z ad w = Uw 1 z = z. c T ψ, ϕ g Bµ = sup µz ψz g ϕz = sup µz ψz f ϕz = T ψ,ϕ f Bµ. Now we retur to prove that 3.2 holds i geeral situatio. I fact, takig the fuctio g w = f w Uw 1. By Propositio 3.1, 3.6 ad 3.8, it follows that µw ψw { Hϕw 1 ϕw 2 α Jϕww,Jϕww } 1/2 = µw ψw 1 ϕw 2 α { H ϕw J ϕww,j ϕww } 1/2 C T ψ, ϕ g w Bµ = C T ψ,ϕ f w Bµ C. This meas that 3.2 holds. Next we prove 3.1. Set the fuctio { } 1 ϕw 2 α h w z = 1 z,ϕw 2, w B. Sice sup1 z 2 α 1 z 2 α h w z sup 1 z α 1 ϕw 2 α 4 α, 1 ϕw it follows that h w Hα ad h w H α C. Moreover h w ϕw = 1/1 ϕw 2 α ad ϕ 1 w h w ϕw = 2α 1 ϕw 2 α+1,..., ϕ w 1 ϕw 2 α+1. Thus 3.9 T ψ,ϕ h w Bµ µw Rψh w ϕw From 3.2 ad h w ϕw we have 3.10 = µw Rψwh w ϕw + ψwrh w ϕw µw Rψw 1 ϕw 2 α µw ψw Rh w ϕw. µw ψw Rh w ϕw = µw ψw h w ϕw,jϕww = 2αµw ψw ϕw,jϕww 1 ϕw 2 α+1 2αµw ψw 1 ϕw 2 α { Hϕw Jϕww,Jϕww } 1/2 CM 2 <.

8 840 Y. X. Liag, Z. H. Zhou ad X. T. Dog Combiig 3.9 ad 3.10 we obtai 3.1. This completes the proof of Theorem 3.1. Theorem 3.2. Suppose that 0 < α <, µ is a ormal fuctio o [0,1, ϕ SB ad ψ HB. The T ψ,ϕ : H α B µ is compact if ad oly if the followigs are all satisfied: a ψ B µ ad ψϕ l B µ for l {1,...,}; b 3.11 lim c 3.12 lim ϕz 1 ϕz 1 µz Rψz 1 ϕz 2 α = 0; µz ψz 1 ϕz 2 α {H ϕzjϕzz,jϕzz} 1/2 = 0. Proof. First suppose that a, b ad c hold. The from b ad c we have for ay ε > 0, there is a δ > 0, such that 3.13 ad 3.14 µz Rψz 1 ϕz 2 α < ε µz ψz 1 ϕz 2 α {H ϕzjϕzz,jϕzz} 1/2 < ε, whe ϕz > δ. Let { f k } k N be ay sequece which coverges to 0 uiformly o compact subsets of B satisfyig f k H α 1. The f k ad f k coverge to 0 uiformly o K = {w B : w δ}. Sice 3.15 sup µz RT ψ,ϕ f k z = sup µz RT ψ,ϕ f k z ϕz K + sup µz RT ψ,ϕ f k z. ϕz B \K If ϕz > δ ad Jϕzz 0, by Lemma 2.4, Lemma 2.3 ad Lemma 2.2 we have 3.16 Whe Jϕzz = 0, 3.17 µz RT ψ,ϕ f k z µz ψz R f k ϕz + µz Rψz f k ϕz Cµz ψz {H ϕzjϕzz,jϕzz} 1/2 f k ϕz,jϕzz 1 ϕz 2 α G v ϕz Jϕzz,Jϕzz + ε f k H α Cε f k B1 r 2 α+1 + ε f k H α Cε. µz RT ψ,ϕ f k z ε f k H α ε. Combiig 3.16 ad 3.17 we obtai that 3.18 If ϕz δ, it follows from a that µz RT ψ,ϕ f k z sup µz RT ψ,ϕ f k z Cε. ϕz B \K µz ψz R f k ϕz + µz Rψz f k ϕz µz ψz f k ϕz,jϕzz + f k ϕz ψ Bµ

9 3.19 Weighted Compositio Operator from Bers-Type Space to Bloch-Type Space o the Uit Ball 841 f k ϕz f k ϕz l=1 l=1 µz ψz Rϕ l z + f k ϕz ψ Bµ µz ψz Rϕ l z µz Rψz ϕ l z + µz Rψz + f k ϕz ψ Bµ f k ϕz µz ψzrϕ l z + Rψzϕ l z + µz Rψz l=1 + f k ϕz ψ Bµ f k ϕz ψϕ l Bµ + ψ Bµ + f k ϕz ψ Bµ 0, k. l=1 The from 3.15, 3.18, 3.19 ad Lemma 2.5 we get the compactess of T ψ,ϕ : H α B µ. For the coverse directio, we assume that T ψ,ϕ : H α B µ is compact. The the boudedess of T ψ,ϕ : H α B µ is obvious. Takig f z = 1 H α, we obtai T ψ,ϕ f Bµ = sup µz RT ψ,ϕ f z = sup µz Rψz f ϕz + ψzr f ϕz = sup µz Rψz <. This shows that ψ B µ. O the other had, for l {1,...,}, takig the fuctios f z = z l H α, we ca obtai T ψ,ϕ f Bµ = sup µz Rψz f ϕz + ψzr f ϕz = sup µz Rψzϕ l z + ψzrϕ l z = sup µz Rψϕ l z. The we obtai that ψϕ l B µ for l {1,...,}. The desired result a follows. Next we prove Let {z k } k N be a sequece i B such that ϕz k 1 as k If such a sequece does ot exist the 3.12 obviously holds. We ca suppose that ϕz k = r k e 1, where r k = ϕz k, e 1 is the vector 1,0,0,...,0. Thus r k 1 as k. If 1 rk 2 η η 2 η 1, where Jϕz k z k = η 1,...,η T. Defiig the fuctio { 1 r 2 } α f k z = k z1 r k 1 r k z 1 2, k N. 1 r k z 1 From Theorem 3.1 we kow that f k Hα with f k H α C, ad otice that f k coverges to 0 uiformly o compact subsets of B whe k. From the compactess of T ψ,ϕ : Hα B µ, we have that lim T ψ,ϕ f k Bµ = 0. The from the similar proof of 3.3 i Theorem k 3.1 we have 3.20 µz k ψz k η 1 1 r 2 k α+1 T ψ,ϕ f k Bµ 0, k. Ad by the similar proofs of 3.6 ad 3.20 we have µz k ψz k 1 ϕz k 2 α { Hϕzk Jϕz k z k,jϕz k z k } 1/2

10 842 Y. X. Liag, Z. H. Zhou ad X. T. Dog 2µzk ψz k η rk 2 0, k. α+1 O the other had, if 1 rk 2 η η 2 > η 1. For j = 2,...,, let θ j = argη j ad a j = e iθ j, whe η j 0 or a j = 0 whe η j = 0. Takig f k z = a 2z a z 1 r 2 k 1 r k z 1 α+2. The from Theorem 3.1 we kow f k Hα,k N ad f k coverges to 0 uiformly o compact subsets of B whe k. Sice the compactess of T ψ,ϕ : Hα B µ, we have that lim k T ψ,ϕ f k Bµ = 0. We otice that f k ϕz k = 0 ad f w ϕz k = 0, Usig the similar proof of 3.7 we obtai 3.22 Ad by usig 3.8 we obtai 3.23 a 2 1 r 2 k α+1,..., a 1 r 2 k α+1 µz k ψz k η η 1 r 2 k α+1 T ψ,ϕ f k Bµ 0, k. µz k ψz k { 1 ϕz k 2 α Hϕzk Jϕz k z k,jϕz k z k } 1/2 C µz k ψz k 21 rk 2 η η 1 rk 2 0, k. α+1 Combiig 3.21 ad 3.23 we obtai 3.12 uder two cases. For the geeral situatio, we ca use the uitary trasform U k to make ϕz k = r k e 1 U k ad we ca prove 3.12 by takig the fuctio g k = f k Uk 1. Next we prove We still let {z k } k N be a sequece i B such that ϕz k 1 as k If such a sequece does ot exist the 3.11 obviously holds. Choosig { 1 ϕzk 2 } α h k z = 1 z,z k 2. From Theorem 3.1 we kow that h k H α ad h k 0 uiformly o the compact subsets of B whe k. By the compactess of T ψ,ϕ : H α B µ ad by the similar proof of 3.9 we obtai that 3.24 T ψ,ϕ h k Bµ µz k Rψz k 1 ϕz k 2 α µz k ψz k Rh k ϕz k. Sice from 3.12 we have that 3.25 µz k ψz k Rh w ϕz k 2αµz k ψz k 1 ϕz k 2 α { Hϕzk Jϕz k z k,jϕz k z k } 1/2 0, k. Combiig 3.24 ad 3.25 we obtai Remark 3.1. Whe ψz 1,T ψ,ϕ = C ϕ, we obtai the ext two Corollaries about compositio operator from Theorems 3.1 ad 3.2.

11 Weighted Compositio Operator from Bers-Type Space to Bloch-Type Space o the Uit Ball 843 Corollary 3.1. Suppose that 0 < α <, µ is a ormal fuctio i [0,1 ad ϕ SB. The C ϕ : H α B µ is bouded if ad oly if µz{h ϕz sup Jϕzz,Jϕzz}1/2 1 ϕz 2 α <. Corollary 3.2. Suppose that 0 < α <, µ is a ormal fuctio i [0,1 ad ϕ SB. The C ϕ : H α B µ is compact if ad oly if ad ϕ l B µ for ay l {1,...,}. µz{h ϕz Jϕzz,Jϕzz} 1/2 lim ϕz 1 1 ϕz 2 α = 0. Remark 3.2. Whe ϕz z,t ψ,ϕ = M ψ, we obtai the ext two Corollaries about multiplicatio operator from Theorems 3.1 ad 3.2. Corollary 3.3. Suppose that 0 < α <, µ is a ormal fuctio o [0,1 ad ψ HB. The M ψ : H α B µ is bouded if ad oly if µz Rψz µz ψz sup 1 z 2 < ad sup α 1 z 2 <. α+1 Corollary 3.4. Suppose that 0 < α <, µ is a ormal fuctio o [0,1 ad ψ HB. The M ψ : Hα B µ is compact if ad oly if a ψ B µ ad z l ψ B µ for ay l {1,...,}; b µz Rψz lim z 1 1 z 2 α = 0; c µz ψz lim z 1 1 z 2 = 0. α+1 Ackowledgmet. The authors thak the editor ad referees for may useful commets ad suggestios. This paper was supported i part by the Natioal Natural Sciece Foudatio of Chia Grat Nos , Refereces [1] H. Che ad P. Gauthier, Boudedess from below of compositio operators o α-bloch spaces, Caad. Math. Bull , o. 2, [2] C. C. Cowe ad B. D. MacCluer, Compositio operators o spaces of aalytic fuctios, Studies i Advaced Mathematics, CRC, Boca Rato, FL, [3] Z. Hu, Exteded Cesàro operators o mixed orm spaces, Proc. Amer. Math. Soc , o. 7, [4] X. Lü, Weighted compositio operators from Fp, q, s spaces to Bers-type spaces i the uit ball, Appl. Math. J. Chiese Uiv. Ser. B , o. 4, [5] K. Madiga ad A. Matheso, Compact compositio operators o the Bloch space, Tras. Amer. Math. Soc , o. 7, [6] S. Oho, K. Stroethoff ad R. Zhao, Weighted compositio operators betwee Bloch-type spaces, Rocky Moutai J. Math , o. 1, [7] S. Stević, Weighted compositio operators betwee mixed orm spaces ad H α spaces i the uit ball, J. Iequal. Appl. 2007, Art. ID 28629, 9 pp.

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