Research Article Norm and Essential Norm of an Integral-Type Operator from the Dirichlet Space to the Bloch-Type Space on the Unit Ball

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1 Hindawi Publishing Corporation Abstract and Applid Analysis Volum 2010, Articl ID , 9 pags doi: /2010/ Rsarch Articl Norm and Essntial Norm of an Intgral-Typ Oprator from th Dirichlt Spac to th Bloch-Typ Spac on th Unit Ball Stvo Stvić Mathmatical Institut of th Srbian Acadmy of Scincs, Knz Mihailova 36/III, Bograd, Srbia Corrspondnc should b addrssd to Stvo Stvić, sstvic@ptt.rs Rcivd 1 August 2010; Accptd 1 Octobr 2010 Acadmic Editor: Irna Lasicka Copyright q 2010 Stvo Stvić. This is an opn accss articl distributd undr th Crativ Commons Attribution Licns, which prmits unrstrictd us, distribution, and rproduction in any mdium, providd th original work is proprly citd. Oprator norm and ssntial norm of an intgral-typ oprator, rcntly introducd by this author, from th Dirichlt spac to th Bloch-typ spac on th unit ball in C n ar calculatd hr. 1. Introduction Lt B n B b th opn unit ball in C n, B 1 D th opn unit disk in C, H B th class of all holomorphic functions on B, andh B, th spac consisting of all f H B such that f z B f z <. For an f H B with th Taylor xpansion f z α a αz α,lt Rf z α α a α z α 1.1 b th radial drivativ of f, whr α α 1,...,α n is a multi-indx, α α 1 α n and z α z α 1 1 zα n n.ltα! α 1! α n!. Th Dirichlt spac D 2 B D 2 contains all f z α a αz α H B, such that f 2 : D f 0 2 α α! 2 α! a α 2 <. α 1.2

2 2 Abstract and Applid Analysis Th quantity f D 2 is a norm on D 2 which for n 1 is qual to usual norm f 0 f D 2 D 2 whr da z 1/π rdrdθis th normalizd ara masur on D. Th innr product, btwn two functions D f z 2 da z, 1.3 f z α a α z α, g z α b α z α, 1.4 on D 2 is dfind by f, g : f 0 g 0 α α α! α! a αb α. 1.5 For α / 0, lt α! α z α α! zα, z B, 1.6 and 0 z 1, thn it is asy to s that th family { α } is an orthonormal basis for D 2,and hnc th rproducing krnl K w z for D 2 is givn by 1 as follows: K w z 1 α! α α! zα w α 1 1 ln 1 z, w, 1.7 α / 0 whr z, w n j 1 z jw j is th innr product in C n. Clarly for ach f D 2 and w B, th nxt rproducing formula holds: f w f, K w. 1.8 Not that for f K w from 1.8, wobtain K w w K w 2 ln D 2 1 w Also, by th Cauchy-Schwarz inquality and 1.9, w hav that, for ach f D 2 and w B, f w f, Kw f D 2 K w D 2 f D 2 ln w 2 Not that inquality 1.10 is xact sinc it is attaind for f K w.

3 Abstract and Applid Analysis 3 Th wightd-typ spac H μ B H μ 2, 3 consists of all f H B such that f H μ : z B μ z f z <, 1.11 whr μ is a positiv continuous function on B wight. Th Bloch-typ spac B μ B B μ consists of all f H B such that f Bμ : f 0 z B μ z Rf z <, 1.12 whr μ is a wight. Lt g H D, g 0 0, and ϕ b a holomorphic slf-map of B, thn th following intgral-typ oprator: P g ϕ f ) z 1 0 f ϕ tz ) g tz dt, z B, f H B 1.13 t has bn rcntly introducd in 4 and considrably studid s,.g., 5 8 and th rlatd rfrncs thrin. For som rlatd oprators, s also 9 16 and th rfrncs thrin. Usual problm in this rsarch ara is to provid function-thortic charactrizations for whn ϕ and g induc boundd or compact intgral-typ oprator on spacs of holomorphic functions. Majority of paprs only find asymptotics of oprator norm of linar oprators. Somwhat concrt but prhaps mor intrsting problm is to calculat oprator norm of ths oprators btwn spacs of holomorphic functions on various domains. Som rsults on this problm can b found, for xampl, in 3, for rlatd rsults s also 7, Having publishd papr 3, w startd with systmatic invstigation of mthods for calculating oprator norms of concrt oprators btwn spacs of holomorphic function. Hr, w calculat th oprator norm as wll as th ssntial norm of th oprator P g ϕ : D 2 B μ, considrably xtnding our rcnt rsult in not Auxiliary Rsults In this sction, w quot svral auxiliary rsults which ar usd in th proofs of th main rsults. Lmma 2.1 s 4. Lt g H B, g 0 0, and ϕ b a holomorphic slf-map of B, thn RP g ϕ f ) z g z f ϕ z ). 2.1 Th nxt Schwartz-typ lmma 34 can b provd in a standard way. Hnc, w omit its proof.

4 4 Abstract and Applid Analysis Lmma 2.2. Assum that g H B, g 0 0, μ is a wight, and ϕ is an analytic slf-map of B,thn P g ϕ : D 2 B μ is compact if and only if P g ϕ : D 2 B μ is boundd and for any boundd squnc f k k N in D 2 convrging to zro uniformly on compacts of B as k, on has lim P g ) ϕ fk Bμ k Lmma 2.3. Assum that g H B, g 0 0, μ is a wight, ϕ is an analytic slf-map of B such that ϕ < 1, and th oprator P g ϕ : D 2 B μ is boundd, thn P g ϕ : D 2 B μ is compact. Proof. First not that sinc P g ϕ : D 2 B μ is boundd and f 0 z 1 D 2 by Lmma 2.1, it follows that RP g ϕ f 0 g H μ. Now, assum that f k k N is a boundd squnc in D 2 convrging to zro on compacts of B as k, thn w hav P g ϕ fk ) Bμ g H μ w ϕ B f k w 0, 2.3 as k,sincϕ B is containd in th ball w ϕ which is a compact subst of B, according to th assumption ϕ < 1. Hnc, by Lmma 2.2, th oprator P g ϕ : D 2 B μ is compact. 3. Oprator Norm of P g ϕ : D 2 B μ In this sction, w calculat th oprator norm of P g ϕ : D 2 B μ. Thorm 3.1. Assum that g H B, g 0 0, and ϕ is a holomorphic slf-map of B,thn P g ϕ D 2 B μ z B μ z g z ln 1 : L. 3.1 ϕ z 2 Proof. Using Lmma 2.1, rproducing formula 1.8, th Cauchy-Schwarz inquality, and finally 1.9, w gt that, for ach f D 2 and w B, μ w RP g ϕ w μ w ) g w f ϕ w μ w g w f, K ϕ w μ w g w f D 2 K ϕ w D f D 2μ w g w ln 1. ϕ w 2

5 Abstract and Applid Analysis 5 Taking th rmum in 3.2 ovr w B as wll as th rmum ovr th unit ball in D 2 and using th fact P g ϕ f 0 0, for ach f H B, which follows from th assumption g 0 0, w gt P g ϕ D 2 B μ L. 3.3 Now assum that th oprator P g ϕ : D 2 for ach w B, B μ is boundd. From 1.9, wobtainthat, ln 1 P g ϕ w 2 ϕ D 2 B μ K ϕ w D 2 P g ϕ D 2 B μ P g ϕ Kϕ w ) Bμ z B μ z g z Kϕ w ϕ z ) 3.4 μ w g w K ϕ w ϕ w ). From 1.9 and 3.4, it follows that L P g ϕ D 2 B μ. 3.5 Hnc, if P g ϕ : D 2 B μ is boundd, thn from 3.3 and 3.5 w obtain 3.1. In th cas P g ϕ : D 2 B μ is unboundd, th rsult follows from inquality Essntial Norm of P g ϕ : D 2 B μ Lt X and Y b Banach spacs, and lt L : X Y b a boundd linar oprator. Th ssntial norm of th oprator L : X Y, L,X Y, is dfind as follows: L,X Y inf { L K X Y : K is compact from X to Y }, 4.1 whr X Y dnot th oprator norm. From this and sinc th st of all compact oprators is a closd subst of th st of boundd oprators, it follows that L is compact if and only if L,X Y Hr, w calculat th ssntial norm of th oprator P g ϕ : D 2 B μ.

6 6 Abstract and Applid Analysis Thorm 4.1. Assum that g H B, g 0 0,μis a wight, ϕ is a holomorphic slf-map of B, and P g ϕ : D 2 B μ is boundd. If ϕ < 1, thn P g ϕ,d 2 B μ 0, and if ϕ 1, thn P g ϕ,d 2 B μ lim ϕ z 1 μ z g z ln 1 ϕ z Proof. Sinc P g ϕ : D 2 B μ is boundd, for th tst function f z 1, w gt g Hμ.If ϕ < 1, thn from Lmma 2.3 it follows that P g ϕ : D 2 B μ is compact which is quivalnt with P g ϕ,d 2 B μ 0. On th othr hand, it is clar that in this cas th condition ϕ z 1 is vacuous, so that 4.3 is vacuously satisfid. Now assum that ϕ 1, and that ϕ z k k N is a squnc in B such that ϕ z k 1ask. For w B fixd, st ln / 1 z, w f w z ln / 1 w 2))), z B. 1/2 4.4 By 1.9, w hav that f w D 2 1, for ach w B. Hnc, th squnc f ϕ zk k N is such that f ϕ zk D 2 1, for ach k N, and clarly it convrgs to zro uniformly on compacts of B. From this and by 35, Thorms 6.19, it asily follows that f ϕ zk 0 wakly in D 2,as k. Hnc, for vry compact oprator K : D 2 B μ, w hav that lim k Kf ϕ z k Bμ Thus, for vry such squnc and for vry compact oprator K : D 2 B μ, w hav that P g ϕ K D 2 B μ P g ϕ f ϕ zk Bμ Kf ϕ zk Bμ lim k f ϕ zk D 2 lim P g ϕ f ϕ zk Bμ k lim k lim n μ z k g zk f ϕ zk ϕ zk ) μ z k g zk ln 1 ϕ z k Taking th infimum in 4.6 ovr th st of all compact oprators K : D 2 B μ,w obtain P g ϕ,d 2 B μ lim n μ z k g zk ln 1 ϕ zk 2, 4.7 from which an inquality in 4.3 follows.

7 Abstract and Applid Analysis 7 In th squl, w prov th rvrs inquality. Assum that r l l N is a squnc of positiv numbrs which incrasingly convrgs to 1. Considr th oprators dfind by ) P g r l ϕf z 1 0 f r l ϕ tz ) g tz dt, l N. 4.8 t Sinc r l ϕ < 1, by Lmma 2.3, w hav that ths oprators ar compact. Sinc P g ϕ : D 2 B μ is boundd, thn g H μ. Lt ρ 0, 1 b fixd for a momnt. By Lmma 2.1, wgt P g ϕ P g r l ϕ D 2 B μ μ z ) g z f ϕ z f rl ϕ z ) f D 2 1 z B f D 2 1 ϕ z ρ f D 2 1 ϕ z >ρ g H μ μ z g z f ϕ z ) f rl ϕ z ) f D 2 1 ϕ z ρ f D 2 1 ϕ z >ρ μ z g z f ϕ z ) f r l ϕ z ) f ϕ z ) f r l ϕ z ) μ z g z f ϕ z ) f rl ϕ z ) Furthr, w hav f f r 2 D 2 α α α! α! a α 2 1 r α ) 2 α α! α! a α 2 f D 2. α 4.11 From 1.10, 4.11, and th fact f z f rz D 2,wobtain f z f rz f D 2 ln 1 z 2, 4.12 in particular f ϕ z ) f r l ϕ z ) f D 2 ln 1 ϕ z Lt I l : f D 2 1 ϕ z ρ f ϕ z ) f r l ϕ z ). 4.14

8 8 Abstract and Applid Analysis Th man valu thorm along with th subharmonicity of th moduli of partial drivativs of f, wll-known stimats among th partial drivativs of analytic functions, Thorm 6.2, and Proposition 6.2 in 35, yild I l f D 2 1 ϕ z ρ 1 r l ϕ z f w w ρ C ρ 1 r l n/p j f 0 f D 2 1 j 1 C ρ 1 r l n/p j f 0 f D 2 1 j 1 w 3 ρ /4 C ρ 1 r l n/p j f 0 f D 2 1 j 1 w 1 ρ /2 n/p 1 f w ) n/p 1 f w 2 1 1/2 w 2) 2 n/p 1 dτ w B n/p 1 f w 2 1 w 2) 2 n/p 1 dτ w 1/2 C ρ 1 r l 0, as l, 4.15 whr dτ z dv z / 1 z 2 n 1 and dv z is th Lbsgu volum masur on B. Using 4.13 in 4.10, ltting l in 4.9, using 4.15, and thn ltting ρ 1, th rvrs inquality follows, finishing th proof of th thorm. Rfrncs 1 K. Zhu, Distancs and Banach spacs of holomorphic functions on complx domains, Journal of th London Mathmatical Socity, vol. 49, no. 1, pp , K. D. Birstdt and W. H. Summrs, Biduals of wightd Banach spacs of analytic functions, Journal of Australian Mathmatical Socity A, vol. 54, no. 1, pp , S. Stvić, Norm of wightd composition oprators from Bloch spac to Hμ on th unit ball, Ars Combinatoria, vol. 88, pp , S. Stvić, On a nw oprator from H to th Bloch-typ spac on th unit ball, Utilitas Mathmatica, vol. 77, pp , S. G. Krantz and S. Stvić, On th itratd logarithmic Bloch spac on th unit ball, Nonlinar Analysis. Thory, Mthods & Applications, vol. 71, no. 5-6, pp , S. Stvić, On a nw oprator from th logarithmic Bloch spac to th Bloch-typ spac on th unit ball, Applid Mathmatics and Computation, vol. 206, no. 1, pp , S. Stvić, On a nw intgral-typ oprator from th Bloch spac to Bloch-typ spacs on th unit ball, Journal of Mathmatical Analysis and Applications, vol. 354, no. 2, pp , W. Yang, On an intgral-typ oprator btwn Bloch-typ spacs, Applid Mathmatics and Computation, vol. 215, no. 3, pp , D. Gu, Extndd Csàro oprators from logarithmic-typ spacs to Bloch-typ spacs, Abstract and Applid Analysis, vol. 2009, Articl ID , 9 pags, Z. Hu, Extndd Csàro oprators on Brgman spacs, Journal of Mathmatical Analysis and Applications, vol. 296, no. 2, pp , 2004.

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