Research Article Carleson Measure in Bergman-Orlicz Space of Polydisc

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1 Abstract ad Applied Aalysis Volume 200, Article ID , 7 pages doi:0.55/200/ Research Article arleso Measure i Bergma-Orlicz Space of Polydisc A-Jia Xu, 2 ad Zou Yag 3 Departmet of Mathematics, Zheiag Uiversity, Hagzhou 30027, hia 2 Istitute of Applied Mathematics, hogqig Uiversity of Post ad Telecommuicatio, hogqig , hia 3 Departmet of Mathematics, hogqig Educatio ollege, hogqig , hia orrespodece should be addressed to A-Jia Xu, aia.xu@gmail.com Received 30 Jue 200; Revised 26 August 200; Accepted 2 September 200 Academic Editor: Dumitru Baleau opyright q 200 A.-J. Xu ad Z. Yag. This is a ope access article distributed uder the reative ommos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Let μ be a fiite, positive measure o D, the polydisc i,adletσ be 2-dimesioal Lebesgue volume measure o D. For a Orlicz fuctio, a ecessary ad sufficiet coditio o μ is give i order that the idetity map J : L a D,σ L D,μ is bouded.. Itroductio We deote by D the uit polydisc i ad by T the distiguished boudary of D.We will use σ to deote the 2-dimesioal Lebesgue volume measure o D, ormalized so that σ D. We use R to describe rectagles o T, ad we use S R to deote the coroa associated to these sets. I particular, if I is a iterval o T of legth δ 0, cetered at e i θ 0 δ/2, S I {z D δ<r<,θ 0 <θ<θ 0 δ}.. The, if R I I 2 I T,withI itervals havig legth δ ad havig ceters e i θ0 δ/2, S R is give by S R S I S I 2 S I,adlet α δ ) e i θ 0 δ /2,..2 If V is ay ope set i T, we defie S V γ S R γ where {R γ } rus through all rectagles i V.

2 2 Abstract ad Applied Aalysis A Orlicz fuctio is a real-valued, odecreasig, covex fuctio : 0, 0, such that 0 0ad. To avoid pathologies, we will assume that we work with a Orlicz fuctio havig the followig additioal properties: is cotiuous ad strictly covex hece icreasig, such that x lim x x..3 The Orlicz space L μ is the space of all equivalece classes of measurable fuctios f : Ω for which there is a costat >0 such that Ω ) f w dμ w <,.4 ad the f μ the Luxemburg orm is the ifimum of all possible costat such that this itegral is. It is well kow that L μ is a Baach space uder the Luxemburg orm μ. For f L,let M μ f ) : Ω f ) dμ <..5 The Bergma-Orlicz space L a D,σ cosists of all aalytic fuctios i L D,σ, which is a closed subspace of L D,σ, so it is a aalytic Baach space also. A theorem of arleso, 2 characterizes those positive measure μ o D for which the Hardy space H p orm domiates the L p μ orm of elemets of H p. Sice the, there is a log history of the developmet ad applicatio of arleso measures, see 3. Thisrich area of research cotais a large body of literature o characterizatios of differet classes of operators i differet spaces ad their applicatios. hag 4 has characterized the bouded measures o L p T 2 usig a two-lie proof referrig to a result of Stei. haracterizatio of the bouded idetity operators o Hardy spaces is a immediate cosequece of hag s proof usig stadard argumets. Hastigs 5 has give a similar result for uweighted Bergma spaces. Macluer 6 has obtaied a arleso measure characterizatio of the idetity operators o Hardy spaces of the uit ball i usig the well-kow results of Hormader. Lefèvre et al. 7 have itroduced a adapted versio of arleso measure i Hardy-Orlicz spaces. Xiao 8, Ortiz, ad Feradez 9 have got a characterizatio of the arleso measure i Bergma-Orlicz spaces of the uit disc. A fiite, positive measure μ o D is called a arleso measure if there is a costat such that μ S I ))) /δ 2.6 for every rectagle I T. I this paper, we provetheorem 2.4.

3 Abstract ad Applied Aalysis 3 2. Mai Results ad Proofs Lemma 2.. For α α,...,α D,letu α z,...,z α 2 2 / α z 4.The u α z,...,z L a D, ad )). /δ 2 2. Proof. It is easy to see that u α z α / α 2 2 δ /δ 2. Sice 0 0, the covexity of implies ax a x for 0 a. Hece, for every >0, we have D δ / 2 δ ) 2 uα z dσ δ ) 2 ) u α z dσ 2 δ D δ 2 δ ) 2 ) u α z, 2.2 but δ / 2 δ 2 u α z / if ad oly if / 2 δ /δ 2 / u α z,thatis, ) ). 2 / uα δ /δ z Moreover, u α z D α 2) 2 α z dσ z 4 D α 2) 2 ) α 2 2 D α z dσ ) 4 z α z 4 dσ z α 2) 2 ) ) 2 ) 2 dσ z D α z α z 2.4 α 2) 2 ) α 2 2. So, we have ) ) 2 )). 2 δ /δ /δ 2 2.5

4 4 Abstract ad Applied Aalysis For m m,...,m Z,k k,...,k with k 2 m 4,, let { ) T mk r e iθ,...,r e iθ 2 m r < 2 m, 2k π 2 θ 2k ) } π m <,, 4 2 m let z mk z mk,...,z mk, where z mk 2 m ) e 2 k /2 πi/2 m 4,, 2.7 let { U mk z,...,z z z mk 7 } 8 2 m,. 2.8 Lemma 2.2. For fixed m 0 m 0,...,m0 ad the correspodig k 0 k 0,...,k0, T m 0 k 0 itersect U mk for at most N 5.57 choices of the pair m, k. Proof. See 5. Lemma 2.3. If f L a D,the f z,...,z ) U mk f ) dσ 2.9 for 6/8 2 m ρ 7/8 2 m ad ay z T mk. Proof. It is clear that f z,...,z is a -subharmoic fuctio i D. Repeated applicatio of Harack s iequality yields f z,...,z ) 2π ρ z z mk ρ z z mk 2π 0 2π )) f z mk ρ e iθ,...,z mk ρ e iθ dθ dθ 0 2π 2π )) f 2 2π z mk ρ e iθ,...,z mk ρ e iθ dθ dθ

5 Abstract ad Applied Aalysis 5 Hece, for z T mk, f z,...,z ) 2 4 m m 7/8 2 m 6/8 2 m ) f dσ. U mk 7/8 2 m f z ) ρ ρ dρ dρ 6/8 2 m 2. Theorem 2.4 Mai theorem. Let μ be a fiite, positive measure o D, ad suppose that is a Orlicz fuctio. The, the idetity map J : L a D,σ L D,μ ) 2.2 is bouded if ad oly if μ is a arleso measure. Proof. Suppose that there exists a costat such that J g σ g μ, 2.3 for all g L a D. By Lemma 2., u α z ) α 2 2 ) 4 L a D. 2.4 α z However, for z S I, we have α z α e i θ 0 δ /2 α e i θ 0 δ /2 α z δ ) δ e i θ 0 δ /2 z ) z z z z δ ) δ δ z ) 2.5 δ 2δ δ ) 3δ, so, u α z,...,z α ) 2 2 α z 4 3 ) 2 δ δ

6 6 Abstract ad Applied Aalysis Therefore, D S I )) /δ 2 u α z dμ )) /δ 2 3 dμ )) /δ 2 3 μ S I, 2.7 that is, μ S I ))), /δ with /3. oversely, suppose that f z L a D, we have D f z,...,c ) Nμ D dμ fσ m m,...,m,m 0 k k,...,k, k 2 m 4 μ T mk m k { μ D m k { μ D m,km 0,k 0 { μ D μ D Nμ D m 0,k 0 m,k {N m0,k0 { D U mk U mk T mk f z,...,c ) Nμ D dμ fσ f z,...,c ) Nμ D dσ fσ f z,...,c T m 0 k 0 U mk T m 0 k 0 U mk T m 0 k 0 Nμ D f σ )dσ } f z,...,c Nμ D f σ f z,...,c Nμ D f σ f z,...,c Nμ D f σ )dσ } f z,...,c } Nμ D )dσ, fσ )dσ } )dσ } 2.9 ad the proof is complete.

7 Abstract ad Applied Aalysis 7 orollary 2.5. Let μ be a fiite, positive measure o D, ad suppose that is a Orlicz fuctio. The μ is a arleso measure if ad oly if there exists some > such that )), /δ for every rectagle I D. Proof. As a fact, for ay measure μ ad Orlicz fuctio, we have D uα z ) dμ 3 ) μ S I 2.2 by the proof of the Mai theorem. So, μ S I /3 ), 2.22 ad the corollary follows. Ackowledgmet This paper was supported by the NNSF of hia ad the Youth Foudatio of QUPT A Refereces L. arleso, A iterpolatio problem for bouded aalytic fuctios, America Mathematics, vol. 80, pp , L. arleso, Iterpolatios by bouded aalytic fuctios ad the coroa problem, Aals of Mathematics. Secod Series, vol. 76, pp , J. B. Garett, Bouded Aalytic Fuctios, Academic Press, New York, NY, USA, S.-Y. A. hag, arleso measure o the bi-disc, Aals of Mathematics. Secod Series, vol. 09, o. 3, pp , W. W. Hastigs, A arleso measure theorem for Bergma spaces, Proceedigs of the America Mathematical Society, vol. 52, pp , B. D. Macluer, ompact compositio operators o H p B N, The Michiga Mathematical Joural, vol. 32, o. 2, pp , P. Lefèvre,D.Li,H.Queffélec, ad L. Rodríguez-Piazza, ompositio operators o Hardy-Orlicz spaces, Memoirs of the America Mathematical Society, vol. 207, o. 974, J. Xiao, The arleso measure o aalytic Orlicz spaces, Ahui Normal Uiversity. Natural Sciece, vol. 6, o. 3, pp. 6 0, 993 hiese. 9 A. G. Ortiz ad J. Feradez, arleso coditio i Bergma-Orlicz spaces, omplex Aalysis ad Operator Theory, vol. 4, o. 2, pp , 200.

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