Research Article Carleson Measure in Bergman-Orlicz Space of Polydisc

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1 Hidawi Publishig orporatio Abstract ad Applied Aalysis Volume 00, Article ID , 7 pages doi:0.55/00/ Research Article arleso Measure i Bergma-Orlicz Space of Polydisc A-Jia Xu, ad Zou Yag 3 Departmet of Mathematics, Zheiag Uiversity, Hagzhou 3007, hia 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 00; Revised 6 August 00; Accepted September 00 Academic Editor: Dumitru Baleau opyright q 00 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 -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 ad,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 -dimesioal Lebesgue volume measure o D, ormalized so that σ D. We use R to describe rectagles o T, ad we use SR to deote the coroa associated to these sets. I particular, if I is a iterval o T of legth δ 0, cetered at e iθ 0δ/, SI {z D δ<r<,θ 0 <θ<θ 0 δ.. The, if R I I I T,withI itervals havig legth δ ad havig ceters e iθ0 δ/, SR is give by SR SI SI SI,adlet α δ ) e iθ 0 δ /,.. If V is ay ope set i T, we defie SV γ SR γ where {R γ rus through all rectagles i V.

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 ) fw dμw <,.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 ad,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, 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 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 μsi ))) /δ.6 for every rectagle I T. I this paper, we provetheorem.4.

3 Abstract ad Applied Aalysis 3. Mai Results ad Proofs Lemma.. For α α,...,α D,letu α z,...,z α / α z 4.The u α z,...,z L ad, ad u α z σ )). /δ. Proof. It is easy to see that u α z α / α δ /δ. Sice 0 0, the covexity of implies ax ax for 0 a. Hece, for every >0, we have D δ / δ ) uα z dσ δ ) ) u α z dσ δ D δ δ ) ) u αz,. but δ / δ u α z / if ad oly if / δ /δ / u α z,thatis, u α z σ ) ). / uα.3 δ /δ z Moreover, u α z D α ) α z dσ z 4 D α ) ) α D α z dσ ) 4 z α z 4 dσ z α ) ) ) ) dσ z D α z α z.4 α ) ) α. So, we have u α z σ ) ) )). δ /δ /δ.5

4 4 Abstract ad Applied Aalysis For m m,...,m Z,k k,...,k with k m 4,, let { ) T mk r e iθ,...,r e iθ m r < m, k π θ k ) π m <,, 4 m 4.6 let z mk z mk,...,z mk, where z mk m ) e k /πi/ m 4,,.7 let { z,...,z z z mk 7 8 m,..8 Lemma.. For fixed m 0 m 0,...,m0 ad the correspodig k 0 k 0,...,k0, T m 0 k 0 itersect for at most N 5.57 choices of the pair m, k. Proof. See 5. Lemma.3. If f L ad,the fz,...,z ) f ) dσ.9 for 6/8 m ρ 7/8 m ad ay z T mk. Proof. It is clear that fz,...,z is a -subharmoic fuctio i D. Repeated applicatio of Harack s iequality yields fz,...,z ) π ρ z z mk ρ z z mk π 0 π )) f z mk ρ e iθ,...,z mk ρ e iθ dθ dθ 0 π π )) f π z mk ρ e iθ,...,z mk ρ e iθ dθ dθ

5 Abstract ad Applied Aalysis 5 Hece, for z T mk, fz,...,z ) 4 m 3 4 m 7/8 m 6/8 m ) f dσ. 7/8 m fz ) ρ ρ dρ dρ 6/8 m. Theorem.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 ad,σ L D,μ ). is bouded if ad oly if μ is a arleso measure. Proof. Suppose that there exists a costat such that Jg σ g μ,.3 for all g L ad. By Lemma., u α z ) α ) 4 L ad..4 α z However, for z SI, we have α z α e iθ 0δ / α e iθ 0δ / α z δ ) δ e iθ 0δ / z ) z z z z δ ) δ δ z ).5 δ δ δ ) 3δ, so, u α z,...,z α ) α z 4 3 ) δ δ 3..6

6 6 Abstract ad Applied Aalysis Therefore, D SI )) /δ u α z dμ )) /δ 3 dμ )) /δ 3 μsi,.7 that is, μsi ))), /δ.8 with /3. oversely, suppose that fz L ad, we have D fz,...,c ) NμD dμ fσ mm,...,m,m 0 kk,...,k, k m 4 μt mk m k { μd m k { μd m,km 0,k 0 { μd μd NμD m 0,k 0 m,k {N m0,k0 { D T mk fz,...,c ) NμD dμ fσ fz,...,c ) NμD dσ fσ fz,...,c T m 0 k 0 T m 0 k 0 T m 0 k 0 NμD f σ )dσ fz,...,c NμD f σ fz,...,c NμD f σ fz,...,c NμD f σ )dσ fz,...,c NμD )dσ, fσ )dσ )dσ.9 ad the proof is complete.

7 Abstract ad Applied Aalysis 7 orollary.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 u α z σ )), /δ.0 for every rectagle I D. Proof. As a fact, for ay measure μ ad Orlicz fuctio, we have D uα z u α z σ ) dμ 3 u α z σ ) μsi. by the proof of the Mai theorem. So, μsi /3 u α z σ ),. 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 Joural of Mathematics, vol. 80, pp , 958. 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. 5, pp. 37 4, B. D. Macluer, ompact compositio operators o H p B N, The Michiga Mathematical Joural, vol. 3, o., pp , P. Lefèvre,D.Li,H.Queffélec, ad L. Rodríguez-Piazza, ompositio operators o Hardy-Orlicz spaces, Memoirs of the America Mathematical Society, vol. 07, o. 974, J. Xiao, The arleso measure o aalytic Orlicz spaces, Joural of 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., pp , 00.