Journal of Mathematical Research with Applications Sept., 018, Vol. 38, No. 5, pp. 458 464 OI:10.3770/j.issn:095-651.018.05.003 Http://jmre.dlut.edu.cn Composition Operators from Hardy-Orlicz Spaces to Bloch-Orlicz Type Spaces Zhonghua HE School of Financial Mathematics and Statistics, Guangdong University of Finance, Guangdong 51051, P. R. China Abstract Let φ be an analytic self-map of. The composition operator C φ is the operator defined on H) by C φ f) = f φ. In this paper, we investigate the boundedness and compactness of the composition operator C φ from Hardy-Orlicz spaces to Bloch-Orlicz type spaces. Keywords composition operator; Hardy-Orlicz space; Bloch-Orlicz space MR010) Subject Classification 30H05; 47G10 1. Introduction Let be the unit disk in the complex plane C and H) the space of all holomorphic functions on. A function f H) is called a µ-bloch function, if f µ := µz) f z) <, denoted as f B µ, where µ is a bounded continuous positive function on. And B µ is a Banach space with the norm f B µ := f0) + f µ. Recently, the Bloch-Orlicz type space was introduced by Ramos Fernández in [1] using Y- oung s functions. More precisely, let ψ : [0, + ) [0, + ) be a strictly increasing convex function such that ψ0) = 0 and lim t + ψt) = +. The Bloch-Orlicz type space associated with the function ψ, denoted by B ψ, is the class of all functions f H) such that 1 z )ψλ f z) ) <, for some λ > 0 depending on f. The Minknowki s functional f ψ = inf{k > 0 : S ψ f k ) 1} defines a seminorm for B ψ, which, in this case, is known as Luxemburg s seminorm, where S ψ f) := 1 z )ψ fz) ). Received September 8, 017; Accepted June 16, 018 Supported by the National Natural Science Foundation of China Grant No. 11501130), the Natural Science Foundation of Guangdong Province Grant No. 017A030310636) and the Science and Technology Project of Guangzhou City Grant No. 0170701016). E-mail address: zhonghuahe010@163.com
Composition operators from Hardy-Orlicz spaces to Bloch-Orlicz type spaces 459 Moreover, B ψ is a Banach space with the norm f B ψ = f0) + f ψ. Also, Ramos Fernández in [1] got that the Bloch-Orlicz type space was isometrically equal to µ-bloch space, where µz) = 1 ψ 1 1, z. 1.1) 1 z ) Then, for f B ψ f Bψ = f0) + µz) f z). It is obvious to see that if ψt) = t p with p 1, then the space B ψ coincides with the α-bloch space B α see []), where α = 1/p. log-bloch space [3, 4]. Also, if ψt) = t log1 + t), then B ψ coincides with the The Hardy-Orlicz space H ψ ) = H ψ is the space of all f H) such that f H ψ := ψ frξ) )dσξ) <, 0<r<1 where is the boundary of the unit disk and σ is the normalized Lebesgue measure on. On H ψ is defined the next quasi-norm f H ψ = f r L ψ, 0<r<1 where f r ξ) = frξ), 0 r < 1, ξ and g L ψ is the Luxembourg quasi-norm defined by g L ψ := inf{λ > 0 : ψ gξ) )dσξ) 1}. λ If ψt) = t p with p > 0, then H ψ is the classical Hardy space H p see [5]), consisting of all f H) such that f p H p = frξ) p dσξ) <. 0<r<1 Let φ be an analytic self-map of, and the composition operator C φ be the operator defined on H) by C φ f)z) := f φ)z) = fφz)). The function φ is called the symbol of C φ. Composition operators between various spaces of holomorphic functions on different domains have been studied by numerous authors [1,4,6 1] and the references therein. This paper is devoted to characterizing the boundedness and compactness of composition operators from Hardy-Orlicz spaces to Bloch-Orlicz type spaces. Throughout this paper, we will use the letter C to denote a generic positive constant that can change its value at each occurrence. The notation a b means that there is a positive constant C such that a Cb. If both a b and b a hold, then one says that a b.. Auxiliary results
460 Zhonghua HE Here we quote some auxiliary results which will be used in the proofs of the main results in this paper. Proposition.1 For every f H ψ and z, we have fz) ψ 1 1 z ) f Hψ..1) Proof Since f is analytic, employing [13, Corollary 4.5] to the functions f r z), we get for z frz) P z, ζ) f r ζ) dσζ),.) where is the invariant Poison Kernel [13]. Using the inequality P z, ζ) = 1 z P z, ζ) 1 z and applying Jensen s inequality to.) with f r replaced by f r / f r H ψ, we obtain ψ frz) ) P z, ζ)ψ f rζ) )dσζ) f r H ψ f r H ψ ψ f rζ) )dσζ) 1 z f r H ψ 1 z..3) From.3) we obtain frz) ψ 1 1 z ) f r H ψ, letting r 1, then inequality.1) follows. The proof is completed. Proposition. For every f H ψ and z, we have Proof By [13, Proposition 4.], we have fz) = Then ifferentiating.4) yields and hence f 1 z) 1 z ψ 1 1 z ) f H ψ. f z) = f z) 1 z ) f z) 1 z ) P z, ζ)fζ)dσζ)..4) ζfζ) dσζ). 1 z ζ) fζ) dσζ), fζ) dσζ) = 1 z fζ) dσζ).
Composition operators from Hardy-Orlicz spaces to Bloch-Orlicz type spaces 461 Applying Jensen s inequality, we obtain That is, The proof is completed. ψ 1 z ) f z) ) f H ψ Lemma.3 For each a, the function belongs to H ψ. Moreover 1 z. 1 z fζ) ψ )dσζ) f H ψ f 1 z) 1 z ψ 1 1 z ) f H ψ. f a z) = 1 4 ψ 1 a )1 1 a 1 zā ) f a H ψ 1. a Proof Using Jensen s inequality for the fact 1 1 a ) 4 1 zā ψ f a rζ) )dσζ) = ψ 1 4 ψ 1 1 4 1 a 1 rζā 1 a dσζ) 1 a dσζ) 1. 1 rζā From this the lemma follows. The proof is completed. 1, we have 1 a ) 1 a 1 rζā )dσζ) The following compactness criteria can be proved similar to [11, Proposition 3.11]. Lemma.4 The bounded operator T : H ψ B ψ is compact if and only if for every bounded sequence {f j } j N in H ψ which converges to zero uniformly on any compact subset of as j, it follows that lim T f j B ψ = 0. 3. Boundedness and compactness In this section, we characterize the boundedness and compactness of the operators C φ : H ψ B ψ. Theorem 3.1 Let φ be an analytic self-map of. Then C φ : H ψ B ψ is bounded if and only if Moreover µz) φ z) M := 1 φz) ψ 1 ) <. 3.1) 1 φz) C φ Hψ Bψ M. 3.)
46 Zhonghua HE Proof Suppose that the condition 3.1) holds. For an arbitrary f H ψ, by Proposition., we have µz) C φ f) z) = µz) f φz)) φ z) µz)ψ 1 1 φz) ) 1 1 φz) φ z) f H ψ M f H ψ. Then C φ : H ψ B ψ is bounded. Moreover, the above proof gets that C φ Hψ B = C φ f B ψ ψ f H ψ \{0} f H ψ M. 3.3) Conversely, pose that C φ : H ψ B ψ is bounded. Then there is a positive constant C such that for any f H ψ, C φ f B ψ C f H ψ. By Lemma.3, we have the following functions are uniformly bounded in H ψ ifferentiating 3.4) we have We easily obtain that It follows that Then φz) >1/ f a z) = 1 4 ψ 1 a )1 1 a 1 zā ). 3.4) f az) = 1 ψ 1 1 a )ā1 a ) 1 zā) 3. Iz) := µz)ψ 1 1 φz) ) φz) φ z) 1 φz) µz) φ z) 1 φz) ψ 1 C φ f a B ψ C φ Hψ B ψ. 1 φz) ) Iz) C φ Hψ Bψ <. 3.5) Let fz) = z H ψ. Applying the boundedness of C φ : H ψ B ψ, we have µz) φ z) = C φ f B ψ C φ Hψ B <. ψ µz) φ z) 0< φz) 1/ 1 φz) ψ 1 1 φz) ) 4 3 ψ 1 1) µz) φ z) <. 3.6) From 3.5) and 3.6) we get that 3.1) holds. Moreover M C φ Hψ Bψ. 3.7) Therefore, from 3.3) and 3.7) the asymptotic expression 3.) is obtained. The proof is completed. Theorem 3. Let φ be an analytic self-map of. Then C φ : H ψ B ψ is compact if and only if φ B ψ and µz) φ z) lim φz) 1 1 φz) ψ 1 ) = 0. 3.8) 1 φz)
Composition operators from Hardy-Orlicz spaces to Bloch-Orlicz type spaces 463 Proof Suppose that C φ : H ψ B ψ is compact. Then C φ : H ψ B ψ is bounded, from the proof of Theorem 3.1 we have obtained that φ B ψ. Consider a sequence {φz j )} j N in such that lim φz j ) = 1. If such sequence does not exist, then 3.8) obviously holds. Using this sequence, we define the functions f j z) = 1 4 ψ 1 1 φz j ) )1 φz j) 1 zφz j ) ), j N. By Lemma.3 we know that the sequence {f j } j N is uniformly bounded in H ψ. From the proof of [11, Theorem 3.6], it follows that the sequence {f j } j N uniformly converges to zero on any compact subset of as j. Hence, by Lemma.4 From this, we have This implies that lim C φf j B ψ = 0. µz j ) φ z j ) 1 φz j ) ψ 1 1 φz j ) ) µz) φ z) 1 φz) ψ 1 1 φz) ) C φ f j B ψ. µz) φ z) lim φz) 1 1 φz) ψ 1 1 φz) ) = 0. Now pose that φ B ψ and 3.8) holds. We first check that C φ : H ψ B ψ is bounded. We observe that 3.8) implies that for every ϵ > 0, there is a 0 < δ < 1 such that for any z with φz) > δ Since for z with 0 < φz) δ we have µz) φ z) 1 φz) ψ 1 ) < ϵ. 3.9) 1 φz) µz) φ z) 1 φz) ψ 1 1 φz) ) φ 1 B ψ 1 δ ψ 1 1 δ ), µz) φ z) 1 φz) ψ 1 1 φz) ) µz) φ z) 0< φz) δ 1 φz) ψ 1 1 φz) ) + µz) φ z) φz) >δ 1 φz) ψ 1 1 φz) ) 1 φ B ψ 1 δ ψ 1 1 δ ) + ϵ. This proves that C φ : H ψ B ψ is bounded. By Lemma.4, in order to prove that C φ : H ψ B ψ is compact, we just need to prove that if the sequence {f j } j N is uniformly bounded in H ψ and uniformly converges to zero on any compact subset of as j, then lim C φf j B ψ = 0.
464 Zhonghua HE have For any ϵ > 0 and the associated δ in 3.9), by using again that φ B ψ and Lemma.3, we C φ f j B ψ 0< φz) δ φ B ψ = µz) f jφz)) φ z) µz) f jφz)) φ z) + 0< φz) δ 0, as j, φz) >δ f jφz)) + ϵ f j H ψ j N µz) φ z) 1 φz) ψ 1 1 φz) ) f j H ψ where we have used the fact that from f j 0 as j uniformly on compact subsets of, it follows that f j 0 as j uniformly on compact subsets of. Hence lim C φf j B ψ = 0, which follows that C φ : H ψ B ψ is compact. The proof is completed. Acknowledgements We thank the referees for their time and suggestions. References [1] J. C. RAMOS FERNÁNEZ. Composition operators on Bloch-Orlicz type spaces. Appl. Math. Comput., 010, 177): 339 340. [] Kehe ZHU. Bloch type spaces of analytic functions. Rocky Mountain J. Math., 1993, 33): 1143 1177. [3] S. STEVIĆ. On new Bloch-type spaces. Appl. Math. Comput., 009, 15): 841 849. [4] S. STEVIĆ, Renyu CHEN, Zehua ZHOU. Weighted composition operators between Bloch type spaces in the polydisc. Mat. Sb., 010, 01): 89 319. [5] P. L. UREN. Theory of H p Spaces. Academic Press, New York, NY, USA, 1970. [6] C. C. COWEN, B.. MACCLUER. Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton, FL, 1995. [7] Zhangjian HU, Shushi WANG. Composition operators on Bloch-type spaces. Proc. Roy. Soc. Edinburgh Sect. A, 005, 1356): 19 139. [8] Zhijie JIANG. Generalized product-type operators from weighted Bergman-Orlicz spaces to Bloch-Orlicz spaces. Appl. Math. Comput., 015, 68: 966 977. [9] Lifang LIU, Guangfu CAO, Xiaofeng WANG. Composition operators on Hardy-Orlicz spaces. Acta Math. Sci. Ser. B Engl. Ed.), 005, 51): 105 111. [10] B. SEHBA, S. STEVIĆ. On some product-type operators from Hardy-Orlicz and Bergman-Orlicz spaces to weighted-type spaces. Appl. Math. Comput., 014, 33: 565 581. [11] A. K. SHARMA, S.. SHARMA. Compact composition operators on Hardy-Orlicz spaces. Mat. Vesnik, 008, 603): 15 4. [1] Maofa WANG, Shaobo ZHOU. Weighted Compostion operators Between Hardy-Orlicz Spaces. Acta. Math. Sci. Ser. A Chin. Ed., 005, 5: 509 515. [13] Kehe ZHU. Spaces of Holomorphic Functions in the Unit Ball. Graduate Texts in Mathematics, Springer- Verlag, New York, 005.