Composition Operators from Hardy-Orlicz Spaces to Bloch-Orlicz Type Spaces

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1 Journal of Mathematical Research with Applications Sept., 018, Vol. 38, No. 5, pp OI: /j.issn: 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 ), the Natural Science Foundation of Guangdong Province Grant No. 017A ) and the Science and Technology Project of Guangzhou City Grant No ). address: zhonghuahe010@163.com

2 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

3 460 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).

4 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 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 = 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.)

5 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 ) = ) 1 φz)

6 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.

7 464 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): [] Kehe ZHU. Bloch type spaces of analytic functions. Rocky Mountain J. Math., 1993, 33): [3] S. STEVIĆ. On new Bloch-type spaces. Appl. Math. Comput., 009, 15): [4] S. STEVIĆ, Renyu CHEN, Zehua ZHOU. Weighted composition operators between Bloch type spaces in the polydisc. Mat. Sb., 010, 01): [5] P. L. UREN. Theory of H p Spaces. Academic Press, New York, NY, USA, [6] C. C. COWEN, B.. MACCLUER. Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton, FL, [7] Zhangjian HU, Shushi WANG. Composition operators on Bloch-type spaces. Proc. Roy. Soc. Edinburgh Sect. A, 005, 1356): [8] Zhijie JIANG. Generalized product-type operators from weighted Bergman-Orlicz spaces to Bloch-Orlicz spaces. Appl. Math. Comput., 015, 68: [9] Lifang LIU, Guangfu CAO, Xiaofeng WANG. Composition operators on Hardy-Orlicz spaces. Acta Math. Sci. Ser. B Engl. Ed.), 005, 51): [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: [11] A. K. SHARMA, S.. SHARMA. Compact composition operators on Hardy-Orlicz spaces. Mat. Vesnik, 008, 603): [1] Maofa WANG, Shaobo ZHOU. Weighted Compostion operators Between Hardy-Orlicz Spaces. Acta. Math. Sci. Ser. A Chin. Ed., 005, 5: [13] Kehe ZHU. Spaces of Holomorphic Functions in the Unit Ball. Graduate Texts in Mathematics, Springer- Verlag, New York, 005.