Maximal functions and properties of the weighted composition operators acting on the Korenblum, α-Bloch and α-Zygmund spaces

 

Gabriel M. Antón Marval¹, René E. Castillo², Julio C. Ramos-Fernández³

¹ Area de Matemáticas, Universidad Nacional Experimental de Guayana, Puerto Ordaz 8050, Estado Bolívar, Venezuela.

² Departamento de Matemáticas, Universidad Nacional de Colombia, AP360354 Bogotá, Colombia.

³ Departamento de Matemáticas, Universidad de Oriente, Cumaná 6101, Estado Sucre, Venezuela.

gabman@gmail.com, recastillo@unal.edu.co, jcramos@udo.edu.ve

---

ABSTRACT

Using certain maximal analytic functions, we obtain new characterizations of the continuity and compactness of the weighted composition operators when acts between Korenblum spaces, α-Bloch spaces and when acts from certain weighted Banach spaces of analytic functions with a logarithmic weight into α-Bloch spaces. As consequence of our results, we obtain a new characterization of the continuity and compactness of composition operators acting between α-Zygmund spaces.

Keywords and Phrases: Weighted Banach spaces of analytic functions, Bloch space, weighted composition operators.

2010 AMS Mathematics Subject Classification: 30D45, 47B33.

---

RESUMEN

Usando ciertas funciones analíticas maximales, obtenemos nuevas caracterizaciones de la continuidad y compacidad de operadores de composición con pesos cuando actúan entre espacios de Korenblum, espacios α-Bloch y cuando actúan desde ciertos espacios de Banach de funciones analíticas con un peso logarítmico en espacios α-Bloch. Como consecuencia de nuestros resultados, obtenemos una nueva caracterización de la continuidad y la compacidad de operadores de composición actuando entre espacios α-Zygmund.

---

 

References

[1] J. M. Anderson and J. Duncan, Duals of Banach spaces of entire functions. Glasgow Math. J. 32 (1990), no. 2, 215–220.

[2] K. D. Bierstedt, J. Bonet and J. Taskinen, Associated weights and spaces of holomorphic functions. Studia Math. 127 (1998), 137–168.

[3] K. D. Bierstedt, R. Meise and W. H. Summers, A projective description of weighted inductive limits. Trans. Amer. Math. Soc. 272 (1982), 107–160.

[4] J. Bonet, P. Domaski and M. Lindström, Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic function. Canad. Math. Bull. 42 (1999),
139–148.

[5] R. E. Castillo, C. E. Marrero-Rodríguez and J. C. Ramos-Fernández, On a Criterion for Continuity and Compactness of Composition Operators on the Weighted Bloch Space. Mediterr. J. Math. 12, No. 3, pp. 1047-1058.

[6] R. E. Castillo, J. C. Ramos-Fernández and E. M. Rojas, A new essential norm estimate of composition operators from weighted Bloch space into μ-Bloch spaces. J. Funct. Spaces Appl. 2013, Art. ID 817278, 5 pp.

[7] F. Colonna, New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space. Cent. Eur. J. Math. 11 (2013), no. 1, 55-73.

[8] F. Colonna and S. Li, Weighted composition operators from Hardy spaces into logarithmic Bloch spaces. J. Funct. Spaces Appl. 2012, Art. ID 454820, 20 pp.

[9] M. Contreras and A. Hernández-Díaz, Weighted composition operators in weighted Banach spaces of analytic functions. J. Austral. Math. Soc. Ser. A, 69 (2000), 41–60.

[10] C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995.

[11] J. Giménez, R. Malavé and J. C. Ramos-Fernández, Composition operators on μ-Bloch type spaces. Rend. Circ. Mat. Palermo, 59 (2010), 107-119.

[12] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman spaces. Graduate Texts in Mathematics, 199. Springer-Verlag, New York, 2000.

[13] O. Hyvärinen and M. Lindström, Estimates of essential norms of weighted composition operators between Bloch-type spaces. J. Math. Anal. Appl. 393 (2012), 38–44.

[14] K. Madigan and A. Matheson, Compact composition operators on the Bloch space. Trans. Amer. Math. Soc. 347, (1995), 2679–2687.

[15] M. T. Malavé-Ramírez and J. C. Ramos-Fernández, On a criterion for continuity and compactness of composition operators acting on α-Bloch spaces. C. R. Math. Acad. Sci. Paris, Ser. I, 351 (2013), 23–26.

[16] A. Montes-Rodríıguez, Weighted composition operators on weighted Banach spaces of analytic functions. J. London Math. Soc. 61 (2000), no. 2, 872–884.

[17] S. Ohno, K. Stroethoff y R. Zhao, Weighted composition operators between Bloch-type spaces. Rocky Mountain J. Math. 33 (2003), no. 1, 191–215.

[18] J. C. Ramos-Fernández, Composition operators between μ-Bloch spaces. Extracta Math. 26 (2011), no. 1, 75–88.

[19] J. C. Ramos-Fernández, A new essential norm estimate of composition operators from α-Bloch spaces into μ-Bloch spaces. Internat. J. Math. 24 (2013), no. 14, 1350104, 7 pp.

[20] J. H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, New York, 1993.

[21] M. Tjani, Compact composition operators on some M¨obius invariant Banach space. Thesis (Ph.D.)Michigan State University. ProQuest LLC, Ann Arbor, MI, 1996. 68 pp. ISBN: 978-0591-27288-8.

[22] M. Tjani, Compact composition operators on Besov spaces. Trans. Amer. Math. Soc. 355, (2003), no. 11, 4683–4698.

[23] K. Zhu, Operator theory in function spaces. Marcel Dekker. New York, 1990.

[24] K. Zhu, Bloch type spaces of analytic functions. Rocky Mountain J. Math. 23 (1993), 1143– 1177.