Abstract
In this article, we give some characterizations of the inner functions in the space of \(W_{\alpha }\). Meanwhile, the zero sets in \(W_{\alpha }\) are also studied.
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Blasi, D., Pau, J.: A characterization of Besov-type spaces and applications to Hankel-type operators. Michigan Math. J. 56, 401–417 (2008)
Dyakonov, K.M.: Division and multiplication by inner functions and embedding theorems for star-invariant subspaces. Am. J. Math. 115, 881–902 (1993)
Dyakonov, K.M.: Smooth functions and coinvariant subspaces of the shift operator. St. Petersburg Math. J. 4, 933–959 (1993)
Dyakonov, K.M.: Besov spaces and outer functions. Michigan Math. J. 45, 143–157 (1998)
Dyakonov, K.M.: Holomorphic functions and quasiconformal mappings with smooth moduli. Adv. Math. 187, 146–172 (2004)
Dyakonov, K.M.: Self-improving behaviour of inner functions as multipliers. J. Funct. Anal. 240, 429–444 (2006)
Dyakonov, K.M., Girela, D.: On \(Q_p\) spaces and pseudoanalytic extension. Ann. Acad. Sci. Fenn. Ser. A Math. 25, 477–486 (2000)
Marshall, D.E., Sundberg, C.: Interpolating sequences for the multipliers of the Dirichlet space, manuscript (1994)
McDonald, G., Sundberg, C.: Toeplitz operators on the disc. Indiana Univ. J. Math. 28, 595–611 (1979)
Pau, J., Pelez, J.: On the zeros of functions in Dirichlet-type spaces. Trans. Am. Math. Soc. 363, 1981–2002 (2011)
Pelez, J.: Inner functions as improving multipliers. J. Funct. Anal. 255, 1403–1418 (2008)
Rochberg, R., Wu, Z.: A new characterization of Dirichlet type spaces and applications. Ill. J. Math. 37, 101–122 (1993)
Shapiro, H.S., Shields, A.L.: On the zeros of functions with finite Dirichlet integral and some related function spaces. Math. Z. 80, 217–229 (1962)
Wu, Z.: The predual and second predual of Wa. J. Fund. Anal. 116, 314–334 (1993)
Wu, Z.: Carleson measures and multipliers for Dirichlet spaces. J. Funct. Anal. 169, 148–163 (1999)
Xiao, J.: Holomorphic \({\cal{Q}}\) classes. Lecture notes in mathematics 1767. Springer, Berlin (2001)
Xiao, J.: Geometric \({\cal{Q}}\) functions. Birkhäuser, Basel (2006)
Zhao, R.: Distances from Bloch functions to some Möbius invariant spaces. Ann. Acad. Sci. Fenn. Math. 33, 303–313 (2008)
Zhu, K.: Operator theory in function spaces. Marcel Dekker, New York (1990)
Zhu, K.: Operator theory in function spaces. American Mathematical Society, Providence (2007)
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Communicated by Klaus Gürlebeck.
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Du, J., Yang, L. Inner functions in \(W_{\alpha }\) as improving multipliers. Ann. Funct. Anal. 11, 555–566 (2020). https://doi.org/10.1007/s43034-019-00037-w
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DOI: https://doi.org/10.1007/s43034-019-00037-w
Keywords
- Dirichlet space \(\mathcal {D}_{\alpha }\)
- \(W_{\alpha }\) space
- Inner functions
- Carleson–Newman sequences
- Zero set
- Improving