Abstract
In this paper, we determine the range of a Cesàro-like operator acting on \(H^\infty \) by describing characterizations of Carleson type measures on [0, 1). A special case of our result gives an answer to a question posed by P. Galanopoulos, D. Girela and N. Merchán recently.
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References
Aulaskari, R., Lappan, P.: Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal, Complex analysis and its applications, Pitman Res. Notes in Math., 305, Longman Sci. Tech., Harlow, pp. 136–146, (1994)
Aulaskari, R., Xiao, J., Zhao, R.: On subspaces and subsets of \(BMOA\) and \(UBC\). Analysis 15, 101–121 (1995)
Aulaskari, R., Girela, D., Wulan, H.: Taylor coefficients and mean growth of the derivative of \({\cal{Q} }_p\) functions. J. Math. Anal. Appl. 258, 415–428 (2001)
Aulaskari, R., Stegenga, D., Xiao, J.: Some subclasses of \(BMOA\) and their characterization in terms of Carleson measures. Rocky Mt. J. Math. 26, 485–506 (1996)
Baernstein, A., II.: Aspects of Contemporary Complex Analysis. In: Analytic functions of bounded mean oscillation, pp. 3–36. Academic Press, Canbridge (1980)
Bao, G., Wulan, H.: Hankel matrices acting on Dirichlet spaces. J. Math. Anal. Appl. 409, 228–235 (2014)
Bao, G., Wulan, H., Ye, F.: The range of the Cesàro operator acting on \(H^\infty \). Canad. Math. Bull. 63, 633–642 (2020)
Bao, G., Ye, F., Zhu, K.: Hankel measures for Hardy spaces. J. Geom. Anal. 31, 5131–5145 (2021)
Blasco, O.: Operators on weighted Bergman spaces (\(0<p\le 1\)) and applications. Duke Math. J. 66, 443–467 (1992)
Bourdon, P., Shapiro, J., Sledd, W.: Fourier series, mean Lipschitz spaces, and bounded mean oscillation, in: Analysis at Urbana, vol. I, Urbana, IL, 1986–1987, in: London Math. Soc. Lecture Note Ser., vol. 137, pp. 81–110, (1989)
Chatzifountas, C., Girela, D., Peláez, J.: A generalized Hilbert matrix acting on Hardy spaces. J. Math. Anal. Appl. 413, 154–168 (2014)
Danikas, N., Siskakis, A.: The Cesàro operator on bounded analytic functions. Analysis 13, 295–299 (1993)
Duren, P.: Theory of \(H^p\) Spaces. Academic Press, New York (1970)
Essén, M., Xiao, J.: Some results on \({\cal{Q} }_p\) spaces, \(0<p<1\). J. Reine Angew. Math. 485, 173–195 (1997)
Garnett, J.: Bounded analytic functions. Springer, New York (2007)
Galanopoulos, P., Girela, D., Merchán, N.: Cesàro-like operators acting on spaces of analytic functions. Anal. Math. Phys., 12 (2022), Paper No. 51
Girela, D.: Analytic functions of bounded mean oscillation. In: Complex Function Spaces, Mekrijärvi 1999 Editor: R. Aulaskari. Univ. Joensuu Dept. Math. Rep. Ser., 4, Univ. Joensuu, Joensuu, (2001) pp. 61–170
Girela, D., Merchán, N.: A Hankel matrix acting on spaces of analytic functions. Integr. Equ. Oper. Theory 89, 581–594 (2017)
Jin, J., Tang, S.:Generalized Cesàro Operators on Dirichlet-Type Spaces. Acta Math. Sci. Ser. B (Engl. Ed.), 42, 212–220(2022)
Merchán, N.: Mean Lipschitz spaces and a generalized Hilbert operator. Collect. Math. 70, 59–69 (2019)
Miao, J.: The Cesàro operator is bounded on \(H^p\) for \(0<p<1\). Proc. Amer. Math. Soc. 116, 1077–1079 (1992)
Ortega, J., Fàbrega, J.: Pointwise multipliers and corona type decomposition in \(BMOA\). Ann. Inst. Fourier (Grenoble) 46, 111–137 (1996)
Siskakis, A.: Composition semigroups and the Cesàro operator on \(H^p\). J. London Math. Soc. 36, 153–164 (1987)
Siskakis, A.: The Cesàro operator is bounded on \(H^1\). Proc. Amer. Math. Soc. 110, 461–462 (1990)
Xiao, J.: Carleson measure, atomic decomposition and free interpolation from Bloch space. Ann. Acad. Sci. Fenn. Ser. A I Math., 19, 35–46 (1994)
Xiao, J.: Holomorphic \({{\cal{Q} }}\) classes. Springer, Berlin (2001)
Xiao, J.: Geometric \({\cal{Q} }_p\) functions. Birkhäuser Verlag, Basel-Boston-Berlin (2006)
Zhu, K.: Operator theory in function spaces. American Mathematical Society, Providence, RI (2007)
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The work was supported by NNSF of China (Nos. 11720101003 and 12271328) and Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515012117).
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Bao, G., Sun, F. & Wulan, H. Carleson measures and the range of a Cesàro-like operator acting on \(H^\infty \). Anal.Math.Phys. 12, 142 (2022). https://doi.org/10.1007/s13324-022-00752-z
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DOI: https://doi.org/10.1007/s13324-022-00752-z