A Korenblum Maximum Principle for weighted Hilbert spaces of entire Dirichlet series with real frequencies
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Abstract
In this paper, we study a Korenblum Maximum Principle for weighted Hilbert spaces of entire Dirichlet series with real frequencies. We investigate dominating sets for which the Korenblum Maximum Principle must hold. The results obtained imply that a dominating set, if exists, must be a left half-plane. This provides a new perspective for studying Korenblum Maximum Principle on function spaces containing the entire Dirichlet series.
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References
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[2] Apostol, Tom M., Modular functions and Dirichlet series in numbertheory, Graduate Texts in Mathematics 41 (1990), Springer-Verlag
[3] Berndt, Bruce C., The quarterly reports of S. Ramanujan, Amer. Math.Monthly 90 (1983), no. 8, 505–516.
[4] Hardy, G.H., Riesz, M., The general theory of Dirichlet’s series, 1964, Cambridge Tracts in Math. and Math. Phys., No. 18, Stechert-Hafner, Inc., New York.
[5] Hayman, W.K., On a conjecture of Korenblum, Analysis (Munich) 19 (1999), no. 2, 195–205.
[6] Le H.K., Le T., Composition operators on entire Dirichlet series with real frequencies, 205 pages, World Scientific, 2025 (https://doi.org/10.1142/14137)
[7] Korenblum B., A maximum principle for the Bergman space, Publ. Mat., 35 (1991), no. 2, 479–486.
[8] Korenblum B., O’Neil R., Richards K., Zhu K., Totally monotone functions with applications to the Bergman space, Trans. Amer. Math. Soc. 337 (1993), no. 2, 795–806.
[9] Korenblum, B., Richards K., Majorization and domination in the Bergman space, Proc. Amer. Math. Soc. 117 (1993), no. 1, 153–158.
[10] Matero, J., On Korenblum’s maximum principle for the Bergman space, Arch. Math. (Basel) 64 (1995), no. 4, 337–340.
[11] Schuster, A., The maximum principle for the Bergman space and the M¨obius pseudodistance for the annulus, Proc. Amer. Math. Soc. 134 (2006), no. 12, 3525–3530.
[12] Schwick, W., On Korenblum’s maximum principle, Proc. Amer. Math. Soc. 125 (1997), no. 9, 2581–2587.
[13] Wang, C., Some results on Korenblum’s maximum principle, J. Math. Anal. Appl. 373 (2011), no. 2, 393–398.
[14] Wee J.J., Le H.K. , On Korenblum constants for some weighted function spaces, Thang Long J. Math. & Math. Sci. 2 (2023), no. 1, 87–116.
[15] Wee J.J., Le H.K., Korenblum constants for various weighted Fock spaces, Complex Variables & Elliptic Equations 68 (2023), no. 8, 1385–1406.
[16] Wee J.J., Le H.K., Korenblum constants for some function spaces, Proc. Amer. Math. Soc. 148 (2020), no. 3, 1175–1185.
[2] Apostol, Tom M., Modular functions and Dirichlet series in numbertheory, Graduate Texts in Mathematics 41 (1990), Springer-Verlag
[3] Berndt, Bruce C., The quarterly reports of S. Ramanujan, Amer. Math.Monthly 90 (1983), no. 8, 505–516.
[4] Hardy, G.H., Riesz, M., The general theory of Dirichlet’s series, 1964, Cambridge Tracts in Math. and Math. Phys., No. 18, Stechert-Hafner, Inc., New York.
[5] Hayman, W.K., On a conjecture of Korenblum, Analysis (Munich) 19 (1999), no. 2, 195–205.
[6] Le H.K., Le T., Composition operators on entire Dirichlet series with real frequencies, 205 pages, World Scientific, 2025 (https://doi.org/10.1142/14137)
[7] Korenblum B., A maximum principle for the Bergman space, Publ. Mat., 35 (1991), no. 2, 479–486.
[8] Korenblum B., O’Neil R., Richards K., Zhu K., Totally monotone functions with applications to the Bergman space, Trans. Amer. Math. Soc. 337 (1993), no. 2, 795–806.
[9] Korenblum, B., Richards K., Majorization and domination in the Bergman space, Proc. Amer. Math. Soc. 117 (1993), no. 1, 153–158.
[10] Matero, J., On Korenblum’s maximum principle for the Bergman space, Arch. Math. (Basel) 64 (1995), no. 4, 337–340.
[11] Schuster, A., The maximum principle for the Bergman space and the M¨obius pseudodistance for the annulus, Proc. Amer. Math. Soc. 134 (2006), no. 12, 3525–3530.
[12] Schwick, W., On Korenblum’s maximum principle, Proc. Amer. Math. Soc. 125 (1997), no. 9, 2581–2587.
[13] Wang, C., Some results on Korenblum’s maximum principle, J. Math. Anal. Appl. 373 (2011), no. 2, 393–398.
[14] Wee J.J., Le H.K. , On Korenblum constants for some weighted function spaces, Thang Long J. Math. & Math. Sci. 2 (2023), no. 1, 87–116.
[15] Wee J.J., Le H.K., Korenblum constants for various weighted Fock spaces, Complex Variables & Elliptic Equations 68 (2023), no. 8, 1385–1406.
[16] Wee J.J., Le H.K., Korenblum constants for some function spaces, Proc. Amer. Math. Soc. 148 (2020), no. 3, 1175–1185.