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Date Published:
10/09/2023
Online Published:
10/09/2023
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Views PDF: 36
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Wee, . J., & Le, . H. K. (2023). On korenblum constants for some weighted function spaces. Thang Long Journal of Science: Mathematics and Mathematical Sciences, 2(8). https://science.thanglong.edu.vn/index.php/volc/article/view/89
On korenblum constants for some weighted function spaces
Main Article Content
Abstract
In this paper, we survey the results on the Korenblum Maximum Principle for some weighted function spaces. Progress and results discussed include the upper bounds and lower bounds of Korenblum constants, as well as the failure of the principle for weighted Bergman spaces, weighted Hardy spaces, weighted Bloch spaces, weighted Fock spaces, and mixed norm spaces. Existing and new open questions are provided.
Article Details
Keywords
Fock space, Hardy space, Bergman space, Bloch space, Korenblum constant, Ramanujan’s Master Theorem, Mellin transform, Dirichlet series.
References
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[4] Karapetrovi´c B., Korenblum maximum principle in mixed norm spaces, Arch. Math. (Basel), 118 (2022), no. 5, 497{507.
[5] Butzer Paul L. and Jansche S., A direct approach to the Mellin transform, J. Fourier Anal. Appl., 3 (1997), no. 4, 325{376.
[6] Cahen E., Sur la fonction ζ(s) de Riemann et sur des fonctions analogues, Ann. Sci. Ecole Norm. Sup. (3), ´ 11 (1894), 75{164 (French).
[7] Danikas N. and Hayman W. K., Domination on sets and in Hp, Results Math., 34(1998), no. 1-2, 85{90, Dedicated to Paul Leo Butzer.
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[11] Hayman W. K., On a conjecture of Korenblum, Analysis (Munich), 19 (1999), no. 2,195{205.
[12] Hinkkanen A., On a maximum principle in Bergman space, J. Anal. Math., 79 (1999),335{344.
[13] Jianhui H. and Zengjian L., The Korenblum’s maximum principle in Fock spaces with small exponents, J. Math. Anal. Appl., 470 (2019), no. 2, 770{776.
[14] Liangying J., Gabriel T. P. and Ruhan Z., On Korenblum’s maximum principle for some function spaces, The first NEAM, Theta Ser. Adv. Math., vol. 22, Editura Fundat¸iei Theta, Bucharest, (2018), 59{80.
[15] Korenblum B., A maximum principle for the Bergman space, Publ. Mat., 35 (1991), no. 2, 479{486.
[16] Korenblum B. and Richards K., Majorization and domination in the Bergman space, Proc. Amer. Math. Soc., 117 (1993), no. 1, 153{158.
[17] Korenblum B., O’Neil R., Richards K. and Zhu K., Totally monotone functions with applications to the Bergman space, Trans. Amer. Math. Soc., 337 (1993), no. 2, 795{806.
[18] Matero J., On Korenblum’s maximum principle for the Bergman space, Arch. Math. (Basel), 64 (1995), no. 4, 337{340.
[19] Perron O., Zur Theorie der Dirichletschen Reihen, J. Reine Angew. Math., 134 (1908), 95{143 (German).
[20] Rainville E. D., Special functions, Chelsea Publishing Co., Bronx, N.Y., (1971).
[21] 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.
[22] Schwick W., On Korenblum’s maximum principle, Proc. Amer. Math. Soc., 125(1997), no. 9, 2581{2587.
[23] Shen C., A slight improvement to Korenblum’s constant, J. Math. Anal. Appl., 337 (2008), no. 1, 464{465.
[24] Chunjie W., Refining the constant in a maximum principle for the Bergman space, Proc. Amer. Math. Soc., 132 (2004), no. 3, 853{855.
[25] Chunjie W., An upper bound on Korenblum’s constant, Integral Equations Operator Theory, 49 (2004), no. 4, 561{563.
[26] Chunjie W., On Korenblum’s constant, J. Math. Anal. Appl., 296 (2004), no. 1, 262{264.
[27] Chunjie W., On Korenblum’s maximum principle, Proc. Amer. Math. Soc., 134 (2006), no. 7, 2061{2066.
[28] Chunjie W., Behavior of the constant in Korenblum’s maximum principle, Math. Nachr., 281 (2008), no. 3, 447{454.
[29] Chunjie W., Domination in the Bergman space and Korenblum’s constant, Integral Equations Operator Theory, 61 (2008), no. 3, 423{432.
[30] Chunjie W., Some results on Korenblum’s maximum principle, J. Math. Anal. Appl., 373 (2011), no. 2, 393{398.
[31] JunJie W., The Korenblum Maximum Principle for some Function Spaces, Bachelor’s Thesis, Nanyang Technological University, 2019. Available at URI hdl.handle. net/10356/77142.
[32] JunJie W. and Khoi L. H., Korenblum constants for some function spaces, Proc. Amer. Math. Soc., 148 (2020), no. 3, 1175{1185.
[33] JunJie W. and Khoi L. H., Korenblum constants for various weighted Fock spaces, Complex Variables and Elliptic Equations, posted on June 2022, 1-22, DOI 10.1080/17476933.2022.2052862.
[34] Kehe Z., Analysis on Fock spaces, Graduate Texts in Mathematics, vol. 263, Springer, New York, (2012).
[2] Berndt Bruce C. The quarterly reports of S. Ramanujan, Amer. Math. Monthly, 90(1983), no. 8, 505{516.
[3] Boˇzin V. and Karapetrovi´c B., Failure of Korenblum’s maximum principle inBergman spaces with small exponents, Proc. Amer. Math. Soc., 146 (2018), no. 6, 2577{2584.
[4] Karapetrovi´c B., Korenblum maximum principle in mixed norm spaces, Arch. Math. (Basel), 118 (2022), no. 5, 497{507.
[5] Butzer Paul L. and Jansche S., A direct approach to the Mellin transform, J. Fourier Anal. Appl., 3 (1997), no. 4, 325{376.
[6] Cahen E., Sur la fonction ζ(s) de Riemann et sur des fonctions analogues, Ann. Sci. Ecole Norm. Sup. (3), ´ 11 (1894), 75{164 (French).
[7] Danikas N. and Hayman W. K., Domination on sets and in Hp, Results Math., 34(1998), no. 1-2, 85{90, Dedicated to Paul Leo Butzer.
[8] Flajolet P., Gourdon X. and Dumas P., Mellin transforms and asymptotics: harmonic sums. Special volume on mathematical analysis of algorithms, Theoret. Comput.Sci. , 144 (1995), no. 1-2, 358.
[9] Hardy G. H. Ramanujan. Twelve lectures on subjects suggested by his life and work, Cambridge University Press, Cambridge, England; Macmillan Company, New York,
(1940).
[10] Hardy G. H. and Riesz M., The general theory of Dirichlet’s series, CambridgeTracts in Mathematics and Mathematical Physics, No. 18, Stechert-Hafner, Inc., New
York, (1964).
[11] Hayman W. K., On a conjecture of Korenblum, Analysis (Munich), 19 (1999), no. 2,195{205.
[12] Hinkkanen A., On a maximum principle in Bergman space, J. Anal. Math., 79 (1999),335{344.
[13] Jianhui H. and Zengjian L., The Korenblum’s maximum principle in Fock spaces with small exponents, J. Math. Anal. Appl., 470 (2019), no. 2, 770{776.
[14] Liangying J., Gabriel T. P. and Ruhan Z., On Korenblum’s maximum principle for some function spaces, The first NEAM, Theta Ser. Adv. Math., vol. 22, Editura Fundat¸iei Theta, Bucharest, (2018), 59{80.
[15] Korenblum B., A maximum principle for the Bergman space, Publ. Mat., 35 (1991), no. 2, 479{486.
[16] Korenblum B. and Richards K., Majorization and domination in the Bergman space, Proc. Amer. Math. Soc., 117 (1993), no. 1, 153{158.
[17] Korenblum B., O’Neil R., Richards K. and Zhu K., Totally monotone functions with applications to the Bergman space, Trans. Amer. Math. Soc., 337 (1993), no. 2, 795{806.
[18] Matero J., On Korenblum’s maximum principle for the Bergman space, Arch. Math. (Basel), 64 (1995), no. 4, 337{340.
[19] Perron O., Zur Theorie der Dirichletschen Reihen, J. Reine Angew. Math., 134 (1908), 95{143 (German).
[20] Rainville E. D., Special functions, Chelsea Publishing Co., Bronx, N.Y., (1971).
[21] 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.
[22] Schwick W., On Korenblum’s maximum principle, Proc. Amer. Math. Soc., 125(1997), no. 9, 2581{2587.
[23] Shen C., A slight improvement to Korenblum’s constant, J. Math. Anal. Appl., 337 (2008), no. 1, 464{465.
[24] Chunjie W., Refining the constant in a maximum principle for the Bergman space, Proc. Amer. Math. Soc., 132 (2004), no. 3, 853{855.
[25] Chunjie W., An upper bound on Korenblum’s constant, Integral Equations Operator Theory, 49 (2004), no. 4, 561{563.
[26] Chunjie W., On Korenblum’s constant, J. Math. Anal. Appl., 296 (2004), no. 1, 262{264.
[27] Chunjie W., On Korenblum’s maximum principle, Proc. Amer. Math. Soc., 134 (2006), no. 7, 2061{2066.
[28] Chunjie W., Behavior of the constant in Korenblum’s maximum principle, Math. Nachr., 281 (2008), no. 3, 447{454.
[29] Chunjie W., Domination in the Bergman space and Korenblum’s constant, Integral Equations Operator Theory, 61 (2008), no. 3, 423{432.
[30] Chunjie W., Some results on Korenblum’s maximum principle, J. Math. Anal. Appl., 373 (2011), no. 2, 393{398.
[31] JunJie W., The Korenblum Maximum Principle for some Function Spaces, Bachelor’s Thesis, Nanyang Technological University, 2019. Available at URI hdl.handle. net/10356/77142.
[32] JunJie W. and Khoi L. H., Korenblum constants for some function spaces, Proc. Amer. Math. Soc., 148 (2020), no. 3, 1175{1185.
[33] JunJie W. and Khoi L. H., Korenblum constants for various weighted Fock spaces, Complex Variables and Elliptic Equations, posted on June 2022, 1-22, DOI 10.1080/17476933.2022.2052862.
[34] Kehe Z., Analysis on Fock spaces, Graduate Texts in Mathematics, vol. 263, Springer, New York, (2012).