Abstract
We completely characterize the sampling measures \(\mu \) for a family of Fock space \(F^p_\phi (0<p<\infty )\) induced by a non-radial doubling weight by refining \(\widehat{\mu }_r\) and related dominating sets. Using this characterization, we illustrate the reason why \(\widehat{\mu }_r\), \(1/\widehat{\mu }_r\in L^\infty \) or \(\tilde{\mu }\), \(1/\tilde{\mu }\in L^\infty \) does not make \(\mu \) be sampling measures on Fock spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blandignères, A., Fricain, E., Gaunard, F., Hartmann, A., Ross, W.T.: Direct and reverse Carleson measures for H(b) spaces. Indiana Univ. Math. J. 64, 1027–1057 (2015)
Christ, M.: On the \(\overline{\partial }\) equation in weighted \(L^2\) norms in \({\mathbb{C} }^{1}\). J. Geom. Anal. 1, 193–230 (1991)
Constantin, O., Ortega-Cerdà, J.: Some spectral properties of the canonical solution operator to \(\overline{\partial }\) on weighted Fock spaces. J. Math. Anal. Appl. 377, 353–361 (2011)
Dall’Ara, G.M.: Pointwise estimates of weighted Bergman kernels in several complex variables. Adv. Math. 285, 1706–1740 (2015)
Fricain, E., Hartmann, A., Ross, W.T.: A survey on reverse Carleson measures. Harmonic analysis, function theory, operator theory and their applications. Theta Ser. Adv. Math. 19, 91–123 (2017)
Hartmann, A., Massaneda, X., Nicolau, A., Ortega-Cerdà, J.: Reverse Carleson measures in Hardy spaces. Collect. Math. 65, 357–365 (2014)
Hong, R.C.: Toeplitz operators on generalized Fock spaces. Bull. Korean Math. Soc. 53, 711–722 (2016)
Hu, Z., Lv, X.: Hankel operators on weighted Fock spaces (in Chinese). Sci. China Math. 46, 141–156 (2016)
Hu, Z., Lv, X.: Positive Toeplitz operators between different doubling Fock spaces. Taiwan. J. Math. 21, 467–487 (2017)
Hu, Z., Virtanen, J.A.: Fredholm Toeplitz operators on doubling Fock spaces. J. Geom. Anal. 32, Paper No. 106 (2022)
Korhonen, T., Rättyä, J.: Zero sequences, factorization and sampling measures for weighted Bergman spaces. Math. Z. 291, 1145–1173 (2019)
Luecking, D.: Closed ranged restriction operators on weighted Bergman spaces. Pac. J. Math. 110, 145–160 (1984)
Luecking, D.: Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives. Am. J. Math. 107, 85–111 (1985)
Luecking, D.: Inequalities on Bergman spaces. Illinois J. Math. 25, 1–11 (1981)
Luecking, D.: Representation and duality in weighted spaces of analytic functions. Indiana Univ. Math. J. 34, 319–336 (1985)
Luecking, D.: Sampling measures for Bergman spaces on the unit disk. Math. Ann. 316, 659–679 (2000)
Lyubarski\(\breve{1}\), Y.I., Seip, K.: Sampling and interpolation of entire functions and exponential systems in convex domains. Ark. Mat. 32, 157–193 (1994)
Lou, Z., Zhu, K., Zhuo, Z.: Atomic decomposition and duality for a class of Fock spaces. Complex Var. Elliptic Equ. 64, 1905–1931 (2019)
Lou, Z., Zhuo, Z.: A Class of reverse Carleson measures on doubling Fock Spaces. Complex Anal. Oper. Theory 13, 1795–1809 (2019)
Marco, N., Massaneda, X., Ortega-Cerdà, J.: Interpolating and sampling sequences for entire functions. Geome. Funct. Anal. 13, 862–914 (2003)
Marzo, J., Ortega-Cerdà, J.: Pointwise estimates for the Bergman kernel of the weighted Fock space. J. Geom. Anal. 19, 890–910 (2008)
Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces, Fractals and Rectifiability. Cambridge University Press, Cambridge (1995)
Ortega-Cerdà, J.: Sampling measures. Publ. Mat. 42, 559–566 (1998)
Oliver, R., Pascuas, D.: Toeplitz operators on doubling Fock spaces. J. Math. Anal. Appl. 435, 1426–1457 (2015)
Seip, K.: Density theorems for sampling and interpolation in the Bargmann-Fock space. Bull. Am. Math. Soc. 26, 322–328 (1992)
Seip, K.: Density theorems for sampling and interpolation in the Bargmann-Fock space I. J. Reine Angew. Math. 429, 91–106 (1992)
Seip, K.: Interpolation and sampling in small Bergman spaces. Collect. Math. 64, 61–72 (2013)
Seip, K.: Interpolation and Sampling in Spaces of Analytic Functions, volume 33 of University Lecture Series. American Mathematical Society, Providence, RI (2004)
Tong, C., Li, J., He, H., Arroussi, H.: Reverse Carleson measures on the weighted Bergman spaces with invariant weight. Ann. Funct. Anal. 12, Paper No. 56 (2021)
Wang, Z., Zhao, X.: Invertibility of Fock Toeplitz operators with positive symbols. J. Math. Anal. Appl. 435, 1335–1351 (2015)
Zhu, K.: Analysis on Fock Spaces. Springer, New York (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Research was supported by NNSF of China(12071272), Guangdong Basic and Applied Basic Research Foundation (2020A1515110493), the Natural Science Research Project of Guangdong Education Department, China (2020KQNCX042) and Guangdong Polytechnic Normal University Science Foundation(2021SDKYA154).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Zhuo, Z., Lou, Z. Sampling Measure on Doubling Fock Spaces. J Geom Anal 33, 313 (2023). https://doi.org/10.1007/s12220-023-01380-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12220-023-01380-0