A Gaussian version of Littlewood’s theorem for random power series
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- by Guozheng Cheng, Xiang Fang, Kunyu Guo and Chao Liu PDF
- Proc. Amer. Math. Soc. 150 (2022), 3525-3536 Request permission
Abstract:
We prove a Littlewood-type theorem for random analytic functions associated with not necessarily independent Gaussian processes. We show that if we randomize a function in the Hardy space $H^2(\mathbb {D})$ by a Gaussian process whose covariance matrix $K$ induces a bounded operator on $l^2$, then the resulting random function is almost surely in $H^p(\mathbb {D})$ for any $p>0$. The case $K=\mathrm {Id}$, the identity operator, recovers Littlewood’s theorem. A new ingredient in our proof is to recast the membership problem as the boundedness of an operator. This reformulation enables us to use tools in functional analysis and is applicable to other situations.References
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Additional Information
- Guozheng Cheng
- Affiliation: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, People’s Republic of China
- MR Author ID: 795828
- Email: gzhcheng@dlut.edu.cn
- Xiang Fang
- Affiliation: Department of Mathematics, National Central University, Chungli, Taiwan (Republic of China)
- MR Author ID: 711208
- ORCID: 0000-0001-9949-7552
- Email: xfang@math.ncu.edu.tw
- Kunyu Guo
- Affiliation: School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China
- Email: kyguo@fudan.edu.cn
- Chao Liu
- Affiliation: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, People’s Republic of China
- Email: 2020024050@dlut.edu.cn
- Received by editor(s): July 21, 2021
- Received by editor(s) in revised form: November 13, 2021
- Published electronically: May 13, 2022
- Additional Notes: The first author was supported by NSFC (11871482).
The second author was supported by MOST of Taiwan (108-2628-M-008-003-MY4) and NSFC (11571248) during his visits to Soochow University in China.
The third and fourth authors were supported by NSFC (11871157). - Communicated by: Javad Mashreghi
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 3525-3536
- MSC (2020): Primary 30B20, 47B38
- DOI: https://doi.org/10.1090/proc/15922
- MathSciNet review: 4439474