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The Dirichlet operator and its mapping properties

Xiong, Jue
http://rave.ohiolink.edu/etdc/view?acc_num=osu1554944747702051

Abstract Details

2019, Doctor of Philosophy, Ohio State University, Mathematics.
My research is focused on finding a natural operator that creates holomorphic functions and has better mapping properties than the Bergman projection $\mathfrak{B}_{\Omega}: L^2(\Omega)\rightarrow A^2(\Omega)$, which is the orthogonal projection of square integrable functions onto the subspace of holomorphic functions. The failure of $L^1$ boundedness of $\mathfrak{B}_{\Omega}$ is demonstrated directly for the unit ball $U^n\subset\C^n$. A proof for the weak type (1,1) estimates of $\mathfrak{B}_{U^n}$ on $L^1$ is also established. We restudy the classical Dirichlet space on the unit disc $U\subset \C$, which is a reproducing kernel Hilbert space when equipped with a suitable norm. The naturally derived Dirichlet operator yields an appropriate substitute for the Bergman projection. We investigate the Dirichlet operator on simply connected domains in $\C$ and obtain its mapping properties via the Bergman operator. We further define the Dirichlet space on $U^n$ as well as on smooth bounded complete Reinhardt domains in $\C^n$ and analyze the mapping properties of the Dirichlet operator, respectively. A substitute result is obtained for the failure of norm convergence of partial sums of $H^1(U)$ functions. The phenomenon observed for the pair $H^1(U)$ and $A^1(U)$ suggests an algebraic scheme for a chain of Banach spaces on $U$ yet to be discovered.
Jeffery McNeal (Advisor)
Liz Vivas (Committee Member)
Kenneth Koenig (Committee Member)
127 p.
Mathematics
Dirichlet operator; Dirichlet kernel; weak type estimates; partial sums of holomorphic functions on the unit disc

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Citations

  • Xiong, J. (2019). The Dirichlet operator and its mapping properties [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1554944747702051

    APA Style (7th edition)

  • Xiong, Jue. The Dirichlet operator and its mapping properties. 2019. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1554944747702051.

    MLA Style (8th edition)

  • Xiong, Jue. "The Dirichlet operator and its mapping properties." Doctoral dissertation, Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1554944747702051

    Chicago Manual of Style (17th edition)

osu1554944747702051
374
© 2019, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.