| Project/Area Number |
14540168
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Gifu University |
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Principal Investigator |
YAMADA Masahiro Gifu University, Education, Mathematics, professor, 教育学部, 助教授 (00263666)
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| Co-Investigator(Kenkyū-buntansha) |
ISHIWATA Tetsuya Gifu University, Education, Mathematics, Assistant professor, 教育学部, 助教授 (50334917)
AIKI Toyohiko Gifu University, Education, Mathematics, Assistant professor, 教育学部, 助教授 (90231745)
TAKUCHI Sigeru Gifu University, Education, Mathematics, professor, 教育学部, 教授 (30021330)
YONEDA Rikio Aichi University of Education, Education, Mathematics, Assistant, 教育学部, 助手 (70342475)
SHIMOMURA Tetsu Hiroshima University, Education, Mathematics, Assistant professor, 教育学部, 助教授 (50294476)
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| Project Period (FY) |
2002 – 2003
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| Keywords | Berman space / integral operator / Carleson inequality / Toeplitz operator |
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| Research Abstract |
We study boudedness of Toeplitz operators. Let $H$ be the upper half-space of the $n$-dimensional Euclidean space. For $O<p<\infty$, let $b^{p}=b^{p}(H,dV)$ be the class of aliharmonic functions $u$ on $H$. The class $b^{p}$ is called the harmonic Bergman space. We show the following results. Suppose that $\mu$ is a $\sigma$-finite positive Borel measure on $H$, $d\nu=\omega dV$ and $\omega$ satisfies the $(A_{q})_{\partial}$-condition for some $1<q<\infty$. There is a constant $C>0$ such that $$ \int_{H} |D^{\alpha}u|^{p} d \mu \le C\int {H}|D^{m}_{y}u|^{p} d\nu $$ for all $u \in b^{p}$ and multi-indices $\alpha$ of order $\ell$ if andonly if There are constants $K>O$ and $0<\varepsilon<1$ such that $\mu(S(w)) \le K t^{(\Yell-m)p}\nu(D_{\varepsilon}(w))$for all $w=(s,t) \in H$. Moreover, let $\mu$ be a $\sigma$-finite positive Borel measure on $\mathbb{R}^{n+1}_{+}$, $\mathbb{N} {0}=\inathbb{N} \cup \{0 Y}$ and $\mathbb{N}^{n}_{O}=\mathbb{N} {0} \times \cdots \times \mathbb{N} [O}$ ($
… More
n$ factors). For a multi-index $\gamnma \in \mathbb{N}^{n}_{01$, $\partia;^{\gamma}_{x}$ denotes the differential monomial $\partialil^{|\gamma|}/\partiali^|gamma_{1}_{x_{1}}\dots \partial^{\gamma_{n}}_{x_{n}}$ and let $\partial_{t}=\partial/\partial_{t}$. We consider conditions for $\mu$ in order that there exists a constant $C>0$ such that $$\int_{\mathbb{R}^{n+1}_{+}}| \partial^{\gamma}_{x} \partial^{\ell}_{t} u|^{p}^-d \mu \le C \int {\mathbb{R}^{n+1}_{+}t^{\lambda}|\partial^{m}_{t} u|^{p}^-dV$$ for all $u \in b^{p}_{\alpha}$, where $\ell,m \in \mathbb{N}_{0}$, and $\lambda \in \mathbb{R}$. Let $D$ be the open unit disk in the complex plane and $H^{p}$ be the classical Hardy spaces on $D$. Carleson proved that a finite positive Borel measure $\mu$ on $D$ satisfies $\int_{D}|f|^{p}d \inu \le C \parallel f \parallel^{p}_{H}^p}} $ for all $f \in H^{p}$ if and only if there exists a constant $K>0$ with $\mu(S(I)) \le K |I|$ for any interval $I \subset \partial D$, where $S(I)$ is the corresponding Carleson square over $I$. We stud y conditions for $\mu$ satisfying such inequalities for parabolic Bergman functions on the upper half space. Less
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