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Study of integral operators on function spaces.

函数空间上的积分算子的研究。

基本信息

批准号：
14540168
负责人：
YAMADA Masahiro
金额：
$ 2.11万
依托单位：
Gifu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540168/
关键词：
Berman space integral operator Carleson inequality Toeplitz operator テープリッツ作加素

项目摘要

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

研究了Toeplitz算子的有界性。设$H$是$n$维欧氏空间的上半空间。对于$O&lt；p&lt；\infty$，设$b^{p}=b^{p}(H，dv)$是$H$上的仿调和函数类$u$。类$b^{p}$称为调和Bergman空间。我们给出了以下结果。设$Mu$是$H$上的$\sigma$-有限正Borel测度，$d\nu=\omega DV$，且$\omega$满足$(A_(Q))_{\Partial}$-条件。存在一个常数$C&gt；0$使得$$\int_{H}|D^{\pha}u|^{p}d\u\int{H}|D^{m}_{y}u|^{p}d\nu$$对b^{p}$中的所有$u\和阶为$\ell$的多个索引$\α$当且仅当存在常数$K&>O$和$0&lt；1$使得$m(S(W))\le K t^{(yell-m)p}\n u(D_{varepsilon}(W))$对所有$w=(S，t)\在H$中。此外，设$\Mu$是$\mathbb{R}^{n+1}_(+)$上的$\sigma$-有限正Borel测度，$\mathbb{N}{0}=\inathbb{N}\cup\0 Y}$\mathbb{N}^{n}_{O}=\mathbb{N}{0}\次\cdots\Times\mathbb{N}[O}$($…更多n$个因素)。对于多指标$\Gammma\in\mathbb{N}^{n}_{01$，$\Partial；^{\Gamma}_{x}$表示微分单项$\partialil^{|\gamma|}/\partiali^|gamma_{1}_{x_{1}}\dots\Partial^{\Gamma_{n}}_{x_{n}}$，令$\Partial_{t}=\Partial/\Partial_{t}$。我们考虑$\MU$的条件，以便存在一个常数$C&GT；$$使得$$\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$$，其中$\ell，m\in\mathbb{N}_{0}$，和$\lambda\in\mathbb{R}$。设$D$是复平面上的开单位圆盘，$H^{p}$是$D$上的经典Hardy空间。Carleson证明了对于H^{p}$中的所有$f，$D$上的有限正Borel测度$\MU$满足$INT_{D}|f|^{p}d\Inu\C\PARALLEL f\PARALLEL^{p}_{H}^p}}$当且仅当对任意区间$i\子集\部分D$存在常数$K&gt；0$，其中$S(I)$是$I$上相应的Carleson平方.研究了上半空间上抛物型Bergman函数满足这类不等式的条件。较少