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

我们研究Toeplitz运营商的辉煌。令$ h $为$ n $维欧几里得空间的上半空间。对于$ o <p <\ infty $，让$ b^{p} = b^{p}（h，dv）$是$ h $上的aliharmonic函数$ u $的类。 $ b^{p} $类称为Harmonic Bergman空间。我们显示以下结果。假设$ \ mu $是$ h $，$ h $，$ d \ nu = \ omega dv $和$ \ omega $上的$ \ sigma $ -finite阳性borel borel测量值满足了$（a_ {q}）_ {\ partial} $ - 对于某些$ 1 <q <q <q <q iffty $。有一个常数$ c> 0 $，以至于$ \ int_ {h} | d^{\ alpha} u |^{p} d \ mu \ mu \ mu \ le c \ le c \ le c \ int { $ \ ell $如果有常数$ k> o $，而$ 0 <\ varepsilon <1 $，以至于$ \ mu（s（w））\ le k t^{（\ yell-m）p} \ nu（d _ {\ varepsilon}（w）$ for $ w =（s n $ w =（s s s s s s s，t）此外，让$ \ mu $成为$ \ sigma $ -finite阳性borel borel测量，$ \ mathbb {r}^{n+1} _ {+} $，$ \ mathbb {n} $ \ mathbb {n}^{n} _ {o} = \ mathbb {n} {0} {0} \ times \ cdots \ times \ times \ times \ mathbb {n} [o} [o} $（$……更多n $ factor）。对于多索引$ \ gamnma \ in \ mathbb {n}^{n} _ {01 $，$ \ partia;^{\ gamma} _ {x} $表示差分单元$ \ partialil^{| \ gamma |}/\ partialli^| gamma_ {1} _ {x_ {1}} \ dots \ dots \ partial^{\ gamma_ {n}} _ {n} _ {x _ {n}}} $ and liet $ part $ part $ part $ part}我们考虑$ \ mu $的条件，以便存在常数$ c> 0 $，以便$ \ int _ {\ mathbb {r}^{n+1} _ {+}} | \ partial^{\ gamma} _ {x} \ partial^{\ ell} _ { {\ Mathbb {r}^{n+1} _ {+} \ mathbb {n} _ {0} $，$ \ lambda \ in \ mathbb {r} $。令$ d $为复杂平面中的开放单元磁盘，而$ h^{p} $是$ d $上的经典hardy空间。卡莱森证明了$ d $上的有限的积极borel量$ \ mu $满足$ \ int_ {d} | f |^{p} d \ inu \ inu \ inu \ inu \ le c \ le c \ parallel f \ parallel f \ parallel^{p} _ {对于任何间隔$ i \ subset \ partial d $，$ \ mu（s（i））\ le k | i | $，其中$ s（i）$是$ i $的相应的carleson广场。我们研究了$ \ mu $的条件，以满足上半部空间上抛物线伯格曼功能的这种不平等现象。较少的