Study of integral operators on function spaces.

我们研究Toeplitz算子的有界性。让 $H$ 的上半部分 $n$维欧几里得空间。因为 $O<p<\infty$，让 $b^{p}=b^{p}(H,dV)$ 是次调和函数的类 $u$ on $H$． 班级 $b^{p}$ 称为调和伯格曼空间。我们显示了以下结果。假设 $\mu$ 是？ $\sigma$-有限正Borel测度 $H$， $d\nu=\omega dV$ 和 $\omega$ 满足 $(A_{q})_{\partial}$-condition for some $1<q<\infty$． 有一个常数 $C>0$ 这样 $$ \int_{H} |D^{\alpha}u|^{p} d \mu \le C\int {H}|D^{m}_{y}u|^{p} d\nu $$ 对所有人 $u \in b^{p}$ 还有多指标 $\alpha$ 有序的 $\ell$ 当且仅当存在常数 $K>O$ 和 $0<\varepsilon<1$ 这样 $\mu(S(w)) \le K t^{(\Yell-m)p}\nu(D_{\varepsilon}(w))$对所有人 $w=(s,t) \in H$． 此外，让 $\mu$ 做一个 $\sigma$-有限正Borel测度 $\mathbb{R}^{n+1}_{+}$， $\mathbb{N} {0}=\inathbb{N} \cup \{0 Y}$ 和 $\mathbb{N}^{n}_{O}=\mathbb{N} {0} \times \cdots \times \mathbb{N} [O}$ （$ … More n$ 因素)。对于多索引 $\gamnma \in \mathbb{N}^{n}_{01$， $\partia;^{\gamma}_{x}$ 表示微分单项式 $\partialil^{|\gamma|}/\partiali^|gamma_{1}_{x_{1}}\dots \partial^{\gamma_{n}}_{x_{n}}$ 让 $\partial_{t}=\partial/\partial_{t}$． 我们考虑条件 $\mu$ 为了存在一个常数 $C>0$ 这样 $$\int_{\mathbb{R}^{n+1}_{+}}| \partial^{\gamma}_{x} \partial^{\ell}_{t} u|^{p}^-d \mu \le C \int {\mathbb{R}^{n+1}_{+}t^{\lambda}|\partial^{m}_{t} u|^{p}^-dV$$ 对所有人 $u \in b^{p}_{\alpha}$，其中 $\ell,m \in \mathbb{N}_{0}$，和 $\lambda \in \mathbb{R}$． 让 $D$ 为复平面上的开单元盘 $H^{p}$ 是哈代的经典空间 $D$． Carleson证明了有限正Borel测度 $\mu$ on $D$ 满足 $\int_{D}|f|^{p}d \inu \le C \parallel f \parallel^{p}_{H}^p}} $ 对所有人 $f \in H^{p}$ 当且仅当存在一个常数 $K>0$ 有 $\mu(S(I)) \le K |I|$ 对于任意区间 $I \subset \partial D$，其中 $S(I)$ 对应的Carleson平方除以 $I$． 我们研究了条件 $\mu$ 在上半空间上满足抛物型Bergman函数的不等式。更少

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