Study of integral operators on function spaces.

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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