加权解析帐篷空间与混合范数空间的极大定理
Maximal theorems for weighted analytic tent and mixed norm spaces
AI总结：
该研究在加权解析帐篷空间与混合范数空间框架下，证明非切向极大算子的有界性，推导多项式稠密性、空间恒等式及Bergman投影有界性的Bekollé-Bonami型刻画。
摘要：
设$\omega$为径向权，$0<p,q<\infty$，且对$\xi\in\overline{\mathbb{D}}$，定义$\Gamma(\xi)=\left\{z\in\mathbb{D}:|\arg z-\arg\xi|<(|\xi|-|z|)\right\}$。平均径向可积空间$L^q_p(\omega)$由单位圆盘$\mathbb{D}$上的复值可测函数$f$构成，满足$$\\|f\\|^q_{L^q_p(\omega)}=\frac{1}{2π}\int_{0}^{2π}\left(\int_{0}^{1}|f(re^{iθ})|^pω(r)r\\,dr\right)^{\frac{q}{p}}dθ<\infty,$$而帐篷空间$T^q_p(\omega)$是满足下式的函数$f$的集合：$$\\|f\\|^q_{T_{p}^{q}(ω)}=\frac{1}{2π}\int_{\partial{\mathbb{D}}}\left(\int_{Γ(ξ)}|f(z)|^pω(z)\frac{dA(z)}{1-|z|}\right)^{\frac{q}{p}}\\,|dξ|<\infty.$$记$\mathcal{H}(\mathbb{D})$为$\mathbb{D}$上的解析函数空间。研究证明，非切向极大算子$$f\mapsto N(f)(ξ)=\sup_{z\inΓ(ξ)}|f(z)|,\quad ξ\in \mathbb{D},$$分别从$AL^q_p(\omega)=L^q_p(\omega)\cap\mathcal{H}(\mathbb{D})$和$AT^q_p(\omega)=T^q_p(\omega)\cap\mathcal{H}(\mathbb{D})$到$L^q_p(\omega)$和$T^q_p(\omega)$有界。利用这些关键不等式进一步得到了多项式在$AL^q_p(\omega)$与$AT^q_p(\omega)$中的稠密性，以及对于满足单侧积分倍增条件的权，成立恒等式$AL^q_p(\omega)=AT^q_p(\omega)$。此外还证明，由标准权$(γ+1)(1-|z|^2)^γ$诱导的经典Bergman投影$P_γ$在$1<q,p<\infty$时于$L^q_p(\omega)$和$T^q_p(\omega)$上的有界性与$q$无关，且可由一类Bekollé-Bonami型条件刻画。
英文摘要：
Let $ω$ be a radial weight, $0<p,q<\infty$ and $Γ(ξ)=\left\{z\in\mathbb{D}:|\arg z-\argξ|<(|ξ|-|z|)\right\}$ for $ξ\in\overline{\mathbb{D}}$ . The average radial integrability space $L^q_p(ω)$ consists of complex-valued measurable functions $f$ on the unit disc $\mathbb{D}$ such that $$\|f\|^q_{L^q_p(ω)}=\frac{1}{2π}\int_{0}^{2π}\left(\int_{0}^{1}|f(re^{iθ})|^pω(r)r\,dr\right)^{\frac{q}{p}}dθ<\infty,$$ and the tent space $T^q_p(ω)$ is the set of those $f$ for which $$\|f\|^q_{T_{p}^{q}(ω)}=\frac{1}{2π}\int_{\partial{\mathbb{D}}}\left(\int_{Γ(ξ)}|f(z)|^pω(z)\frac{dA(z)}{1-|z|}\right)^{\frac{q}{p}}\,|dξ|<\infty.$$ Let $\mathcal{H}(\mathbb{D})$ denote the space of analytic functions in $\mathbb{D}$. It is shown that the non-tangential maximal operator $$f\mapsto N(f)(ξ)=\sup_{z\inΓ(ξ)}|f(z)|,\quad ξ\in \mathbb{D},$$ is bounded from $AL^q_p(ω)=L^q_p(ω)\cap\mathcal{H}(\mathbb{D})$ and $AT^q_p(ω)=T^q_p(ω)\cap\mathcal{H}(\mathbb{D})$ to $L^q_p(ω)$ and $T^q_p(ω)$, respectively. These pivotal inequalities are used to establish further results such as the density of polynomials in $AL^q_p(ω)$ and $AT^q_p(ω)$, and the identity $AL^q_p(ω)=AT^q_p(ω)$ for weights admitting a one-sided integral doubling condition. It is also shown that the boundedness of the classical Bergman projection $P_γ$, induced by the standard weight $(γ+1)(1-|z|^2)^γ$, on $L^q_p(ω)$ and $T^q_p(ω)$ with $1<q,p<\infty$ is independent of $q$, and is described by a Bekollé-Bonami type condition.
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