径向加权Besov空间与Pick性质
Radially weighted Besov spaces and the Pick property
摘要：
对于$s\in \mathbb R$，$\mathbb C^d$的单位球$\mathbb B_d$上的加权Besov空间定义为$B^s_ω=\{f\in \operatorname{Hol}(\mathbb B_d): \int_{\mathbb B_d}|R^sf|^2 ωdV<\infty\}.$这里$R^s$是径向导数算子$R= \sum_{i=1}^d z_i\frac{\partial}{\partial z_i}$的幂，$V$表示Lebesgue测度，$ω$是不支持任何半径为$< 1$的球的径向权函数。我们的结果意味着，对于所有这样的权重$ω$和$ν$，每个有界列乘法算子$B^s_ω\to B^t_ν\otimes \ell^2$诱导有界行乘法器$B^s_ω\otimes \ell^2 \to B^t_ν$。此外，我们表明，如果一个重量$ω$满足，对于一些$α>-1$的比例$ω(z)/(1-|z|^2)^α$是非减$t_0<|z|<1$，则$B^s_ω$是一个完整的Pick空间，每当$s\ge (α+d)/2$。
英文摘要：
For $s\in \mathbb R$ the weighted Besov space on the unit ball $\mathbb B_d$ of $\mathbb C^d$ is defined by $B^s_ω=\{f\in \operatorname{Hol}(\mathbb B_d): \int_{\mathbb B_d}|R^sf|^2 ωdV<\infty\}.$ Here $R^s$ is a power of the radial derivative operator $R= \sum_{i=1}^d z_i\frac{\partial}{\partial z_i}$, $V$ denotes Lebesgue measure, and $ω$ is a radial weight function not supported on any ball of radius $< 1$. Our results imply that for all such weights $ω$ and $ν$, every bounded column multiplication operator $B^s_ω\to B^t_ν\otimes \ell^2$ induces a bounded row multiplier $B^s_ω\otimes \ell^2 \to B^t_ν$. Furthermore we show that if a weight $ω$ satisfies that for some $α>-1$ the ratio $ω(z)/(1-|z|^2)^α$ is nondecreasing for $t_0<|z|<1$, then $B^s_ω$ is a complete Pick space, whenever $s\ge (α+d)/2$.
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