Radially weighted Besov spaces and the Pick property

Abstract

For $s\in \mathbb{R}$ the weighted Besov space on the unit ball ${\mathbb{B}}_d$ of ${\mathbb{C}}^d$ is defined by $B_{\omega }^s=\{f\in \mathrm{Hol}({\mathbb{B}}_d):{\int }_{{\mathbb{B}}_d}|R^{s}f{|}^2\omega dV<\infty \}.$ 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 $\omega$ is a radial weight function not supported on any ball of radius $<1$. Our results imply that for all such weights $\omega$ and $\nu$, every bounded column multiplication operator $B_{\omega }^s\to B_{\nu }^t\otimes {\ell }^2$ induces a bounded row multiplier $B_{\omega }^s\otimes {\ell }^2\to B_{\nu }^t$. Furthermore we show that if a weight $\omega$ satisfies that for some $\alpha >-1$ the ratio $\omega (z)/(1-|z{|}^2{)}^{\alpha }$ is nondecreasing for $t_0<|z|<1$, then $B_{\omega }^s$ is a complete Pick space, whenever $s\ge (\alpha +d)/2$.

32 pages