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Radially weighted Besov spaces and the Pick property

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 $s\ge (α+d)/2$.