Radially weighted Besov spaces and the Pick property

TL;DR

This paper studies weighted Besov spaces on the unit ball in complex space, establishing conditions under which these spaces have the Pick property and analyzing the boundedness of certain multiplication operators.

Contribution

It introduces new conditions on weights that ensure Besov spaces are complete Pick spaces and characterizes the boundedness of column and row multipliers between these spaces.

Findings

Bounded column multipliers induce bounded row multipliers in these spaces.

Certain weight conditions guarantee the Besov spaces have the complete Pick property.

The paper characterizes when weighted Besov spaces are complete Pick spaces based on weight behavior.

Abstract

For $s\in \mathbb R$ the weighted Besov space on the unit ball $\mathbb B_d$ of $\mathbb C^d$ is defined by $B^s_\omega=\{f\in \operatorname{Hol}(\mathbb B_d): \int_{\mathbb B_d}|R^sf|^2 \omega dV<\infty\}.$ 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 $B^s_\omega \to B^t_\nu \otimes \ell^2$ induces a bounded row multiplier $B^s_\omega \otimes \ell^2 \to B^t_\nu$. 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 $B^s_\omega$ is a complete Pick space, whenever $s\ge (\alpha+d)/2$.