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

$$\displaystyle B^s_\omega =\{f\in \operatorname {Hol}(\mathbb {B}_d): \int _{\mathbb {B}_d}|R^sf|{ }^2 \omega dV<\infty \}. $$

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 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 $B^s_\omega \to B^t_\nu \otimes \ell ^2$ induces a bounded row multiplier $B^s_\omega \otimes \ell ^2 \to B^t_\nu$. Furthermore we show that if a weight ω satisfies that for some α > −1 the ratio ω(z)∕(1 −|z|²)^α is nondecreasing for t ₀ < |z| < 1, then $B^s_\omega$ is a complete Pick space, whenever s ≥ (α + d)∕2.

M.H. was partially supported by a Feodor Lynen Fellowship. J.M. was partially supported by National Science Foundation Grant DMS 1565243.