Radially weighted Besov spaces and the Pick property [article]

Alexandru Aleman, Michael Hartz, John E. McCarthy, Stefan Richter
2018 arXiv   pre-print
For s∈ R the weighted Besov space on the unit ball B_d of C^d is defined by B^s_ω={f∈Hol( B_d): ∫_ B_d|R^sf|^2 ω dV<∞}. Here R^s is a power of the radial derivative operator R= ∑_i=1^d z_i∂/∂ 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→ B^t_ν⊗ℓ^2 induces a bounded row multiplier B^s_ω⊗ℓ^2 → B^t_ν. Furthermore we show that if a
more » ... ght 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> (α+d)/2.
arXiv:1807.00730v1 fatcat:7mub6dlyarc3lnmcan7o6d7424