A Carleson type measure and a family of M\"obius invariant function spaces

Abstract

For $0<s<1$, let $\{z_n\}$ be a sequence in the open unit disk such that $\sum_n (1-|z_n|^2)^s \delta_{z_n}$ is an $s$-Carleson measure. In this paper, we consider the connections between this $s$-Carleson measure and the theory of M\"obius invariant $F(p, p-2, s)$ spaces by the Volterra type operator, the reciprocal of a Blaschke product, and second order complex differential equations having a prescribed zero sequence.