Embedding Theorems for Star-Invariant Subspaces Generated by Smooth Inner Functions

Given an inner functionθon the unit disk D, letKpθ≔Hp∩θzHpbe the corre- sponding star-invariant subspace of the Hardy spaceHp. We are concerned with embedding theorems of the formKpθ⊂Lq(μ), whereμis a measure on D, and some related norm inequalities. In particular, assuming thatθhas a certain smoothness property, we characterize theμʹs such that everyf∈K1θsatisfies∫D |f|q dμ1/q⩽const ∫∂D |f| |θ′|α |dz|,where 1⩽q<∞ and 0<α<1. This result sheds some light on the nature of Carleson-type measures forKpθ.