Imbedding theorems for the invariant subspaces of the backward shift operator

Abstract

For subspaces K ^p_θ of the form$K_\theta ^p = H^p \cap \theta \overline {H_o^p }$ of the Hardy space H^p and for measures μ with support in the closed unit circleclos $\mathbb{D}$, one finds conditions that ensure the imbedding K_θ⊂L^p(μ). One considers measures with support inclos $\mathbb{D}$, satisfying the following condition: for some number ε>0 and for all circles Δ with center on the circumference, intersecting the set$\left\{ {z \in \mathbb{D}:\left| {\theta \left( z \right)} \right|< \varepsilon } \right\}$, we have the inequality μ(Δ)⩽Cℓ(Δ). Here C does not depend on Δ, while ℓ(Δ) is the radius of the circle Δ. For such measures one has the imbedding K ^p_θ ⊂L^p(μ). From here one derives a criterion for the imbedding K ²_A ⊂L²(μ), found by B. Cohn for inner functions θ, such that the set$\left\{ {z \in \mathbb{D}:\left| {\theta \left( z \right)} \right|< \varepsilon } \right\}$ is connected for some positive ɛ. In the paper one also proves that a condition on μ, necessary and sufficient for the imbedding of K ^p_θ into L^p(μ), must depend on p.