Imbedding theorems for the invariant subspaces of the backward shift operator

For subspaces K ^p of the form of the Hardy space H^p and for measures with support in the closed unit circleclos , one finds conditions that ensure the imbedding KL^p(). One considers measures with support inclos , satisfying the following condition: for some number >0 and for all circles with center on the circumference, intersecting the set , we have the inequality ()C(). Here C does not depend on , while () is the radius of the circle . For such measures one has the imbedding K ^p L^p(). From here one derives a criterion for the imbedding K _A ² L²(), found by B. Cohn for inner functions , such that the set is connected for some positive . In the paper one also proves that a condition on , necessary and sufficient for the imbedding of K ^p into L^p(), must depend on p.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 38–51, 1986.