A note on Carleson measures for Hardy spaces

We investigate Carleson measures mu on (D) over bar where D is the open unit disk in C, along with functional analytic properties of the formal identity of the Hardy space H-P(D) into the Lebesgue space L-q(mu), for any previously fixed 0 < p, q < infinity. Our corresponding characterizations do not only extend the classical results for measures concentrated on I but also provide different proofs for the latter ones. Among the applications are generalizations to formal identities as above of several results which have been known for composition operators only.