We investigate Carleson measures µ on D where D is the open unit disk in C, along with functional analytic properties of the formal identitiy of the Hardy space H p (D) into the Lebesgue space Lq (µ), for any previously fixed 0 < p, q < ∞. Our corresponding characterizations do not only extend the classical results for measures concentrated on D 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.
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Cruz et al. (2005) studied this question.
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