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.