Carleson measures and embeddings of abstract Hardy spaces into function lattices

We apply interpolation techniques to study behaviour of the canonical inclusion maps of quasi-Banach spaces of analytic functions on the open unit disk of the plane into (quasi)-Banach function lattices on the closed or open unit disk equipped with a Borel measure. These results are applied to abstract Hardy spaces generated by symmetric spaces. We investigate relationships between boundedness or compactness of the canonical inclusion maps and generalized variants of Carleson measures and show applications to composition operators on abstract Hardy spaces. We specialize our results to Hardy–Lorentz spaces.