A sharp iteration principle for higher-order Sobolev emebeddings

Optimal higher-order Sobolev type embeddings are shown to follow via iteration of lower-order ones. As a consequence, the optimal target spaces in the relevant Sobolev embeddings can be determined both in standard and in non-standard classes of function spaces and underlying measure spaces. In particular, our results are applied to any-order Sobolev embeddings in regular (John) domains of the Euclidean space, in Maz'ya classes of (possibly irregular) Euclidean domains described in terms of their isoperimetric function, and in families of product probability spaces, of which the Gauss space is a classical instance.