A sharp iteration principle for higher-order Sobolev embeddings
We survey results from the paper [CPS] in which we developed a new sharp iteration method and applied it to show that the optimal Sobolev embeddings of any order can be derived from isoperimetric inequalities. We prove thereby that the well-known link between first-order Sobolev embeddings and isoperimetric inequalities translates to embeddings of any order, a fact that had not been known before. We show a general reduction principle that reduces Sobolev type inequalities of any order involving arbitrary rearrangement-invariant norms on open sets in $\mathbb R^n$, possibly endowed with a measure density and satisfying an isoperimetric inequality of fairly general type, to considerably simpler one-dimensional inequalities for suitable integral operators depending on the isoperimetric function of the relevant sets. As a direct application of the reduction principle we determine the optimal target space in the relevant Sobolev embeddings both in standard and in non-standard classes of function spaces and underlying measure spaces. In particular, the results apply to any-order Sobolev embedding on regular (John) domains, on Maz'ya classes of (possibly irregular) Euclidean domains described in terms of their isoperimetric function, and on families of product probability spaces, of which the Gauss space and the exponential measure space are classical instances.