Adams’ Inequality and Limiting Sobolev Embeddings into Zygmund Spaces

Abstract

We exhibit sharp embedding constants for Sobolev spaces of any order into Zygmund spaces, obtained as the product of sharp embedding constants for second order Sobolev space into Lorentz spaces. As a consequence, we derive a new proof of Adams’ inequality, which holds in the larger hypotheses of homogenoeous Navier boundary contidions.