Optimal Sobolev embeddings on $\mathbb{R}^n$

Abstract

We study Sobolev-type embeddings involving rearrangement-invariant norms. In particular, we focus on the question when such embeddings are optimal. We concentrate on the case when the functions involved are defined on $\mathbb R\sp n$. This subject has been studied before, but only on bounded domains. We first establish the equivalence of the Sobolev embedding to a new type of inequality involving two integral operators. Next, we show this inequality to be equivalent to the boundedness of a certain Hardy operator on a specific new type of cone of positive functions. This Hardy operator is then used to provide optimal domain and range rearrangement-invariant norm in the embedding inequality. Finally, the limiting case of the Sobolev embedding on Rn is studied in detail.