Optimal Sobolev Type Inequalities in Lorentz Spaces

It is well known that the classical Sobolev embeddings may be improved within the framework of Lorentz spaces L p,q : the space D 1,p (R n ) , 1 < p < n, embeds into L p ∗ ,q (R n ) , p ≤ q ≤ ∞. However, the value of the best possible embedding constants in the corresponding inequalities is known just in the case L p ∗ ,p (R n ) . Here, we determine optimal constants for the embedding of the space D 1,p (R n ) , 1 < p < n, into the whole Lorentz space scale L p ∗ ,q (R n ) , p ≤ q ≤ ∞, including the limiting case q = p of which we give a new proof. We also exhibit extremal functions for these embedding inequalities by solving related elliptic problems.