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\u2009<\u2009p\u2009<\u2009n, embeds into L p 17 ,q (R n ) , p\u2009 64\u2009q\u2009 64\u2009 1e. However, the value of the best possible embedding constants in the corresponding inequalities is known just in the case L p 17 ,p (R n ) . Here, we determine optimal constants for the embedding of the space D 1,p (R n ) , 1\u2009<\u2009p\u2009<\u2009n, into the whole Lorentz space scale L p 17 ,q (R n ) , p\u2009 64\u2009q\u2009 64\u2009 1e, including the limiting case q\u2009=\u2009p of which we give a new proof. We also exhibit extremal functions for these embedding inequalities by solving related elliptic problems
