An elementary proof of sharp Sobolev embeddings (Q2758994)

Let \(\Omega\) be an open subset of \(\mathbb R^n\), \(n\geq 2\), \(1\leq p\leq n\) and set \(p^*=np/(n-p)\) if \(1\leq p<n\). Classical Sobolev embeddings state that \(W_0^{1,p}(\Omega)\hookrightarrow L^{p^*}(\Omega)\) for \(1\leq p<n\) and that \(W_0^{1,n}(\Omega) \hookrightarrow \exp L^{\frac n{n-1}}(\Omega)\) (Trudinger inequality). NEWLINENEWLINENEWLINEThese embeddings are optimal in the class of Orlicz spaces. However, sharper embeddings can be obtained in the context of rearrangement-invariant Banach function spaces. One has \(W_0^{1,p}(\Omega) \hookrightarrow L^{p^*,p}(\Omega)\) (the Lorentz space) for \(1\leq p<n\) and \(W_0^{1,n}(\Omega) \hookrightarrow BW_n(\Omega)\) (this last embedding is due to Brézis and Wainger). Again, these embeddings are sharp in the class of rearrangement-invariant Banach function spaces. NEWLINENEWLINENEWLINEIn the present paper, the authors give an elementary proof of these sharp Sobolev embeddings. The essential tool of this proof is a weak version of the Gagliardo-Nirenberg embedding. One of the features of their approach is that it includes both the case \(1\leq p<n\) and the case \(p=n\). NEWLINENEWLINENEWLINEMoreover, the authors define a new function set, called \(W_p(\Omega)\), which is a strict subset of \(BW_p(\Omega)\). A byproduct of their proofs of sharp Sobolev embeddings is the fact that, when \(\left|\nabla u\right|\in L^n(\Omega)\), \(u\) belongs to \(W_n(\Omega)\). Some properties of \(W_p(\Omega)\) are investigated. In particular, \(W_p(\Omega)\) is not closed under addition.