An Elementary Proof of Sharp Sobolev Embeddings

We present an elementary uniied and self-contained proof of sharp Sobolev embedding theorems. We introduce a new function space and use it to improve the limiting Sobolev embedding theorem due to Brrzis and Wainger. Let be an open subset of R n , where n 2, let 1 p < 1 and let W 1;p (() be the Sobolev space, that is, the set of all functions in L p ((), whose distributional derivatives of the rst order belong to L p ((), too. If p = n we assume that jj < 1. We deene W 1;p 0 (() as the closure of C 1 0 (() in W 1;p ((). We denote p = np n ? p ; 1 p < n: The classical Sobolev theorem 16] asserts that W 1;p 0 (() , ! L p (() when 1 < p < n: (1.1) (As usual, , ! stands for a continuous embedding.) Although p tends to innnity as