A Note on the Sobolev and Gagliardo-Nirenberg inequality when p > N

It is known that the Sobolev space W-1,W-P(R-N) is embedded into LNP/(N-P)(R-N) if p < N and into L-infinity(R-N) if p > N. There is usually a discontinuity in the proof of those two different embeddings since, for p > N, the estimate parallel to u parallel to(infinity) <= C parallel to Du parallel to(N/P)(p)parallel to u parallel to(1-N/p)(p) is commonly obtained together with an estimate of the Holder norm. In this note, we give a proof of the L-infinity-embedding which only follows by an iteration of the Sobolev-Gagliardo-Nirenberg estimate parallel to u parallel to(N/(N-1)) <= C parallel to Du parallel to(1). This kind of proof has the advantage to be easily extended to anisotropic cases and immediately exported to the case of discrete Lebesgue and Sobolev spaces; we give sample results in case of finite differences and finite volumes schemes.