Une inégalité polynômiale en plusieurs variables (Q789628)

Let \(\Omega\) be a bounded open subset of \({\mathbb{R}}^ N\) and m in \(C^ s({\bar \Omega})\). We show that under general assumptions on m and \(\Omega\) there exist two constants C and d such that for any polynomial \(P(x_ 1,...,x_ N)\) we have \(\| P\|_{L^{\infty}(\Omega)}\leq C(d^ 0P)^ d\| Pm\|_{L^{\infty}(\Omega)}\). Furthermore we give the sharp value and the geometric meaning of d.