On the Problem of Sharp Exponents in Multivariate Nikolskii-Type Inequalities Original Research Article

Let Ω⊂RN be a bounded open set, w a weight-function, p>0, and α>0. Assuming some regularity conditions on w and the boundary ∂Ω of Ω we prove that if grad w ≠ 0 on the set Γ where w vanishes and if Γ is transversal to ∂Ω then there exists a positive constant C such that for any polynomial P of degree at most n we have ||P||p,Ω ≤ Cnα||P|w|α||p,Ω ; furthermore the exponent α of n is optimal.