On the best constants in Markov inequalities for the Laplace operator on polynomials with the Hermite and Legendre norm

Abstract

We consider the Laplace operator on the finite-dimensional linear space of algebraic polynomials in N variables such that each variable occurs at most with the power n. This space of polynomials is equipped with the Hermite norm or the Legendre norm. We derive an asymptotic formula for the norm of the Laplace operator on this space in the Hermite case and asymptotic upper and lower bounds in the Legendre case as n goes to infinity.