Sharp estimate of the Laplacian of a polyharmonic function and applications

Abstract:

The classical sharp inequality of Markov estimates the values of the derivative of the polynomial of degree $n$ in the interval $[a,b]$ through the uniform norm of the polynomial in the same interval multiplied by $2{n^2}/(b - a)$. In the present paper we provide an exact estimate for the values of the Laplacian of a polyharmonic function of degree $m$ by the uniform norm of the polyharmonic function multiplied by $2{(m - 1)^2}/{R^2}(x)$ where $R(x)$ is the distance from the point $x$ to the boundary of the domain. The inequality of Markov (and the similar inequality of Bernstein about trigonometric polynomials) finds many applications in approximation theory for functions of one variable. We prove analogues to some of these results in the multivariate case.