Multivariate Bernstein and Markov inequalities
Let \(S\) be a convex set in \(\mathbb{R}^ d\) and let \(0<p\leq\infty\), the paper extends the Markov and Bernstein inequalities for the norms in \(L_ p(S)\), of the derivatives of polynomials in \(d\) variables, of total degree \(n\). The most significant result is the following. For a point \(v\in S\) and a vector \(\xi\in\mathbb{R}^ d\), define the directional distance of \(v\) to the boundary of \(S\) by \[ \tilde d(v,\xi):=\sup_{{\lambda>0\atop v+\lambda\xi\in S}}d(v,v+\lambda\xi)\sup_{{\lambda<0\atop v+\lambda\xi\in S}}d(v,v+\lambda\xi). \] Then it is shown that \[ \left\|\tilde d(\cdot,\xi)\left({\partial\over\partial\xi}\right)^ rP_ n(\cdot) \right\|_{L_ p(S)}\leq C(p,r)n^ r\| P_ n\|_{L_ p(S)}, \] where \(C(p,r)\) is independent of \(S,\xi.n\) and \(P_ n\). There are other important inequalities which we will not quote here.
scientific article; zbMATH DE number 92492