Markov's inequality on locally Lipschitzian compact subsets of \({\mathbb{R}}^ N\) in \(L^ p\)-spaces (Q1101908)

The main goal of the paper is to show that for all \(p: 1\leq p\leq \infty\) Markov type inequalities \(\| D_ iP\|_{L^ p}\leq C(\Omega,P)n^ 2\| P\|_{L^ p({\bar \Omega})}\) where \((i=1,...,N)\), \(D_ i=d/dx_ i\), \(x=(x_ 1,...,x_ N)\leftarrow {\mathbb{R}}^ N\), P is a polynomial in N variables of degree n, hold for all bounded domains \(\Omega\) in \({\mathbb{R}}^ N\) with locally Lipschitz boundaries. The method of the proof is based on obtaining local estimates near the boundary by ``pushing'' small ``cylinders'' inside \(\Omega\) (this is possible in view of the Lipschitz condition) and applying the classical Markov inequalities on line segments generating those cylinders. This result is covered by earlier results of \textit{H. Wallin} [Approximation Theory, 2nd Conf. Endmonton/Alberta 1982, CMS Conf., Proc. 3, 377-388 (1983; Zbl 0545.41046)], where a complete characterization of closed sets in \({\mathbb{R}}^ N\) for which Markov's inequalities hold is obtained.