Comparing norms of polynomials in one and several variables (Q1378757)

The author proves several Nikolskij type inequalities comparing norms of polynomials. If \(P_n\) is a polynomial of degree \(\leq n\), and \(0<p<q \leq\infty\), then \(|P_n |_{L_q (\partial G)} \leq C(p,q)n^k \cdot |P_n|_{L_p (\partial G)}\) where \(k\) depends on \(p,q\) and on the smoothness of the boundary \(\partial G\) of the Jordan domain \(G\). Similarly, area norms \(|P_n |_{L_q (G)}\) and \(|P_n |_{L_p (G)}\) are compared, where \(G\) is assumed to be a quasidisk. These results are extended to polynomials of several complex variables, defined on a product \(H=G_1 \times G_2\times \cdots \times G_m\) of domains.