Multivariate polynomial inequalities of different $${L_{p,W}(V)}$$ L p , W ( V ) -metrics with k-concave weights

Abstract

Let W be a k-concave weight on an open convex set V in \({{\mathbb R}^m}\), \({k \in [0, \infty]}\), and let \({\mu_W}\) be the weighted measure on V generated by W with \({\mu_W(V) < \infty}\). We find lower and upper estimates of a constant A in the inequality (\({0 \leqq p < q \leqq \infty}\)) $$\begin{array}{ll}\bigg(\frac{1}{\mu_W(V)}\int_V \big|P(x)\big|^{q} W(x) \, dx \bigg)^{1/q} \\ \leqq A(n, m, p, q, V, W)\bigg(\frac{1}{\mu_W(V)}\int_V \big|P(x)\big|^p W(x) \, dx\bigg)^{1/p},\end{array}$$ where P is a polynomial of m variables of degree at most n. In the case of log-concave measures (k = 0) we improve estimates of A obtained by A. Brudnyi. For \({k \in (0, \infty]}\) estimates of A are new, and we show that they are sharp with respect to n as \({n \to \infty}\). The proofs are based on distributional inequalities for polynomials obtained by Nazarov, Sodin, Volberg, and Fradelizi. Two new examples for a generalized Jacobi weight on [−1, 1] and a multivariate Gegenbauer-type weight on a convex body are included.