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 (\({0 \leqq p < q \leqq \infty}\))
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.
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Ganzburg, M.I. Multivariate polynomial inequalities of different \({L_{p,W}(V)}\)-metrics with k-concave weights. Acta Math. Hungar. 150, 99–120 (2016). https://doi.org/10.1007/s10474-016-0632-z
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DOI: https://doi.org/10.1007/s10474-016-0632-z
Key words and phrases
- multivariate polynomial
- polynomial inequality of different metrics
- (1/d)-concave measure
- log-concave measure