期刊
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
卷 532, 期 2, 页码 -出版社
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jmaa.2023.127986
关键词
Multivariate polynomials; Cuspidal sets; L- p norm; Bernstein-Markov inequality
The goal of this paper is to establish that the square root of the Euclidean distance to the boundary is a universal measure for obtaining L-p Bernstein type inequalities on general star-like Lip 1 domains. It also explores the case of cuspidal Lip alpha, 0 < alpha < 1 graph domains.
The goal of the present paper is to establish that square root of the Euclidean distance to the boundary is the universal measure suitable for obtaining L-p Bernstein type inequalities on general star like Lip 1 domains. This will be proved for derivatives of any order, every 0 < p < infinity and generalized Jacobi type weights. A converse result will show that the square root of the Euclidean distance to the boundary in general is the best possible measure in the vicinity of any vertex of a convex polytope. In addition we will also consider cuspidal Lip alpha, 0 < alpha < 1 graph domains. It turns out that for such cuspidal domains the situation can change dramatically: instead of taking the square root we need to use the (1/alpha - 1/2 )-th power of the Euclidean distance to the boundary when 0 < alpha < 1, and this measure of the distance to the boundary is in general the best possible, as well.(c) 2023 The Author(s). Published by Elsevier Inc.
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