Lp Bernstein type inequalities for star like Lip α 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.

Automated summary

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.