Lp Bernstein type inequalities for star like Lip α domains

Kroó, András (2024) Lp Bernstein type inequalities for star like Lip α domains. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 532 (2). ISSN 0022-247X

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Abstract

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 Lp Bernstein type inequalities on general star like Lip 1 domains. This will be proved for derivatives of any order, every 0 < p < ∞ 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 Lipα, 0 < α < 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/α − 1/2 )-th power of the Euclidean distance to the boundary when 0 < α < 1, and this measure of the distance to the boundary is in general the best possible, as well.

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