Bernstein inequality on conic domains and triangle

TL;DR

This paper establishes new weighted Bernstein inequalities on conic surfaces and triangles, revealing stronger derivative bounds in polygonal domains than previously known, with implications for approximation theory.

Contribution

The paper introduces novel weighted Bernstein inequalities on conic domains and triangles, showing stronger derivative bounds than classical results, especially in polygonal regions.

Findings

Weighted Bernstein inequalities are established on conic surfaces and solid cones.

New inequalities for derivatives in $x$ variables are stronger than classical ones.

The results reveal a previously unobserved phenomenon in polygonal domains.

Abstract

We establish weighted Bernstein inequalities in $L^p$ space for the doubling weight on the conic surface $\mathbb{V}_0^{d+1} = \{(x,t): \|x\| = t, x \in \mathbb{R}^d, t\in [0,1]\}$ as well as on the solid cone bounded by the conic surface and the hyperplane $t =1$, which becomes a triangle on the plane when $d=1$. While the inequalities for the derivatives in the $t$ variable behave as expected, there are inequalities for the derivatives in the $x$ variables that are stronger than what one may have expected. As an example, on the triangle $\{(x_1,x_2): x_1 \ge 0, \, x_2 \ge 0,\, x_1+x_2 \le 1\}$, the usual Bernstein inequality for the derivative $\partial_1$ states that $\|\phi_1 \partial_1 f\|_{p,w} \le c n \|f\|_{p,w}$ with $\phi_1(x_1,x_2):= x_1(1-x_1-x_2)$, whereas our new result gives $$\| (1-x_2)^{-1/2} \phi_1 \partial_1 f\|_{p,w} \le c n \|f\|_{p,w}.$$ The new inequality is stronger and points out a phenomenon unobserved hitherto for polygonal domains.