Bernstein inequality on conic domains and triangles

We establish weighted Bernstein inequalities in L p space for the doubling weight on the conic surface Vd+1 0 = {(x, t) : parallel to x parallel to = t, x is an element of Rd, t is an element of [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 {(x1, x2) : x1 >= 0, x2 >= 0, x1 + x2 <= 1}, the usual Bernstein inequality for the derivative partial differential 1 states that parallel to phi 1 partial differential 1 f parallel to p,w <= cn parallel to f parallel to p,w with phi 1(x1, x2) := x1(1 - x1 - x2), whereas our new result gives parallel to(1- x2)-1/2 phi 1 partial differential 1f parallel to p,w <= cn parallel to f parallel to p,w. The new inequality is stronger and points out a phenomenon unobserved hitherto for polygonal domains.(c) 2023 Published by Elsevier Inc.