Bernstein inequality on conic domains and triangles

Abstract

We establish weighted Bernstein inequalities in Lp space for the doubling weight on the conic surface V0d+1={(x,t):‖x‖=t,x∈Rd,t∈[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 ∂1 states that ‖ϕ1∂1f‖p,w≤cn‖f‖p,w with ϕ1(x1,x2)≔x1(1−x1−x2), whereas our new result gives ‖(1−x2)−1/2ϕ1∂1f‖p,w≤cn‖f‖p,w.The new inequality is stronger and points out a phenomenon unobserved hitherto for polygonal domains.