2022/05/03 by Yuan Xu, Xu, Yuan
Mathematics · #41A10 #41A63 #42C10 #42C40 #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2205.01320
openalex publication_date 2022/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We establish weighted Bernstein inequalities in Lp space for the doubling weight on the conic surface \mathbbV0d+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 ∂1 f‖p,w ≤ c n ‖f‖p,w with ϕ1(x1,x2):= x1(1-x1-x2), whereas our new result gives ‖ (1-x2)-1/2 ϕ1 ∂1 f‖p,w ≤ c n ‖f‖p,w. The new inequality is stronger and points out a phenomenon unobserved hitherto for polygonal domains.