On Bernstein inequalities on the unit ball

2025/11/12 by Beberok, Tomasz, Xu, Yuan
#33C45 #42C05 #42C10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2511.09713

Abstract

Two types of Bernstein inequalities are established on the unit ball in ℝ^d, which are stronger than those known in the literature. The first type consists of inequalities in L^p norm for a fully symmetric doubling weight on the unit ball. The second type consists of sharp inequalities in L² norm for the Jacobi weight, which are established via a new self-adjoint form of the spectral operator that has orthogonal polynomials as eigenfunctions.

Citations

• Sharp Bernstein inequalities on simplex
• Approximation and localized polynomial frame on conic domains
• Gaussian bounds for the heat kernels on the ball and simplex: Classical approach
• Probabilistic and average linear widths of weighted Sobolev spaces on the ball equipped with a Gaussian measure
• Localized polynomial frames on the ball
• Weighted Approximation of functions on the unit sphere

Related

• Sharp Bernstein inequalities on simplex
• A Bernstein type inequality
• Bernstein inequalities with nondoubling weights
• Some notes on L^p Bernstein inequality when 0
• A Multiplier Version of the Bernstein Inequality on the Complex Sphere