On Bernstein Inequalities on the Unit Ball

Two types of Bernstein inequalities are established on the unit ball in \({\mathbb {R}}^d\) ${\mathbb{R}}^d$ , which are stronger than those known in the literature. The first type consists of inequalities in \(L^p\) $L^p$ norm for a fully symmetric doubling weight on the unit ball. The second type consists of sharp inequalities in \(L^2\) $L^2$ norm for the Jacobi weight, which are established via a new self-adjoint form of the spectral operator that has orthogonal polynomials as eigenfunctions.