Sharp Bernstein inequalities on simplex

We prove several new families of Bernstein inequalities of two types on the simplex. The first type consists of inequalities in $L^2$ norm for the Jacobi weight, some of which are sharp, and they are established via the spectral operator that has orthogonal polynomials as eigenfunctions. The second type consists of inequalities in $L^p$ norm for doubling weight on the simplex. The first type is not necessarily a special case of the second type when $d \ge 3$.