  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 $d \ge 3$.
