Bernstein inequalities are classical estimates for polynomials that bound the supremum of the gradient of a polynomial by the degree times the supremum of the polynomial. It was proposed by Donnelly and Fefferman to study these inequalities for Laplace eigenfunctions, the degree being replaced by the square root of the eigenfunction. An inequality with a dimensional power of the eigenvalue was obtained by Donnelly and Fefferman. With Eugenia Malinnikova we were able to prove a dimension-free Bernstein inequality; I will describe our work. Time permitting, I will also discuss joint work with Eugenia Malinnikova and Fedor Nazarov, still in progress, that allows us to obtain a sharp Bernstein inequality.
