Sharp \(L_p\) Bernstein type inequality for cuspidal domains in \(\mathbb{R}^d\) (Q2037067)

For Bernstein type inequalities bounds of weighted derivatives of polynomials with weights vanishing at the boundary of the domain are given. For example, in the case of the space \(L^p_w[-\pi,\pi ]\) with the Jacobi weight \(w(x)=\sqrt{1-x^2}\) the sharp rate of the derivatives for polynomials of degree \(n\) in Bernstein inequality is \(n\). A new sharp Bernstein type inequality for multivariate algebraic polynomials in \(L_p\) norm for general cuspidal domains in \(\mathbb {R}^d\) are established. In particular, in case of so-called \(\mathrm{Lip }\gamma\) cuspidal graph domains, the exact rate is given by \(n^{2/\gamma -1}\).