About the sharp Jackson–Nikol'skii inequality for algebraic polynomials on a multidimensional Euclidean sphere

The best constant $C_{nm}$ in the Jackson–Nikol'skii inequality between uniform and integral norms of algebraic polynomials of given total degree $n\ge0$ on the unit sphere $\mathbb S^{m-1}$ of the Euclidean space $\mathbb R^m$ is studied. Two-sided estimates for the constant $C_{nm}$ are obtained, which, in particular, give the order $n^{m-1}$ of its behavior with respect to $n$ as $n\to+\infty$ for a fixed $m$.