Jackson–Nikol'skii inequality between the uniform and integral norms of algebraic polynomials on a Euclidean sphere

We study the sharp Jackson–Nikol'skii inequality between the uniform and integral norms of algebraic polynomials of a given (total) degree $n\ge0$ on the unit sphere $\mathbb S^{m-1}$ of the Euclidean space $\mathbb R^m$. We prove that the polynomial $Q_n$ in one variable with the unit leading coefficient, which deviates least from zero in the space $L^\psi(-1,1)$ of functions summable on $(-1,1)$ with the Jacobi weight $\psi(t)=(1-t)^\alpha(1+t)^\beta$, $\alpha=(m-1)/2$, $\beta=(m-3)/2$, as zonal polynomial in one variable $t=x_m$, $x=(x_1,\dots,x_m)\in\mathbb S^{m-1}$, is extremal in the Jackson–Nikol'skii inequality on the sphere $\mathbb S^{m-1}$.