Markov–Bernstein–Nikol'skii constants for polynomials in $L^{p}$-space with the Gegenbauer weight

We study the sharp Markov–Bernstein–Nikol'skii inequality of the form $\|D^{s}u\|_{\infty} \le C_{p}(n;s)\times\\\times\|u\|_{p}$, $p\in [1,\infty]$ for trigonometric and algebraic polynomials $u$ of degree at most $n$ in the weighted space $L^{p}$ with the Gegenbauer–Dunkl differential operator $D$. In particular cases, these inequalities are reduced to the classical inequalities of approximation theory of the Markov, Bernstein, and Nikol'skii type, to which numerous papers are devoted. We apply the results of V.A. Ivanov (1983, 1992), V.V. Arestov and M.V. Deikalova (2013, 2015), F. Dai, D.V. Gorbachev and S.Yu. Tikhonov (2020) for algebraic constants in $L^{p}$ on compact Riemannian manifolds of rank 1 (including the Euclidean sphere) and an interval with Gegenbauer weight, refer to the works of E. Levin and D. Lubinsky (2015), M.I. Ganzburg (2017, 2020), a review of the classic results of G.V. Milovanović, D.S. Mitrinović and Th.M. Rassias (1994).
Earlier we studied the case $s=0$. In this paper, we consider the case $s\ge 0$. Our main result is to prove the existence in the trigonometric case for even $s=2r$ of extremal polynomials $u_{*}$ that are real, even, and $C(n;s)=\frac{|D^{s}u_{*}(0)|}{\|u_{*}\|_{p}}$. With the help of this fact, the relationship with the algebraic constant for the Gegenbauer weight is proved. On the one hand, this relationship allows to automatically characterize extremal algebraic polynomials. On the other hand, well-known algebraic results carry over to a more general trigonometric version. The main method of proof is the application of the Gegenbauer–Dunkl harmonic analysis constructed by D.V. Chertova (2009). As a consequence, we give the explicit constants for $p=2, \infty$ (using the results of V.A. Ivanov), we give the relations of orthogonality and duality (proved by methods of convex analysis from approximation theory), we establish one asymptotic result of the Levin–Lubinsky type (due to the connection with the multidimensional Nikol'skii constant for spherical polynomials).