Sharp Bernstein–Nikolskii inequalities for polynomials and entire functions of exponential type

The classical Bernstein–Nikolskii inequalities of the form $\|Df\|_{q}\le \mathcal {C}_{pq}\|f\|_{p}$ for $f\in Y$, give estimates for the $pq$-norms of the differential operators $D$ on classes $Y$ of polynomials and entire functions of exponential type. These inequalities play an important role in harmonic analysis, approximation theory and find applications in number theory and metric geometry. Both order inequalities and inequalities with sharp constants are studied. The last case is especially interesting because the extremal functions depend on the geometry of the manifold and this fact helps in solving geometric problems.
Historically, Bernstein's inequalities are referred to the case $p=q$, and Nikolskii's inequalities to the estimate of the identity operator for $p<q$. For the first time, an estimate for the derivative of a trigonometric polynomial for $p=\infty$ was given by S.N. Bernstein (1912), although earlier A.A. Markov (1889) gave its algebraic version. Bernstein's inequality was refined by E. Landau, M. Riess, and A. Sigmund (1933) proved it for all $p\ge 1$. For $p<1$, the Bernstein order inequality was found by V.I. Ivanov (1975), E.A. Storozhenko, V.G. Krotov and P. Oswald (1975), and the sharp inequality by V.V. Arestov (1981). For entire functions of exponential type, the sharp Bernstein inequality was proved by N.I. Akhiezer, B.Ya. Levin ($p\ge 1$, 1957), Q.I. Rahman and G. Schmeisser ($p<1$, 1990).
The first one-dimensional Nikolskii inequalities for $q=\infty$ were established by D. Jackson (1933) for trigonometric polynomials and J. Korevaar (1949) for entire functions of exponential type. In all generality for $q\le \infty$ and $d$-dimensional space, this was done by S.M. Nikolskii (1951). The estimates of Nikolskii constants were refined by I.I. Ibragimov (1959), D. Amir and Z. Ziegler (1976), R.J. Nessel and G. Wilmes (1978), and many others. Bernstein–Nikolskii order inequalities for different intervals were studied by N.K. Bari (1954). Variants of inequalities for general multiplier differential operators and weighted manifolds can be found in the works of P.I. Lizorkin (1965), A.I. Kamzolov (1984), A.G. Babenko (1992), A.I. Kozko (1998), K.V. Runovsky and H.-J. Schmeisser (2001), F. Dai and Y. Xu (2013), V.V. Arestov and P.Yu. Glazyrina (2014) and other authors.
For a long time, the theory of Bernstein–Nikolskii inequalities for polynomials and entire functions of exponential type developed in parallel until E. Levin and D. Lubinsky (2015) established that for all $p>0$ the Nikolskii constant for functions is the limit of trigonometric constants. For the Bernstein–Nikolskii constants, this fact was proved by M.I. Ganzburg and S.Yu. Tikhonov (2017) and refined by the author together with I.A. Martyanov (2018, 2019). Multidimensional results of the Levin–Lyubinsky type were proved by the author together with F. Dai and S.Yu. Tikhonov (the sphere, 2020), M.I. Ganzburg (the torus, 2019 and the cube, 2021).
Until now, the sharp Nikolskii constants are known only for $(p,q)=(2,\infty)$. The case of the Nikolskii constant for $p=1$ is intriguing. Advancement in this area was obtained by Ya.L. Geronimus (1938), S.B. Stechkin (1961), L.V. Taikov (1965), L. Hörmander and B. Bernhardsson (1993), N.N. Andreev, S.V. Konyagin and A.Yu. Popov (1996), author (2005), author and I.A. Martyanov (2018), I.E. Simonov and P.Yu. Glazyrina (2015). E. Carneiro, M.B. Milinovich and K. Soundararajan (2019) pointed out applications in the theory of the Riemann zeta function. V.V. Arestov, M.V. Deikalova et al (2016, 2018) characterized extremal polynomials for general weighted Nikolskii constants using duality. Here, S.N. Bernshtein, L.V. Taikov (1965, 1993) and others stood at the origins.
A new direction is the proof of Nikolskii's sharp inequalities on classes of functions with constraints. It reveals a connection with the extremal problems of harmonic analysis of Turan, Delsarte, the uncertainty principle by J. Bourgain, L. Clozel and J.-P. Kahane (2010) and others. For example, the author and coauthors (2020) showed that the sharp Nikolskii constant for nonnegative spherical polynomials gives an estimate for spherical designs by P. Delsarte, J.M. Goethals and J.J. Seidel (1977). Variants of problems for functions lead to famous estimates for the density of spherical packing, and order results are closely related to Fourier inequalities.
These results are presented in the framework of the general theory of Bernstein–Nikolskii inequalities, applications in approximation theory, number theory, metric geometry are presented, open problems are proposed.