On multivariate $L^p$ Bernstein-Markov type inequalities

Abstract:

It was established in a recent paper by Kroó [J. Approx. Theory 281/282 (2022), p. 5] that the following sharp $L^2$ Bernstein-Markov type inequality on the unit ball $B^d$ holds for any polynomial $p$ of degree at most $n$ in $d$ variables, \begin{equation*} \|(1-|x|^2)^{\frac {\mu +1}{2}}D p\|_{L_2(B^d)}\leq M_n(d)\|(1-|x|^2)^{\frac {\mu }{2}}p\|_{L_2(B^d)}, \;\;\mu >-1, \end{equation*} with $M_n(d)=\sqrt {n(n+d+2\mu )}$ or $\sqrt {n(n+d+2\mu )-d+1}$ when $n$ is even or odd, respectively, where $Dp$ denotes the $l^2$ norm of the gradient of $p$. And all of the estimates listed above were sharp with equalities being attained for certain polynomials. In this paper the uniqueness of the corresponding extremal polynomials is verified.

For homogeneous polynomials $h_n\in H^d_n$ of degree $n$ in $d$ variables we will prove sharp $L_2$ Markov type inequalities \begin{equation*} \xi _n\|h_n\|_{L_2(S^{d-1})}\leq \|D h_n\|_{L_2(S^{d-1})}\leq \sqrt {n(2n+d-2)}\|h_n\|_{L_2(S^{d-1})} \end{equation*} with $\xi _n=n$ if $n$ is even and $\xi _n=\sqrt {n^2+d-1}$ if $n$ is odd. The upper bound is attained if and only if $h_n$ is a spherical harmonic while the lower bound is attained if and only if $h_n(x)=c|x|^n$, or $h_n(x)=|x|^{n-1}q(x), q\in H^d_1$ when $n$ is even or odd, respectively.

In addition, we will study possible extensions of these results to the $L^p$ case. In particular it will be established that the $L^p$ Markov factor for homogeneous polynomials of degree $n$ on $C^2$ star like domains with non degenerate outer normals is of asymptotically optimal order $n$.