On multivariate Lp Bernstein-Markov type inequalities

Kroó, András (2024) On multivariate Lp Bernstein-Markov type inequalities. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 152 (12). pp. 5149-5162. ISSN 0002-9939

Text S0002-9939-2024-16925-2.pdf - Published Version Restricted to Repository staff only Download (258kB) | Request a copy

Abstract

It was established in a recent paper by Kroó [J. Approx. Theory 281/282 (2022), p. 5] that the following sharp L2 Bernstein-Markov type inequality on the unit ball Bd holds for any polynomial p of degree at most n in d variables, (Formula presented), with (Formula presented) when n is even or odd, respectively, where Dp denotes the l2 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 hn ∈ Hnd of degree n in d variables we will prove sharp L2 Markov type inequalities (Formula presented) with ξn = n if n is even and ξn = √n2 + d - 1 if n is odd. The upper bound is attained if and only if hn is a spherical harmonic while the lower bound is attained if and only if hn(x) = c|x|n, or hn(x) = |x|n-1q(x), q ∈ H1d when n is even or odd, respectively. In addition, we will study possible extensions of these results to the Lp case. In particular it will be established that the Lp Markov factor for homogeneous polynomials of degree n on C2 star like domains with non degenerate outer normals is of asymptotically optimal order n. © 2024 American Mathematical Society.

Actions (login required)

Edit Item