On Nikol'skii inequalities for domains in $R^d$

TL;DR

This paper investigates Nikol'skii inequalities for polynomials on convex domains, linking geometric boundary properties to inequality exponents, and computes explicit asymptotics for various domains using Christoffel functions.

Contribution

It introduces a geometric approach to determine Nikol'skii inequality exponents for convex domains and computes explicit asymptotics for Christoffel functions across multiple examples.

Findings

Explicit formulas for Nikol'skii exponents on convex domains.

Connection between boundary geometry and inequality sharpness.

Asymptotic behavior of Christoffel functions for various domains.

Abstract

Nikol'skii inequalities for various sets of functions, domains and weights will be discussed. Much of the work is dedicated to the class of algebraic polynomials of total degree $n$ on a bounded convex domain $D$. That is, we study $\sigma:= \sigma(D,d)$ for which \[ \|P\|_{L_q(D)}\le c n^{\sigma(\frac1p-\frac1q)}\|P\|_{L_p(D)},\quad 0<p\le q\le\infty, \] where $P$ is a polynomial of total degree $n$. We use geometric properties of the boundary of $D$ to determine $\sigma(D,n)$ with the aid of comparison between domains. Computing the asymptotics of the Christoffel function of various domains is crucial in our investigation. The methods will be illustrated by the numerous examples in which the optimal $\sigma(D,n)$ will be computed explicitly.