Metric properties in the mean of polynomials on compact isotropy irreducible homogeneous spaces

TL;DR

This paper investigates metric properties of polynomial functions on compact isotropy irreducible homogeneous spaces, providing estimates for averages of norms and measures of level sets that are independent of the specific space or polynomial subspace.

Contribution

It introduces a framework for estimating average metric quantities of polynomials on a broad class of homogeneous spaces, extending known results to new settings.

Findings

Average $L^p$ norms of polynomials are bounded by $ oot{p+1}{e}$ for $p eq 2$

Hausdorff measures of level sets are estimated from above

Results are independent of the specific space and polynomial subspace

Abstract

Let $M=G/H$ be a compact connected isotropy irreducible Riemannian homogeneous manifold, where $G$ is a compact Lie group (may be, disconnected) acting on $M$ by isometries. This class includes all compact irreducible Riemannian symmetric spaces and, for example, the tori $\bbR^n/\bbZ^n$ with the natural action on itself extended by the finite group generated by all transpositions of coordinates and inversions in circle factors. We say that $u$ is a polynomial on $M$ if it belongs to some $G$-invariant finite dimensional subspace $\cE$ of $L^2(M)$. We compute or estimate from above the averages over the unit sphere $\cS$ in $\cE$ for some metric quantities such as Hausdorff measures of level set and norms in $L^p(M)$, $1\leq p\leq\infty$, where $M$ is equipped with the invariant probability measure. For example, the averages over $\cS$ of $\|u\|_{L^p(M)}$, $p\geq2$, are less than $\sqrt{\frac{p+1}{e}}$ independently of $M$ and $\cE$.