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

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 \(\mathbb{R }^n/\mathbb{Z }^n\) with the natural action on itself extended by the finite group generated by all permutations of the 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 \(\mathcal{E }\) of \(L^2(M)\). We compute or estimate from above the averages over the unit sphere \(\mathcal{S }\) in \(\mathcal{E }\) for some metric quantities such as Hausdorff measures of level set and norms in \(L^p(M)\), \(1\le p\le \infty \), where \(M\) is equipped with the invariant probability measure. For example, the averages over \(\mathcal{S }\) of \(\Vert u\Vert _{L^p(M)}\), \(p\ge 2\), are less than \(\sqrt{\frac{p+1}{e}}\) independently of \(M\) and \(\mathcal{E }\).