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

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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 \(\mathcal{E }\).

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Notes

  1. The action is effective if its kernel is trivial and virtually effective if it is finite. The kernel of the action consists of those \(g\in G\) which define the identical transformation of \(M\).

  2. The two-sheeted expanding covering \(z\rightarrow z^2\) of the unit circle \(\mathbb{T }\) in \(\mathbb{C }\) is a simple example of a non-global local homothety.

  3. we assume that the action of \(G\) is virtually effective; see the footnote on page .

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  1. Omsk Branch of Sobolev Institute of Mathematics, Pevtsova, 13, 644099, Omsk, Russia

    V. M. Gichev

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Correspondence to V. M. Gichev.

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Part of the work was done during my stay in the Institut Mittag-Leffler (Djursholm, Sweden), 2011 fall. I thank the Institute for support and hospitality.

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Gichev, V.M. Metric properties in the mean of polynomials on compact isotropy irreducible homogeneous spaces. Anal.Math.Phys. 3, 119–144 (2013). https://doi.org/10.1007/s13324-012-0051-4

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