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arXiv · 2005.01674

Marcinkiewicz-type discretization of $L^p$-norms under the Nikolskii-type inequality assumption

Egor Kosov
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Abstract

The paper studies the sampling discretization problem for integral norms on subspaces of $L^p(\mu)$. Several close to optimal results are obtained on subspaces for which certain Nikolskii-type inequality is valid. The problem of norms discretization is connected with the probabilistic question about the approximation with high probability of marginals of a high dimensional random vector by sampling. As a byproduct of our approach we refine the result of O. Gu$\acute{e}$don and M. Rudelson concerning the approximation of marginals. In particular, the obtained improvement recovers a theorem of J. Bourgain, J. Lindenstrauss, and V. Milman concerning embeddings of finite dimensional subspaces of $L^p[0, 1]$ into $\ell_p^m$. The proofs in the paper use the recent developments of the chaining technique by R. van Handel.

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Egor Kosov. 2020-05-04. Marcinkiewicz-type discretization of $L^p$-norms under the Nikolskii-type inequality assumption. https://arxiv.org/abs/2005.01674

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