St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Spectral subspaces of $ L^p$ for $ p<1$

Author: A. B. Aleksandrov
Translated by: the author
Original publication: Algebra i Analiz, tom 19 (2007), nomer 3.
Journal: St. Petersburg Math. J. 19 (2008), 327-374
MSC (2000): Primary 42B35
Published electronically: March 21, 2008
MathSciNet review: 2340705
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Abstract: Let $ \Omega$ be an open subset of $ \mathbb{R}^n$. Denote by $ L^p_{\Omega}(\mathbb{R}^n)$ the closure in $ L^p(\mathbb{R}^n)$ of the set of all functions $ \varepsilon\in L^1(\mathbb{R}^n)\cap L^p(\mathbb{R}^n)$ whose Fourier transform has compact support contained in $ \Omega$. The subspaces of the form $ L^p_\Omega(\mathbb{R}^n)$ are called the spectral subspaces of $ L^p(\mathbb{R}^n)$. It is easily seen that each spectral subspace is translation invariant; i.e., $ f(x+a)\in L^p_\Omega(\mathbb{R}^n)$ for all $ f\in L^p_\Omega(\mathbb{R}^n)$ and $ a\in\mathbb{R}^n$. Sufficient conditions are given for the coincidence of $ L^p_\Omega(\mathbb{R}^n)$ and $ L^p(\mathbb{R}^n)$. In particular, an example of a set $ \Omega$ is constructed such that the above spaces coincide for sufficiently small $ p$ but not for all $ p\in(0,1)$. Moreover, the boundedness of the functional $ f\mapsto(\mathcal{F} f)(a)$ with $ a\in\Omega$, which is defined initially for sufficiently ``good'' functions in $ L^p_\Omega(\mathbb{R}^n)$, is investigated. In particular, estimates of the norm of this functional are obtained. Also, similar questions are considered for spectral subspaces of $ L^p(G)$, where $ G$ is a locally compact Abelian group.

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Additional Information

A. B. Aleksandrov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia

Keywords: Translation invariant subspace, spectral subspace, Hardy classes, uniqueness set
Received by editor(s): November 8, 2006
Published electronically: March 21, 2008
Additional Notes: Supported in part by RFBR (grant no. 05-01-00924)
Article copyright: © Copyright 2008 American Mathematical Society