Spectral subspaces of $L^p$ for $p<1$
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A. B. Aleksandrov
Translated by: the author - St. Petersburg Math. J. 19 (2008), 327-374
- DOI: https://doi.org/10.1090/S1061-0022-08-01001-7
- Published electronically: March 21, 2008
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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.References
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Bibliographic Information
- A. B. Aleksandrov
- Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
- MR Author ID: 195855
- Email: alex@pdmi.ras.ru
- Received by editor(s): November 8, 2006
- Published electronically: March 21, 2008
- Additional Notes: Supported in part by RFBR (grant no. 05-01-00924)
- © Copyright 2008 American Mathematical Society
- Journal: St. Petersburg Math. J. 19 (2008), 327-374
- MSC (2000): Primary 42B35
- DOI: https://doi.org/10.1090/S1061-0022-08-01001-7
- MathSciNet review: 2340705