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

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.