SPECTRAL SUBSPACES OF L p FOR p < 1

Let Ω be an open subset of Rn. Denote by LpΩ(R n) the closure in Lp(Rn) of the set of all functions ε ∈ L1(Rn)∩Lp(Rn) whose Fourier transform has compact support contained in Ω. The subspaces of the form LpΩ(R n) are called the spectral subspaces of Lp(Rn). It is easily seen that each spectral subspace is translation invariant; i.e., f(x + a) ∈ LpΩ(R) for all f ∈ L p Ω(R n) and a ∈ Rn. Sufficient conditions are given for the coincidence of LpΩ(R n) and Lp(Rn). In particular, an example of a set Ω is constructed such that the above spaces coincide for sufficiently small p but not for all p ∈ (0, 1). Moreover, the boundedness of the functional f → (Ff)(a) with a ∈ Ω, which is defined initially for sufficiently “good” functions in LpΩ(R n), is investigated. In particular, estimates of the norm of this functional are obtained. Also, similar questions are considered for spectral subspaces of Lp(G), where G is a locally compact Abelian group. §