Vol. 99, No. 2, 1982

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Strict local inclusion results between spaces of Fourier transforms

Walter Russell Bloom

Vol. 99 (1982), No. 2, 265–270

Let G denote a noncompact Hausdorff locally compact abelian group, Γ its character group, and write (Ls,lt) for the space of Fourier transforms of functions in the amalgam (Ls,lt). We show that for 1 p < q the local inclusion (L1,lp)loc
⊂(L,lq) is strict, that is, given any nonvoid open subset Ω of Γ there exists f (L,lq) such that f ĝ does not vanish on Ω for any g (L1,lp). If in addition G is assumed to be second countable then we show there exists such an f independent of the choice of Ω. Of special interest is the case, included in the above results, where the amalgams (L1,lq), (L,lp) are replaced by Lp(G), Lq(G) respectively.

Mathematical Subject Classification 2000
Primary: 43A25
Received: 5 December 1979
Revised: 31 December 1980
Published: 1 April 1982
Walter Russell Bloom