Rearrangement-invariant spaces of functions on LCA groups
The author continues his earlier study [e.g., J. Funct. Anal. 146, 62-115 (1997; Zbl 0870.43001)] of kernels and hulls of function spaces. To explain the terminology, let \(R\) be a relation in the space of measurable functions (on a given measure space) and let \(A\) be a space of measurable functions. Then \(f\in \text{hull} (A)\) provided \(fRg\) for some \(g\in A\) and \(f\in \text{kernel} (A)\) if \(fRg\) for all \(g\in A\). In the paper under review the measure space is a locally compact (noncompact and nondiscrete) abelian group and \(fRg\) provided \(f\) is a measurable rearrangement of \(g\). \(A\) is one of the spaces of all functions \(f\in S'\cap L^1_{\text{loc}}\) with \(\sigma\)-finite support, such that the Fourier transform \(\widehat f\) is respectively, in \(L^1_{ \text{loc}}\), in the Lorentz space \(L(p,q)\) or in \(L^\infty\). The kernels and hulls are determined in all cases except for that of the Lorentz space with \(1<p<2\). Some of the descriptions involve a new function space denoted by \(P_0\): a function \(g\in P_0\) provided \(\text{supp} g=\cup E_n\) with \(E_n\) disjoint, measure \((E_n)=1\), \(g\in L^1 (E_n)\) and \(\int_{E_n} g\to 0\) as \(n\to \infty\).
scientific article; zbMATH DE number 1214159
scientific article