An exotic minimal Banach space of functions. (English) Zbl 1019.46025

Authors’ abstract “This note describes a new Banach space \(\mathcal B_0\) of square integrable functions on \(\mathbb R^d\) having many interesting invariance properties. In fact, the Fourier transform, time-frequency shifts, and \(L^2\)-normalized dilations act isometrically on it. For its definition, we make use of a general construction principle for minimal invariant spaces. We demonstrate a variety of properties following immediately from this principle. Furthermore, we give a number of different characterizations, including various atomic decompositions, as well as natural necessary and sufficient conditions for an \(L^2\)-function to belong to this new space. It turns out that this new space is somewhat exotic, since it is neither rearrangement invariant nor solid”.


46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)