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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”.

MSC:

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