## 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:

 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)