Non-isometric translation and modulation invariant Hilbert spaces

Abstract

Let $\mathcal{H}$ be a Hilbert space of distributions on ${\mathbf{R}}^d$ which contains at least one non-zero element in ${\mathcal{D}}^{\prime }({\mathbf{R}}^d)$. If there is a constant $C_0>0$ such that $$ \nm {e^{i\scal \cdo ξ}f(\cdo -x)}{\mathcal H}\le C_0\nm f{\mathcal H}, \qquad f\in \mathcal H ,\ x,ξ\in \mathbf R^d, $$ then we prove that $\maclH = L^2(\mathbf R^d)$, with equivalent norms.

13 pages. This is the third version of the document. We have mainly performed minor corrections compared to version 2. We observe the new title. The old title was "Non-isotropic translation and modulation invariant Hilbert spaces"