Non-isometric translation and modulation invariant Hilbert spaces

Let $\mathcal H$ be a Hilbert space of distributions on $\mathbf R^d$ which contains at least one non-zero element in $\mathscr D '(\mathbf R^d)$. 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.