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 ${\mathcal{D}}^{\mathrm{\prime }}({\mathbf{R}}^d)$. If there is a constant $C_0>0$ such that $$\text{nm}{e}^{i\text{scal}\text{cdo}\xi }f(\text{cdo}-x)\mathcal{H}\le C_0\text{nm}f\mathcal{H},f\in \mathcal{H},x,\xi \in {\mathbf{R}}^d,$$ then we prove that $\text{maclH}=L^2({\mathbf{R}}^d)$, with equivalent norms.