Non-isotropic translation and modulation invariant Hilbert spaces

Let H be a Hilbert space of distributions on Rᵈ which contains at least one non-zero element in D ' (Rᵈ). If there is a constant C₀>0 such that e^{i f (-x) } H C₀ f H, f H, \ x, Rᵈ, then we prove that = L² (Rᵈ), with equivalent norms.