A characterization of translation and modulation invariant Hilbert space of tempered distributions

Let \(\mathcal {S}(\mathbb {R}^n)\) be the Schwartz space and \(\mathcal {S'}(\mathbb {R}^n)\) be the space of tempered distributions on \(\mathbb {R}^n\). In this article, we prove that if \(\mathcal {H} \subseteq \mathcal {S'}(\mathbb {R}^n)\) is a non-zero Hilbert space of tempered distributions which is translation and modulation invariant such that

$$\begin{aligned} |(f,g)| \le C \Vert f\Vert _{\mathcal {H}} \end{aligned}$$

for some \(C>0\) and for all \(f\in \mathcal {H}\), then \(\mathcal {H}=L^2(\mathbb {R}^n)\), where \(g(x) = e^{-x^2}\) for all \(x\in \mathbb {R}^n\) and \((\cdot , \cdot )\) denotes the standard duality pairing between \(\mathcal {S'}(\mathbb {R}^n)\) and \(\mathcal {S}(\mathbb {R}^n)\) with respect to which \((\mathcal {S}(\mathbb {R}^n))^*=\mathcal {S'}(\mathbb {R}^n)\).