Non-isometric translation and modulation invariant Hilbert spaces

Abstract

Let $H$ be a Hilbert space, continuously embedded in $S^{\prime }\left(R^d\right)$, and which contains at least one non-zero element in $S^{\prime }\left(R^d\right)$. If there is a constant $C_0>0$ such that${\Vert e^{i〈\cdot ,\xi 〉}f(\cdot -x)\Vert }_H\le C_0{\Vert f\Vert }_H,f\in H,x,\xi \in R^d,$ then we prove that $H=L^2\left(R^d\right)$, with equivalent norms.