Let $T \in B(L^2(\mathbb{R}^n))$ be a bounded linear operator and $0 \le t \le s$. Suppose there exists $C_1, C_2 > 0$ such that for all $f \in L^2$ \begin{align} \| Tf \|_{H^s} &\le C_1 \| f \|_{L^2}, \tag{1} \\ \| Tf \|_{H^t} &\ge C_2 \| f \|_{L^2} \tag{2}. \end{align}
Optional assumption: $T = \mathcal{F}^{-1} M \mathcal{F}$ for some multiplication operator $M$ and the Fourier-Plancherel transform $\mathcal{F}$, i.e. $T$ is diagonal in Fourier space.
It is easy to see that (1) implies $\mathrm{range}(T) \subset H^s$ and (2) implies that $T$ is injective.
Question:
Can we conclude that $\mathrm{range}(T) \supset H^t$?
The example I have in mind are convolution operators for which the answer is yes but I don't see why this would hold on general.