"Classical inequality" regarding the Bessel potential space $H^{-1}$

Question

While reading the paper "Compensated compactness and Hardy spaces" bt Coifman et al., I stumbled upon the "classical inequality" (as they refer to it): For any cube $Q\subset\mathbb{R}^n$ we have $$\inf_{\lambda\in\mathbb{C}} \left(\int_Q |b-\lambda|^2\right)^\frac{1}{2} \leq C_n \sum_{j=1}^n \left\|\frac{\partial b}{\partial x_j}\right\|_{H^{-1}(Q)},$$ where the Bessel potential space $H^{-1}$ is given by $$H^{-1}(Q):= \{ u\in \mathscr{S}'(Q):\langle D_x\rangle^{-1}u\in L^2(Q) \}. $$ Here $\langle D_x\rangle^{-1}f=\mathscr{F}^{-1}[\langle \xi\rangle^{-1}\hat{f}]$ (where $\mathscr{F}$ denotes the Fourier transform) and $\langle \xi\rangle:=(1+|\xi|^2)^\frac{1}{2}$. I tried to prove this inequality, but couldn't manage to do so. Therefore I would be really grateful if any of you could provide some insight. Thanks a lot.