Let $A \in \mathbb{C}^{m \times N}$ be such that, for some constants $0 , \rho < 1$ and $\tau \geq 0,$ $$\| v_{S} \|_{2} \leq \frac{\rho}{\sqrt{s}} \| v_{\overline{S}}\|_{1} + \tau \|A^{*}A \|_{\infty}~\textrm{for all}~S \subset [N]~\textrm{and all}~v \in \mathbb{C}^N.$$ Here $[N]$ is an index set, $\{ 1,2, \dots , N \}.$ For $x \in \mathbb{C}^N,$ let $y=Ax+e$ for some $e \in \mathbb{C}^m$ with $\| A^{*} e \|_{\infty} \leq \eta.$ Let $\tilde{x}$ be a minimizer of $$\min_{z \in \mathbb{C}^N} \| z \|_1~\textrm{subject to}~\| A^{*}(Az - y) \|_{\infty} \leq \eta.$$ Show that $$\| x - \tilde{x} \|_2 \leq \frac{C \sigma_{s}(x)_1}{\sqrt{s}}+D \eta,$$ for constants $C,D >0$ depending only on $\rho$ and $\tau.$
Here $\sigma_s(x)_{p}=\inf_{\| z \|_{0} \leq s} \| x -z \|_p$ is the $\ell_p - $error of best $s-$term approximation to $x \in \mathbb{C}^N$.
I'm struggling in solving this problem. I'll be thankful if anyone can give me a suggestion.