A Besov norm estimate of analytic function based on Paley-Wiener theorem

Question

Assume that $m$ is a bounded and analytic function over a sector $\Sigma_\omega$ and $m_e=m\circ \exp$. Hence $m_e$ is a bounded and analytic function over the strip $$ \{z\in\mathbb{C}\mid |\Im z|<\omega\}. $$ I need to show that $$ \|m_e\mid_{\mathbb{R}}\|_{B_{\infty,1}^\alpha}\leq C\omega^{-\alpha}\sup_{|\arg z|<\omega}\|m(z)\|, $$ where $B^\alpha_{\infty,1}$ denotes the Besov norm over $\mathbb{R}$. This estimate is from the paper ''Banach spaces operator with a bounded $H^\infty$ calculus''. An important step of the deduction is $$ (\star)\quad\|m_e\star\check{\phi}_n\|_{\infty}\leq \exp(-\omega 2^{|n|-2})\|m\|_\infty, $$ where $$ \phi_{\pm n}(\xi)=\phi(\pm \xi \cdot2^{1-n})\quad \forall \xi\in\mathbb{R}, $$ with $\phi=(1-2|\xi-1|)_++(1/2-|\xi-3/2|)_+$. It claims that it is based on ``proof of Paley-Wiener theroem''. But I don't know how to show $(\star)$ holds.