  In this paper, the indirect signal production system with nonlinear
transmission is considered \[ \left\{ \begin{array}{lll} & u_t = \Delta
u-\nabla\cdot(u \nabla v), \\ \displaystyle & v_t =\Delta v-v+w,\\
\displaystyle & w_t =\Delta w-w+ f(u) \end{array} \right. \] in a bounded
smooth domain $\Omega\subset \mathbb{R}^n$ associated with homogenous Neumann
boundary conditions, where $f\in C^1([0,\infty))$ satisfies $0\le f(s) \le
s^{\alpha}$ with $\alpha>0$. It is known that the system possesses a global
bounded solution if $0<\alpha<\frac 4n$ when $n\ge 4$. In the case $n\le 3$ and
if we consider superlinear transmission, no regularity of $w$ or $v$ can be
derived directly. In this work, we show that if $0<\alpha< \min\{\frac
4n,1+\frac 2n\}$, the solution is global and bounded via an approach based on
the maximal Sobolev regularity.
