  In a $d-$dimensional strip with $d\geq 2$, we study the non-stationary Stokes
equation with no-slip boundary condition in the lower and upper plates and
periodic boundary condition in the horizontal directions. In this paper we
establish a new maximal regularity estimate in the real interpolation norm
  \begin{equation*}
  ||f||_{(0,1)}=\inf_{f=f_0+f_1}\left\{\left\langle\sup_{0<z<1}
|f_0|\right\rangle+
  \left\langle\int_0^{1} |f_1| \frac{dz}{(1-z)z}\right\rangle\right\}\,,
  \end{equation*} where the brackets $\langle\cdot\rangle$ denotes the
horizontal-space and time average. The norms involved in the definition of
$\|\cdot\|_{(0,1)}$ are critical for two reasons: the exponents are borderline
for the Calder\'on-Zygmund theory and the weight $1/z$ just fails to be
Muckenhoupt. Therefore, the estimate is only true under horizontal bandedness
condition, (i. e. a restriction to a packet of wave numbers in Fourier space).
The motivation to express the maximal regularity in such a norm comes from an
application to the Rayleigh-B\'enard problem.
