A maximal regularity estimate for the non-stationary Stokes equation in the strip

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.