Mixed $L^p \ell^q$ -norm Minkowski type inequality?

Question

I'm trying to find whether it is true that for functions $\{ f(x,k)\}$ where $k\in \mathbb{Z}$, and $x \in \mathbb{R}^d$, that lie in $L^q_x$ and $\ell_k^p$ simultaneously that \begin{align*} || f ||_{L^q_x \ell_k^p} \leq || f ||_{\ell_k^p L^q_x} \end{align*} where \begin{align*} || f ||_{L^q_x \ell_k^p} := \left( \int \left( \sum_k |f(x,k)|^p \right)^{q/p} dx \right)^{1/q} \end{align*}

thanks!