Does the norm $\mathcal{H} := \| w(x,y)\log^+ (w(x,y)) \|_{L^1}$ define a reflexive Banach space?

Question

Suppose $\Omega \subset \mathbb{R}^2$ is an open bounded set. Given the norm \begin{equation*} \| w(x,y) \|_{\mathcal{H}(\Omega)} := \| w(x,y) \log^+(w(x,y)) \|_{L^1(\Omega \times \Omega)} \end{equation*} where \begin{equation*} \log^+(x) = \left\{ \begin{array}{ll} \log(x) & \log(x) \geq 0 \\ 0 & \text{else} \end{array} \right. \end{equation*} is the space \begin{equation*} \mathcal{W} = \left\{ w \in L^1(\Omega \times \Omega): w \geq 0, \| w \|_{\mathcal{H}(\Omega)} < \infty \right\} \end{equation*} a reflexive Banach space? I saw this result utilized in a paper without any justification so I just wanted to verify that it is indeed true. Thanks for the help.