Let $X$ be a pointed metric space, with base point 0. The space of Lipschitz function which preserves the base point, $Lip_0(X)=\{f:X\to\mathbb{R} : f(0)=0\}$ consider with the norm $\|f(x)\|=\sup_{x\neq y}\{\frac{|f(x)-f(y)|}{d(x,y)}\}$ is a Banach space. It is well known that $Lip_0(X)$ is the dual space of the so called, the Lipschitz free-space of $X$ or the Arens-Eells space $\mathcal{F} (X)$.
Question: Let $X_1$ and $X_2$ pointed metric spaces and consider $X_1 \times X_2$ with the max-distance. Which properties of the space $\mathcal{F}(X_1\times X_2)$ can be inferred from the spaces $\mathcal F(X_1)$ and $\mathcal F(X_2)$?
For instance, if $\mathcal F(X_1)$ and $\mathcal F(X_2)$ have the approximation property (or are Schur space, or has the Dunford- Pettis property, etc etc.) , Does $\mathcal F(X_1 \times X_2)$ has the approximation property (or are Schur space, or has the Dunford- Pettis property, etc etc.) ?
