Let $f$ and $g$ be submodular functions, and let $\widehat{f}$ and $\widehat{g}$ be the Lovasz extensions of $f$ and $g$, respectively.
What can we say about the Lovasz extension of $f \times g$, i.e. $\widehat{f\times g}$ ?
In particular, is there any result that helps getting and explicit formulation of the Lovasz extension of $f \times g$ ?
Is there any result stating that we can express $\widehat{f\times g}$ in terms of $\widehat{f}$ and $\widehat{g}$ ?
What if we further assume that $f$ is modular ?
Not sure if this is what you had in mind, but in the very special case where $f$ and $g$ are both normalized modular functions, this formula works: $$ \widehat{f \times g} (x) = \sum_i \sum_j \hat{f}(e_i) \hat{g}(e_j) \min(x_i,x_j) $$ In general, you can use the values of the $\hat{f}, \hat{g}$ to get the values of the $f,g$, which gives you the values of $f\times g$, and plug those into a formula for the Lovasz extension to get $\widehat{f \times g}$. For example: $$ \widehat{f \times g}(x) = E_{t \sim U(0,1)} \hat{f}( [x > t] )\hat{g}( [x > t] ) $$ $$ [x > t]_i = \begin{cases} 0 & x_i \le t\\ 1 & x_i > t\end{cases} $$ While it's an explicit formula in terms of values of $\hat{f}, \hat{g}$, it may not be that useful.