Let $f_n$ and $g_n$ two generalized functions such that :
- the product $f_n g_n$ is well defined for all $n\in \mathbb{N}$, which means $WF(f_n)+WF(g_n)$ does not contain an element of the form $(x,0)$. \
( where $WF(u)$ is the wavefront of generalized function $f$)
- $f_n$ converges to $f$ and $g_n$ converges to $g$ (Weak* topologie).
- the product $f g$ is well defined.\
My question is : Can we deduce that $f_n g_n$ converge to $f g$ (weak* topologie) ?