I have two sequences $(f_n)$ and $(g_n)$ that converge weakly in $L^p$ and $L^q$, ($1<p,q<\infty$) respectively to $f,g$ such that $(f_ng_n)$ converges weakly to some $h$ in $L^r$ ($\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$). Notice that $f_n>0$ for all n.
I would like to know if $h-fg$ have a sign, for instance almost everywhere positive or negative. Thanks!
Let $f_n(x) = g_n(x) = \sin(nx)1_{[0,\pi]}(x)$ for all real number $x$. Then $f_n = g_n \to 0$ weakly in $L^2$, but $f_ng_n \to 1/2$ weakly in $L^1$. Multiplying $g_n$ by a fixed bounded measurable function $b$, you can get whatever you want for the sign of the weak limit of $f_ng_n$, which is $(1/2)b$.