If we have a product of functions $fg$ with $f\in L^r$ and $g\in L^s$ for some $s,r>1$ satsfying $1/r+1/s=1$, then we know that $fg\in L^1$.
But if $g$ is a little bit more than $L^s$, say $L^s \log L$ , can we say that $fg$ is a little bit more than $L^1$ ? For instance $L^1 \log L^1$ ?

If we have a product of functions $fg$ with $f\in L^r$ and $g\in L^s$ for some $s,r>1$ satisfying $1/r+1/s=1$, then we know that $fg\in L^1$.
But if $g$ is a little bit more than $L^s$, say $L^s \log L$ , can we say that $fg$ is a little bit more than $L^1$ ? For instance $L^1 \log L^1$ ?