From the Hölder Inequality we know $\lVert fg\rVert_1\le \lVert f\rVert_p\lVert g\rVert_q$ whenever $p,q$ are conjugate. I expect the following (partial) converse is false:
If $\lVert fg \rVert_1$ is finite then $\lVert f\rVert_p\lVert g\rVert_q$ is finite for some conjugate $p,q$
i.e. I do not expect the boundedness of $fg$ in $L^1$ to imply $f,g$ are in some conjugate $L^p$ spaces. I am not sure what a counter-example would look like though, since checking there is no choice of $p,q$ for which we get boundedness seems a daunting task. Is there a simple example I am missing?