If one ask to me to prove that the converse of Hölder inequality for $p=1,\infty$, then what statement should I prove? Do you guys agree that the term "the converse of Hölder inequality for $p=1,\infty$" does make sense?
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I would guess it means something like this:
Let $p$ and $q$ be Hölder conjugates and let $f$ be measurable. Then, if there is some constant $M$ such that
$$ \left\| fg\right\| _1\leq M\left\| g\right\| _q $$
for all $g\in L^q$, then $f\in L^p$.