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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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up vote 2 down vote accepted

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 $f\in L^p$.

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just to complement: ...and $\| f\| _p \leq M$ – Jose27 May 8 '11 at 16:20

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