Counterexamples for Hölder's inequality when $p$ and $q$ are not conjugate. Hölder's inequality shows that, when $$ \frac{1}{p} + \frac{1}{q} = 1,$$ and $f\in L^p$ and $g\in L^q$, then $$\Vert f\,g\Vert_1 \le \Vert f \Vert_p \Vert g \Vert_q.$$
Is there an example of this inequality failing for $p=q=1$?  I have been unable to find one in the usual places (analysis texts, and various combinations of search terms online).

Consider the real line with Lebesgue measure and $f$, $g$ the characteristic functions of the interval $[0,x]$ for a fixed $x$. 
Since  $\lVert f\rVert_p= x^{1/p}$ and similarly for the $q$ norm, the Hölder's inequality would read $x\leqslant  x^{1/p+1/q}$. 
If $p=q=1$, just pick some $x$ such that $x\gt x^2$. 
This proves more in the general case: if $1/p+1/q\neq 1$, there is not universal constant $C$ such that for any $(f,g)\in \mathbb L^p\times\mathbb L^q$, $\lVert fg\rVert_1\leqslant C\lVert f\rVert_p\cdot \lVert g\rVert_q$. 
