Hölder's inequality, equality condition

Question

It is known that equality in Holder's inequality (i.e $|\int_Efg|=||f||_p||g||_q$) holds iff $\|g\|_q^q|f|^p=\|f\|_p^p|g|^q$ a.e. However recently I read somewhere that an additional constraint must me met, that of $\operatorname{sgn}(f)=\operatorname{sgn}(g)$ a.e. Can anyone help me understand why this additional constraint is needed?

Answer 1

Normally, Hölder's inequality is written as $$\int_E|fg|\le \|f\|_p\|g\|_q \tag{1}$$ that is, with absolute value inside the integral. For this version, you don't need the additional constraint on absolute value in the equality case (analyzed in On the equality case of the Hölder and Minkowski inequalites).

The inequality for $\left| \int_Efg\right|$ is obtained by combining $(1)$ with the integral triangle inequality $$\left| \int_Efg\right|\le \int_E |fg|\tag{2}$$ Equality holds in (2) if and only if $fg$ is either nonnegative a.e. or nonpositive a.e. Indeed, the proof of (2) involves observing that $$ \int_E (|fg|-fg) \ge 0$$ and $$ \int_E (|fg|+fg) \ge 0$$ and the equality cases here are clear.