Let $\mu$ be a positive measure and consider the spaces $L^p(\mu)$, with $1\leq p < \infty.$ Denote by $p^*$ the conjugate exponent of $p$, id est, $\frac{1}{p} + \frac{1}{p^*} = 1.$ I remember reading that for every $f \in L^1(\mu)$ there exists $g \in L^p(\mu)$ and $h \in L^{p^*}(\mu)$ such that $f = g\cdot h$, something like the reverse implication of the Hölder inequality.
I would like to have some references to read the proof of this result.
Thank you very much!
The cases $p=1$ and $p=\infty$ being trivial, assume $1 < p < \infty$. Let $g(x) = |f(x)|^{1/p} \text{signum}(f(x))$ and $h(x) = |f(x)|^{1/p^*}$.