Let $f: X \rightarrow \mathbb{R}$ be a measurable function where $(X,\mu)$ is a measure space and say that $f \in L^{p,\infty}$ $\iff$ $[f]_p < \infty$
where $[f]_p = \sup_{t>0} t \mu(\{x : |f(x)| > t \})^{\frac{1}{p}} < \infty$.
In my measure theory class we are calling $[*]_p$ the weak $L^p$ norm.
Now then, suppose $1<p<\infty$ and let $p'$ be the dual exponent of $p$. Prove that:
$[fg]_1\leq p^{\frac{1}{p}}(p')^{(\frac{1}{p'})}[f]_p[g]_{p'}$
My attempt:
I'm first trying to reduce the case to where $[f]_p=[g]_p=1$. I've also noticed that I can prove the above inquality if $p^{\frac{1}{p}}(p')^{(\frac{1}{p'})}=2$, and to get the constant $p^{\frac{1}{p}}(p')^{(\frac{1}{p'})}$ I was going to optimize the function $h(x,y)=x^{-p}+y^{-p}$ subject to the constraint $xy= \lambda$... But I don't know maybe i'm going about this all wrong... Insight appreciated!