It is well known that for two given functions $f,g:\mathbb{R}^d \rightarrow \mathbb{R}^d$ such that $fg \in L^1(\mathbb{R}^d)$ and $f\in L^p(\mathbb{R}^d)$ and $g\in L^q(\mathbb{R}^d)$ with $\frac{1}{p}+\frac{1}{q}=1$, $p,q\geq 1$ then $$\left|\int_{\mathbb{R}^d} f(x)g(x)dx\right|\leq \int_{\mathbb{R}^d} |f(x)g(x)|dx \leq \left(\int_{\mathbb{R}^d} |f(x)|^p\right)^{1/p} \left(\int_{\mathbb{R}^d}|g(x)|^q\right)^{1/q},$$ the latter known as Hölder inequality. This equality also holds when $"p=\infty"$ taking supremum.
$\bullet$ My question is: are there other type of inequalities that allow to "split" $f$ and $g$? Whatever inequality, more general or not, that allows to split functions $f$ and $g$. Any ideas? How or where could I find this information? Preferably one of the functions should be as general as possible, e.g. non-continuous.
Thanks a lot! :)