Hölder bounds the $L^1$ norm of a product by conjugate $L^p$ and $L^q$ norms.Suggest a slogan
Let \((X,\mcm,\mu)\) be a measure space, let \(1\le p,q\le\infty\) satisfy \(\frac 1 p + \frac 1 q = 1\), and let \(f,g\colon X\to\CC\) be measurable. Then \[ \norm{f g}_{1} \leq \norm{f}_{p} \norm{g}_{q} . \] For \(1<p,q<\infty\), this reads \[ \int_X \abs{fg}\dmu \leq \qty{\int_X \abs{f}^p\dmu}^{1/p} \qty{\int_X \abs{g}^q\dmu}^{1/q} . \]