# Hölder inequality valid for surface measure

my question is whether the Hölder inequality is true for the surface measure: So let $\Omega \subset \mathbb{R}^n, n\in \mathbb{N}$ with $C^1$-boundary, $f \in L^p(\partial\Omega)$, $g\in L^q(\partial\Omega)$, $1/p+1/q =1$. Then

$$\int_{\partial\Omega} |fg| ~d\sigma \leq \left(\int_{\partial\Omega} |f|^p~d\sigma\right)^{1/p}\left(\int_{\partial\Omega} |g|^q~d\sigma\right)^{1/q}.$$

Is this true?

• Yes, it's true for any measure. – Kenny Wong Jun 24 '17 at 15:52