General Hölder's Inequality

Let \((\Omega, \Sigma, \mu)\) be a meausurable space . Furthermore, assume \(r\in [1,\infty]\) and \(p_1,\ldots ,p_n\in [1,\infty]\) such that

\begin{equation*} \sum_{k=1}^{n} \frac{1}{p_k}=\frac{1}{r}. \end{equation*}

Then

\begin{equation*} \lVert \prod_{k=1}^{n} f_k\rVert_r \le \prod_{k=1}^{n} \lVert f_k\rVert_{p_k} \end{equation*}

for all \(f_k\in L^{p_k}(\mu)\).