I know the generalized Hölder inequality sounds like as: Let $1\leq p_1,\ldots,p_n<\infty$ and $p>0$ such that $\frac1p=\frac1{p_1}+\cdots+\frac1{p_n}$. Then, for all measurable functions $f_1,\ldots,f_n: (X,\mu) → \mathbb C$ we have $\left\|\prod _{{k=1}}^{n}f_{k}\right\|_{p}\leq \prod _{{k=1}}^{n}\|f_{k}\|_{{p_{k}}}$. (see this or this )
My question is: what are conditions on $f_1,\ldots,f_n$ so that the equality holds, ie.. $\left\|\prod _{{k=1}}^{n}f_{k}\right\|_{p}= \prod _{{k=1}}^{n}\|f_{k}\|_{{p_{k}}}$
Best Answer
Generalized Hölder's inequality is obtained by repeateadly applying Hölder's inequality for a pair of function. For equality to holds, it must hold at every step of the process (no matter in what order it is done), i.e., for all pairs $f_k,f_j$. For a pair, the equality case is well-known.
Conclusion: equality holds iff and only if all functions $|f_k|^{p_k}$ agree a.e., up to multiplicative constants. In other words, $$ \frac{|f_k|^{p_k}}{\|f_k\|_{p_k}^{p_k}} = \frac{|f_j|^{p_j}}{\|f_j\|_{p_j}^{p_j}} \quad \text{a.e.}, \qquad \forall k,j=1,\dots,n $$