A general Young inequality. Generalize Young's inequality given in Problem 15.14 and show that

\[\left\|f_{1} \star f_{2} \star \cdots \star f_{N}ight\|_{r} \leqslant \prod_{j=1}^{N}\left\|f_{j}ight\|_{p}, \quad p=\frac{N r}{(N-1) r+1}\]

for all \(N \in \mathbb{N}, r \in[1, \infty)\) and \(f_{j} \in \mathcal{L}^{p}\left(\lambda^{n}ight)\).

Data from problem 15.14

Young's inequality. Adapt the proof of Theorem 15.6 and show that

\[
\|u \star w\|_{r} \leqslant\|u\|_{p} \cdot\|w\|_{q}
\]

for all \(p, q, r \in[1, \infty), u \in \mathcal{L}^{p}\left(\lambda^{n}ight), w \in \mathcal{L}^{q}\left(\lambda^{n}ight)\) and \(r^{-1}+1=p^{-1}+q^{-1}\).