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 \(r^{-1}+1=p^{-1}+q^{-1}\).

Data from theorem 15.6

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Theorem 15.6 (Young's inequality) Let ue L (X") and vELP(X"), p= [1,00). The convolution uv defines a function in LP (X") and satisfies u* v=v*u, and ||uv|p|u||1||v||p. (15.7)