Young functions. Let (phi:[0, infty) ightarrow[0, infty)) be a strictly increasing continuous function such that (phi(0)=0) and

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Young functions. Let \(\phi:[0, \infty) ightarrow[0, \infty)\) be a strictly increasing continuous function such that \(\phi(0)=0\) and \(\lim _{\xi ightarrow \infty} \phi(\xi)=\infty\). Denote by \(\psi(\eta):=\phi^{-1}(\eta)\) the inverse function. The functions

\[\Phi(A):=\int_{[0, A)} \phi(\xi) \lambda^{1}(d \xi) \quad \text { and } \quad \Psi(B):=\int_{[0, B)} \psi(\eta) \lambda^{1}(d \eta)\]

are called conjugate Young functions. Adapt the proof of Lemma 13.1 to show the following general Young inequality:

\[A B \leqslant \Phi(A)+\Psi(B) \quad \forall A, B \geqslant 0 .\]

[ interpret \(\Phi(A)\) and \(\Psi(B)\) as areas below the graph of \(\phi(\xi)\) resp. \(\psi(\eta)\).]

Data from lemma 13.1

Lemma 13.1 (Young's inequality) Let p, qe (1,00) be conjugate numbers, i.e. 1+1=1, hence q=1. Then B fon-1dn

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