Young’s complement of $ x \mapsto x \, {\log^{+}}(x) $, $ N $-functions and Orlicz spaces

Question

The function $ \Phi: \mathbb{R} \to \mathbb{R} $ is an $ N $-function if and only if it is continuous, even and convex with:

1. $ \displaystyle \lim_{x \to 0} \frac{\Phi(x)}{x} = 0 $.
2. $ \displaystyle \lim_{x \to \infty} \frac{\Phi(x)}{x} = \infty $.
3. $ \Phi(x) > 0 $ if $ x > 0 $.

Young’s complement of $ \Phi $, denoted by $ \Psi $, is defined by $$ \forall x \in \mathbb{R}: \qquad \Psi(x) \stackrel{\text{df}}{=} \int_{0}^{x} {(\Phi')^{- 1}}(t) ~ \mathrm{d}{t}. $$

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Question. Let $ \Phi(x) \stackrel{\text{df}}{=} x \, {\log^{+}}(x) $, where $ {\log^{+}}(x) \stackrel{\text{df}}{=} \max(0,\log(x)) $. Then what is $\Psi(x)$? Or can we describe the Orlicz space $ L^{\Psi} $?

Answer 1

That space should be $L_{\exp}$. Check Bennett and Sharpley's "Interpolation of Operators" Chapter 4.6 or look in Rao and Ren's "Theory of Orlicz Spaces".