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

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 $$\forall x \in \mathbb{R}: \qquad \Psi(x) \stackrel{\text{df}}{=} \int_{0}^{x} {(\Phi')^{- 1}}(t) ~ \mathrm{d}{t}.$$

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 $L^{\Psi}$?

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".