Orlicz space

Summary

Fix a convex growth function $\Phi$. The Orlicz space $L^\Phi(\mu)$ consists of the measurable functions $f$ for which $\Phi(\abs{f})$ is integrable after scaling $f$. The choice $\Phi(t) = t^p$ gives the Lp space $L^p$, so Orlicz spaces generalise the $L^p$ scale. They are Banach spaces with the Luxemburg norm. They describe integrability conditions that lie between the $L^p$ spaces, such as $L \log L$, which appears in the theory of maximal functions and singular integrals.