Young function

In mathematics, certain functions useful in functional analysis are called Young functions.

A function $$\theta : \R \to [0, \infty]$$ is a Young function, iff it is convex, even, lower semicontinuous, and non-trivial, in the sense that it is not the zero function $$x \mapsto 0$$, and it is not the convex dual of the zero function $$x \mapsto \begin{cases} 0 \text{ if } x = 0, \\ +\infty \text{ else.}\end{cases}$$

A Young function is finite iff it does not take value $$\infty$$.

The convex dual of a Young function is denoted $$\theta^*$$.

A Young function $$\theta$$ is strict iff both $$\theta$$ and $$\theta^*$$ are finite. That is, $\frac{\theta(x)} x \to \infty,\quad\text{as }x\to \infty,$

The inverse of a Young function is$$\theta^{-1}(y)=\inf \{x: \theta(x)>y\}$$

The definition of Young functions is not fully standardized, but the above definition is usually used. Different authors disagree about certain corner cases. For example, the zero function $$x \mapsto 0$$ might be counted as "trivial Young function". Some authors (such as Krasnosel'skii and Rutickii) also require $$\lim_{x \downarrow 0} \frac{\theta(x)}{x} = 0$$