Young function

Template:Short description Template:Multiple issues In mathematics, certain functions useful in functional analysis are called Young functions.

A function $\theta :ℝ\to [0,\mathrm{\infty }]$ is a Young function, if 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 $x\mapsto \{\begin{array}{c}0\text{ if }x=0,\\ +\mathrm{\infty }\text{ else.}\end{array}$

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

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, $\frac{\theta (x)}{x}\to \mathrm{\infty },\text{as }x\to \mathrm{\infty },$

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 $\underset{x\downarrow 0}{\lim }\frac{\theta (x)}{x}=0$

References

• Léonard, Christian. "Orlicz spaces." (2007).
• Template:Cite journal. Gives another definition of Young's function.
• Template:Cite book In the book, a slight strengthening of Young functions is studied as "N-functions".
• Template:Cite book

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