Young function

In mathematics, Young functions are a class of functions that arise in functional analysis, especially in the study of Orlicz spaces.

Definition

A function is called a Young function if it is convex, even, lower semicontinuous, and non-trivial, in the sense that it is neither the zero function nor its convex dual

A Young function said to be finite if it does not take the value .

A Young function is strict if both and its convex dual are finite; i.e.,

The inverse of a Young function is given by .

Some authors (such as Krasnosel'skii and Rutickii) also require that

.

Norm

Let be a σ-finite measure on a set , and a Young function. For any measurable function on , we define the Luxemburg norm as

Examples

The following functions are Young functions:

• .

• for all . This function leads to the usual norm on .

References

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