Young function

Short description: Mathematical functions

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

Definition

A function $\theta :\mathbb{ℝ}\to [0,\mathrm{\infty }]$ 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 $x\mapsto 0$ nor its convex dual

$x\mapsto \{\begin{array}{ll}0& \text{ if }x=0,\\ +\mathrm{\infty }& \text{ otherwise.}\end{array}$

A Young function said to be finite if it does not take the value $\mathrm{\infty }$.

A Young function $\theta$ is strict if both $\theta$ and its convex dual ${\theta }^{*}$ are finite; i.e.,

$\underset{x\to \mathrm{\infty }}{\lim }\frac{\theta (x)}{x}=\mathrm{\infty }.$

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

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

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

Norm

Let $\mu$ be a σ-finite measure on a set $X$, and $\theta$ a Young function. For any measurable function $f$ on $X$, we define the Luxemburg norm as

$\Vert f{\Vert }_{\theta }=\inf \{b>0|\underset{X}{\int }\theta (|f|/b)d\mu \le 1\}.$

Examples

The following functions are Young functions:

• ${\theta }_{\text{exp}}(x)=e^{|x|}-1$.

• ${\theta }_p(x)=|s{|}^p/p$ for all $p\ge 1$. This function leads to the usual norm $p^{1/p}\Vert f{\Vert }_{\theta }=\Vert f{\Vert }_p=(\underset{X}{\int }|f{|}^{p}d\mu {)}^{1/p}$ on $L^p(\mu )$.

References

• Léonard, Christian. "Orlicz spaces." (2007).
• O’Neil, Richard (1965). "Fractional integration in Orlicz spaces. I" (in en). Transactions of the American Mathematical Society 115: 300–328. doi:10.1090/S0002-9947-1965-0194881-0. ISSN 0002-9947. https://www.ams.org/tran/1965-115-00/S0002-9947-1965-0194881-0/. . Gives another definition of Young's function.
• Krasnosel'skii, M.A.; Rutickii, Ya B. (1961-01-01) (in English). Convex Functions and Orlicz Spaces (1 ed.). Gordon & Breach. ISBN 978-0-677-20210-5. http://archive.org/details/convexfunctionso0000makr.  In the book, a slight strengthening of Young functions is studied as "N-functions".
• Rao, M.M.; Ren, Z.D. (1991). Theory of Orlicz Spaces. Pure and Applied Mathematics. Marcel Dekker. ISBN 0-8247-8478-2.