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 ${\displaystyle \theta$ :\mathbb {R} \to [0,\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

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

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

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

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

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

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

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

$\|f\|_{\theta }=\inf \left\{b>0\;{\bigg \vert }\int _{X}\theta (|f|/b)\,d\mu \leq 1\right\}.$

Examples

The following functions are Young functions:

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

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

References

• Léonard, Christian. "Orlicz spaces." (2007).
• O’Neil, Richard (1965). "Fractional integration in Orlicz spaces. I". Transactions of the American Mathematical Society. 115: 300–328. doi:10.1090/S0002-9947-1965-0194881-0. ISSN 0002-9947.. Gives another definition of Young's function.
• Krasnosel'skii, M.A.; Rutickii, Ya B. (1961-01-01). Convex Functions and Orlicz Spaces (1 ed.). Gordon & Breach. ISBN 978-0-677-20210-5. {{cite book}}: ISBN / Date incompatibility (help) 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.