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 ${\displaystyle x\mapsto 0}$ nor its convex dual

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

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

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

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

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

Some authors (such as Krasnosel'skii and Rutickii)^{[citation needed]} also require that

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

Norm

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

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

Examples

The following functions are Young functions:

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

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

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.