Young function

Last updated April 09, 2024

In mathematics, certain functions useful in functional analysis are called Young functions.

A function ${\displaystyle \theta$ is a Young function, iff 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 $x\mapsto {\begin{cases}0{\text{ if }}x=0,\\+\infty {\text{ else.}}\end{cases}}$

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

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

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

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References

Léonard, Christian. "Orlicz spaces." (2007).
O’Neil, Richard (1965). "Fractional integration in Orlicz spaces. I". Transactions of the American Mathematical Society. 115 (0): 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. 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.

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