Young function explained

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

A function

\theta : \R \to [ 0 , inf ty] 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 if x = 0 , \ +inf ty else. \end{cases}

A Young function is finite iff it does not take value

inf ty .

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, $\backslash frac\; x\; \backslash to\; \backslash infty,\backslash quad\backslash textx\backslash to\; \backslash infty,$

The inverse of a Young function is$$\backslash theta^(y)=\backslash inf\; \backslash$$

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} (\theta(x))/(x) = 0

References

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