In mathematics, Young functions are a class of functions that arise in functional analysis, especially in the study of Orlicz spaces.
Definition A function θ : R → [ 0 , ∞ ] {\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 ↦ 0 {\displaystyle x\mapsto 0} nor its convex dual
x ↦ { 0 if x = 0 , + ∞ otherwise. {\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 θ {\displaystyle \theta } is strict if both θ {\displaystyle \theta } and its convex dual θ ∗ {\displaystyle \theta ^{*}} are finite; i.e.,
lim x → ∞ θ ( x ) x = ∞ . {\displaystyle \lim _{x\to \infty }{\frac {\theta (x)}{x}}=\infty .}
The inverse of a Young function is given by θ − 1 ( y ) = inf { x : θ ( x ) > y } {\displaystyle \theta ^{-1}(y)=\inf\{x:\theta (x)>y\}} . Some authors (such as Krasnosel'skii and Rutickii) also require that
lim x ↓ 0 θ ( x ) x = 0 {\displaystyle \lim _{x\downarrow 0}{\frac {\theta (x)}{x}}=0} .
Norm Let μ {\displaystyle \mu } be a σ-finite measure on a set X {\displaystyle X} , and θ {\displaystyle \theta } a Young function. For any measurable function f {\displaystyle f} on X {\displaystyle X} , we define the Luxemburg norm as
‖ f ‖ θ = inf { b > 0 | ∫ X θ ( | f | / b ) d μ ≤ 1 } . {\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:
θ exp ( x ) = e | x | − 1 {\displaystyle \theta _{\text{exp}}(x)=e^{|x|}-1} .
θ p ( x ) = | s | p / p {\displaystyle \theta _{p}(x)=|s|^{p}/p} for all p ≥ 1 {\displaystyle p\geq 1} . This function leads to the usual norm p 1 / p ‖ f ‖ θ = ‖ f ‖ p = ( ∫ X | f | p d μ ) 1 / p {\displaystyle p^{1/p}\|f\|_{\theta }=\|f\|_{p}=(\textstyle \int _{X}|f|^{p}d\mu )^{1/p}} on L p ( μ ) {\displaystyle L^{p}(\mu )} .
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