Pear Tree Orlicz Room

A pear tree-Orlicz space (also Orlicz space ) is a term from the mathematical sub-area of functional analysis and a function space that generalizes the L ^p spaces . It is named after the Polish mathematicians Zygmunt Wilhelm Birnbaum and Władysław Orlicz .

definition

Orlicz function

Let be a σ-finite measure on a set . A convex function is called an Orlicz function (also Young function ) if the following applies: ${\displaystyle \mu }$ ${\displaystyle X}$ ${\displaystyle \phi \colon [0,\infty ]\to [0,\infty ]}$

${\displaystyle {\frac {\phi (x)}{x}}\to \infty ,\quad {\text{wenn }}x\to \infty ,}$ and

${\displaystyle {\frac {\phi (x)}{x}}\to 0,\quad {\text{wenn }}x\to 0}$.

Orlicz standard

Now be the right inverse function to , that is, it holds . We define the complementary function to as the integral over its right inverse function: ${\displaystyle \psi }$ ${\displaystyle \phi }$ ${\displaystyle \psi (s)=\sup\{t:\phi (t)\leq s\}}$ ${\displaystyle \phi }$

${\displaystyle Q_{\psi }(x)=\int _{0}^{|x|}\psi (s)\,\mathrm {d} s}$.

The Orlicz norm is then given by:

${\displaystyle \|f\|_{Q}=\sup \left\{\left|\int _{X}fg\,\mathrm {d} \mu \right|:\int _{X}Q_{\psi }(g(t))\,\mathrm {d} \mu (t)\leq 1\right\}}$.

Pear Tree Orlicz Room

The pear tree-orlicz space is defined as

${\displaystyle L_{Q}(X,\mu ):=\left\{f\colon X\to \mathbb {K} :f\,{\rm {ist\ messbar}}\,,\|f\|_{Q}<\infty \right\}}$

(or in short as ), i.e. as the space of all measurable functions that have a finite Orlicz norm. ${\displaystyle L_{Q}}$

Luxembourg norm

An equivalent norm called the Luxemburg norm is obtained by

${\displaystyle \|f\|_{\phi }=\inf \left\{k\in (0,\infty ):\int _{X}\phi (|f|/k)\,\mathrm {d} \mu \leq 1\right\}}$.

The following norm results for a random variable : ${\displaystyle Y}$

${\displaystyle \|Y\|_{\phi }=\inf \left\{k\in (0,\infty ):\mathbb {E} [\phi (|Y|/k)]\leq 1\right\}}$.

It applies to the Luxembourg-standard: . ${\displaystyle \|\emptyset \|_{\phi }=\inf \left\{\emptyset \right\}=\infty }$

properties

• The following applies to inclusion: ${\displaystyle p\in [1,\infty )}$ ${\displaystyle L^{\infty }\supset L_{Q}\supset L^{p}}$

If one takes for , one obtains the L ^p -spaces . ${\displaystyle \phi _{p}(x):=x^{p}}$

• The pear tree orlicz room is a Banach room .

Individual evidence

1. ↑ On the generalization of the concept of mutually conjugated potencies Studia Mathematica 3, pp. 1–67, 1931.