Orlicz space

In mathematics, and especially in harmonic analysis and functional analysis, an Orlicz space is a type of function space which generalizes L^p spaces. Like ${\displaystyle L^{p}}$ spaces, they are Banach spaces. The spaces are named for Władysław Orlicz, who was the first to define them in 1932.

Besides ${\displaystyle L^{p}}$ spaces, a variety of function spaces arising naturally in analysis are Orlicz spaces. One such space is ${\displaystyle L\,\log ^{+}\!L}$, which arises in the study of Hardy–Littlewood maximal functions, consisting of measurable functions ${\displaystyle f}$ such that

${\displaystyle \int _{\mathbb {R} ^{n}}|f(x)|\log ^{+}|f(x)|\,dx<\infty .}$

Here ${\displaystyle \log ^{+}}$ is the positive part of the logarithm. Also included in the class of Orlicz spaces are many of the most important Sobolev spaces. In addition, the Orlicz sequence spaces are examples of Orlicz spaces.