Orlicz space

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

Besides the L^p spaces, a variety of function spaces arising naturally in analysis are Orlicz spaces. One such space L log⁺ L, which arises in the study of Hardy–Littlewood maximal functions, consists of measurable functions f such that the

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

Here 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.