Orlicz space

In the mathematical analysis, and especially in real and harmonic analysis, a Birnbaum–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 and Zygmunt William Birnbaum, who first defined them in 1931.

Besides the L^p spaces, a variety of function spaces arising naturally in analysis are Birnbaum–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 integral

Here log⁺ is the positive part of the logarithm. Also included in the class of Birnbaum–Orlicz spaces are many of the most important Sobolev spaces.

Suppose that μ is a σ-finite measure on a set X, and Φ : [0, ∞) → [0, ∞) is a Young function, i.e., a convex function such that

Let $L_{\Phi }^{\dagger }$ $L_{\Phi }^{\dagger }$ be the set of measurable functions f : X → R such that the integral

is finite, where, as usual, functions that agree almost everywhere are identified.

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