While preparing an upcoming post on interpolation inequalities and interpolation spaces, I noticed that many of the interesting examples came in the form of variations on the notion of an *L*^{p} space such as the weak *L*^{p} spaces and Lorentz spaces. Rather than over-burden that post, I decided to put it off for a bit and first post some preparatory material on such spaces.

These notes owe a lot to my trusty copy of Loukas Grafakos’ 2008 monograph *Classical Fourier Analysis*, and to some (sadly incomplete) notes by Christian Léonard.

Throughout, unless otherwise noted, the setting is that of a general measure space (Ω, ℱ, *μ*) and the functions involved are ℱ-measurable functions *u*: Ω → ℝ or ℂ (although I see no obvious problems with extending all of this to functions taking values in a Banach space, in the style of a Bochner integral). The main examples in a Fourier analysis context are when (Ω, ℱ, *μ*) is

- Euclidean space ℝ
^{n}with its Borel or Lebesgue*σ*-algebra and*μ*= Lebesgue measure; or, more generally - a topological group
*G*such as the circle or torus group with its Borel*σ*-algebra and*μ*= Haar measure.

As with much of this stuff, the main motivation is to come up with clever notions of how “nice” or “nasty” a function *u*: Ω → ℝ is, usually in terms of its integrability properties. The first, simplest, examples, are spaces of functions *u* for which some positive power of (the modulus of) *u* is integrable, the *L*^{p} spaces:

**Recap: Strong L^{p} Spaces**

For 0 < *p* < ∞, define the *L*^{p} “norm” of *u*: Ω → ℝ by

and define

(Here and in the following, I’m using the common probabilistic shorthand notation [|*u*| > *t*] for { x ∈ Ω | |*u*(*x*)| > *t* }.) The **(strong) Lebesgue space** *L*^{p}(Ω, *μ*) is defined to be the vector space of all measurable functions *u* for which the corresponding “norm” is finite, with the convention that functions that are equal *μ*-almost everywhere are identified. When *p* ≥ 1, the *L*^{p} “norm” is a bona fide norm, and *L*^{p} is complete with respect to this norm, so it is a Banach space. When p < 1, the “norm” is only a quasi-norm and satisfies the following modified triangle inequality:

However, *L*^{p} is still complete with respect to this quasi-norm, and so *L*^{p} is a quasi-Banach space.

*L*^{p} spaces crop up all over the place in analysis and, in particular, in probability theory. When (Ω, ℱ, *μ*) is a probability space (i.e. *μ*(Ω) = 1), a measurable function *u*: Ω → ℝ is a real-valued **random variable**. To say that *u* ∈ *L*^{1} is to say that it has a well-defined **expected value**

and to say that *u* ∈ *L*^{2} is to say that it has a well-defined **variance**

An important property of the *L*^{p} spaces is that they satisfy an **interpolation property**: whenever *u* lies in *L*^{p} and in *L*^{q}, it must also lie in *L*^{r} for every *r* between *p* and *q*, and, furthermore, the norms of *u* in these spaces satisfy the inequality

where the exponent *α* satisfies

So, for example, the space *L*^{2} is “between” the spaces *L*^{1} and *L*^{3}, with the “betweenness” quantified by the exponent *α* = ¼ and the estimate

**Weak L^{p} Spaces**

It might pass you by if you first encounter *L*^{p} spaces in a general measure theory course, but if you instead encounter them in a probability theory course, then you’re almost certain to prove Chebyshev’s inequality (or Markov’s inequality, if you prefer) that, whenever *u* ∈ *L*^{p} for some 0 < *p* < ∞ and *t* > 0,

The converse to this inequality is false, though: that is, it’s tempting to think that perhaps *u* ∈ *L*^{p} whenever

but this is actually a strictly weaker notion — and we call it being weakly *L*^{p}.

For 0 < *p* < ∞, define the weak *L*^{p} “norm” of *u*: Ω → ℝ by

and define the **weak L^{p} space**

*L*

^{p,∞}(Ω,

*μ*) to be the vector space of all measurable functions

*u*for which the corresponding “norm” is finite, with the usual identification of functions that are equal

*μ*-almost everywhere. The weak

*L*

^{∞}space

*L*

^{∞,∞}is defined to be the usual strong

*L*

^{∞}space with its usual norm. For finite

*p*,

*L*

^{p,∞}“norm” is only a quasi-norm: it satisfies the modified triangle inequality

Moreover, the weak *L*^{p} spaces are complete with respect to their corresponding quasi-norms: they are quasi-Banach spaces.

Hopefully, it’s not too surprising that, for every 0 < *p* < ∞ and every *u* ∈ *L*^{p},

from which it follows that *L*^{p} is continuously embedded in the larger space *L*^{p,∞}.

An important property of the weak spaces *L*^{p,∞} is that they generalize the interpolation property of the strong *L*^{p} spaces: whenever *u* lies in *L*^{p,∞} and in *L*^{q,∞}, it must also lie in *L*^{r} for every *r* between *p* and *q*, and the respective norms satisfy the inequality

where, as before, *α* satisfies

Even better, the constant *C*_{p,q,r} can be taken to be unity upon replacing the *L*^{r} norm by the weak *L*^{r} quasi-norm:

**Lorentz Spaces**

Whereas the strong and weak *L*^{p} spaces have their “strength” described by a single parameter *p*, the Lorentz spaces are described by two parameters *p* and *q*. For 0 < *p* < ∞ and 0 < *q* ≤ ∞, define the Lorentz (*p*, *q*) “norm” of *u*: Ω → ℝ by

and

The **Lorentz space** *L*^{p,q}(Ω, *μ*) is defined to be the vector space of all measurable functions *u* for which the corresponding “norm” is finite, with the usual identification of functions that are equal *μ*-almost everywhere. The reason for the quotation marks around the word “norm” is that this expression is not, in fact, a norm: the triangle inequality only holds up to a non-unit constant, and so these “norms” are in fact quasi-norms.

Also, the Lorentz spaces are complete with respect to their corresponding quasi-norms: they are quasi-Banach spaces.

A nice property of the Lorentz spaces is that they embed into one another in a nice way: there is a continuous embedding of *L*^{p,q} into *L*^{p,r} when *q* < *r*:

Also, the Lorentz spaces include the weak *L*^{p} spaces as a special case, as suggested by the notation used earlier: the Lorentz space *L*^{p,∞} is exactly the weak *L*^{p} space *L*^{p,∞} of the previous section.

Interpolation between Lorentz spaces is… interesting. I don’t want to spoil the surprise yet, but it will turn out that once the machinery of interpolation spaces and the Reiteration Theorem have been introduced, the interpolants of Lorentz spaces are very easy to work out in terms of Lebesgue *L*^{p} spaces.

**Orlicz Spaces**

Orlicz spaces take a different approach to assessing the integrability of a function. Whereas the Lebesgue and Lorentz spaces use a numerical quantifier of integrability (the indices *p* and *q*), Orlicz spaces use a qualitative measure, a Young function.

To get the idea, it helps to recall a less well-known form of Chebyshev’s inequality: whenever Φ is a non-negative extended-real-valued function that is (weakly) increasing on the range of |*u*|,

This suggests that the integral on the right-hand side could be used as a “Φ-norm” of *u*in the same way as the original Chebyshev inequality was rearranged to define the weak *L*^{p} quasi-norm. To make this idea work properly, we need to use Young functions.

A function Φ: [0, ∞) → [0, ∞] is called a **Young function** if it is convex, lower semi-continuous, and Φ(0) = 0. It is called **non-trivial** if it is neither the zero function, nor the function that is 0 at 0 and ∞ elsewhere. It is called **finite** if it does not take the value ∞, and is called **strict** if it is finite and

Given a Young function Φ, define the *L*^{Φ} **Luxemburg norm** of a measurable function *u*: Ω → ℝ by

The **(large) Orlicz space** *L*^{Φ}(Ω, *μ*) is the vector space of all measurable functions *u* for which this quantity is finite; under the usual identification of functions that agree *μ*-almost everywhere, the Luxemburg norm is a norm on *L*^{Φ}. Indeed, with this norm, *L*^{Φ} is a Banach space.

The standard *L*^{p} spaces, for 1 ≤ *p* < ∞, are Orlicz spaces for Φ_{p}(*t*) := |*t*|^{p} ⁄ *p* with

The space *L*^{∞} is obtained by taking

Other notable choices of Φ include Φ(*t*) := exp(*t*) − 1 and Φ(*t*) := *t* log_{+}(*t*). Also, certain Sobolev embedding theorems are easily stated using Orlicz spaces: for example, **Trudinger’s inequality** shows that for an open, bounded Lipschitz domain Ω ⊆ ℝ^{n}, the Sobolev space *W*_{0}^{1,n}(Ω) embeds continuously into the Orlicz space *L*^{Φ}(Ω) with

Two very useful relations satisfied by the norm in *L*^{Φ} are that

and, in particular,

Orlicz spaces satisfy the following Hölder-type inequality:

where Φ^{∗} denotes the convex conjugate of Φ:

Another norm on *L*^{Φ} is the **Orlicz norm** defined by

or, which is the same thing

The Luxemburg and Orlicz norms are equivalent: they satisfy

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