Strong and weak Lebesgue spaces, Lorentz spaces, and Orlicz spaces

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 Lp space such as the weak Lp 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 ℝn 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 Lp spaces:

Recap: Strong Lp Spaces

For 0 < p < ∞, define the Lp “norm” of u: Ω → ℝ by

$\displaystyle \| u \|_{L^{p}} := \left( \int_{\Omega} |u(x)|^{p} \, \mathrm{d} \mu(x) \right)^{1/p}$

and define

$\displaystyle \| u \|_{L^{\infty}} := \mathop{\mu\mathrm{\,ess\,sup}}_{x \in \Omega} |u(x)| = \inf \{ B > 0 \mid \mu[|u| > B] = 0 \}.$

(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 Lp(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 Lp “norm” is a bona fide norm, and Lp 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:

$\displaystyle \| u + v \|_{L^{p}} \leq 2^{(1-p)/p} \bigl( \| u \|_{L^{p, q}} + \| v \|_{L^{p, q}} \bigr) .$

However, Lp is still complete with respect to this quasi-norm, and so Lp is a quasi-Banach space.

Lp 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 ∈ L1 is to say that it has a well-defined expected value

$\displaystyle \mathbb{E}[u] := \int_{\Omega} u(x) \, \mathrm{d} \mu(x)$

and to say that u ∈ L2 is to say that it has a well-defined variance

$\displaystyle \mathbb{V}[u] := \mathbb{E}[|u - \mathbb{E}[u]|^{2}].$

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

$\displaystyle \| u \|_{L^{r}} \leq \| u \|_{L^{p}}^{\alpha} \| u \|_{L^{q}}^{1 - \alpha},$

where the exponent α satisfies

$\displaystyle \frac{1}{r} = \frac{\alpha}{p} + \frac{1 - \alpha}{q}.$

So, for example, the space L2 is “between” the spaces L1 and L3, with the “betweenness” quantified by the exponent α = ¼ and the estimate

$\displaystyle \| u \|_{L^{2}} \leq \| u \|_{L^{1}}^{1/4} \| u \|_{L^{3}}^{3/4}.$

Weak Lp Spaces

It might pass you by if you first encounter Lp 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 ∈ Lp for some 0 < p < ∞ and t > 0,

$\displaystyle \mu[|u| > t] \leq \dfrac{\| u \|_{L^{p}}^{p}}{t^{p}}$

The converse to this inequality is false, though: that is, it’s tempting to think that perhaps u ∈ Lp whenever

$\displaystyle \sup_{t > 0} t^{p} \mu[|u| > t] < \infty,$

but this is actually a strictly weaker notion — and we call it being weakly Lp.

For 0 < p < ∞, define the weak Lp “norm” of u: Ω → ℝ by

$\displaystyle \| u \|_{L^{p,\infty}} := \sup_{t > 0} t \mu[|u| > t]^{1/p}$

and define the weak Lp space Lp,∞(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, Lp,∞ “norm” is only a quasi-norm: it satisfies the modified triangle inequality

$\displaystyle \| u + v \|_{L^{p, \infty}} \leq \max (2, 2^{1/p}) \bigl( \| u \|_{L^{p, \infty}} + \| v \|_{L^{p, \infty}} \bigr) .$

Moreover, the weak Lp 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 ∈ Lp,

$\displaystyle \| u \|_{L^{p,\infty}} \leq \| u \|_{L^{p}}$

from which it follows that Lp is continuously embedded in the larger space Lp,∞.

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

$\displaystyle \| u \|_{L^{r}} \leq C_{p,q,r} \| u \|_{L^{p,\infty}}^{\alpha} \| u \|_{L^{q,\infty}}^{1 - \alpha},$

where, as before, α satisfies

$\displaystyle \frac{1}{r} = \frac{\alpha}{p} + \frac{1 - \alpha}{q}.$

Even better, the constant Cp,q,r can be taken to be unity upon replacing the Lr norm by the weak Lr quasi-norm:

$\displaystyle \| u \|_{L^{r, \infty}} \leq \| u \|_{L^{p,\infty}}^{\alpha} \| u \|_{L^{q,\infty}}^{1 - \alpha}.$

Lorentz Spaces

Whereas the strong and weak Lp 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 (pq) “norm” of u: Ω → ℝ by

$\displaystyle \| u \|_{L^{p, q}} := p^{1/q} \bigl\| t \mu[ |u| > t ]^{1/p} \bigr\|_{L^{q}(\mathbb{R}_{+}, t^{-1} \mathrm{d} t)}$

and

$\displaystyle \| u \|_{L^{\infty, \infty}} := \| u \|_{L^{\infty}}.$

The Lorentz space Lp,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.

$\displaystyle \| u + v \|_{L^{p, q}} \leq 2^{1/p} \max (1, 2^{(1 - q)/q}) \bigl( \| u \|_{L^{p, q}} + \| v \|_{L^{p, q}} \bigr) .$

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 Lp,q into Lp,r when q < r:

$\displaystyle \| u \|_{L^{p, r}} \leq C_{p,q,r} \| u \|_{L^{p, q}} .$

Also, the Lorentz spaces include the weak Lp spaces as a special case, as suggested by the notation used earlier: the Lorentz space Lp,∞ is exactly the weak Lp space Lp,∞ 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 Lp 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|,

$\displaystyle \mu[|u| > t] \leq \frac1{\Phi(t)} \int_{\Omega} \Phi(u(x)) \, \mathrm{d} \mu (x).$

This suggests that the integral on the right-hand side could be used as a “Φ-norm” of uin the same way as the original Chebyshev inequality was rearranged to define the weak Lp 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

$\displaystyle \lim_{t \to \infty} \frac{\Phi(t)}{t} = \infty.$

Given a Young function Φ, define the LΦ Luxemburg norm of a measurable function u: Ω → ℝ by

$\displaystyle \| u \|_{L^{\Phi}} := \inf \left\{ t > 0 \,\middle|\, \int_{\Omega} \Phi(|u(x)| / t) \, \mathrm{d} \mu(x) \leq 1 \right\}.$

The (large) Orlicz space LΦ(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 LΦ. Indeed, with this norm, LΦ is a Banach space.

The standard Lp spaces, for 1 ≤ p < ∞, are Orlicz spaces for Φp(t) := |t|p ⁄ p with

$\displaystyle p^{1/p} \| u \|_{L^{\Phi_{p}}} = \| u \|_{L^{p}}.$

The space L is obtained by taking

$\displaystyle \Phi_{\infty}(t) := \begin{cases} 0, & 0 \leq t \leq 1 \\ \infty, & 1 < t < \infty. \end{cases}$

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 ⊆ ℝn, the Sobolev space W01,n(Ω) embeds continuously into the Orlicz space LΦ(Ω) with

$\displaystyle \Phi(t) = \exp \bigl( t^{n / (n - 1)} \bigr) - 1.$

Two very useful relations satisfied by the norm in LΦ are that

$\displaystyle 0 < \| u \|_{L^{\Phi}} < \infty \implies \int_{\Omega} \Phi \bigl( u / \| u \|_{L^{\Phi}} \bigr) \, \mathrm{d} \mu \leq 1$

and, in particular,

$\displaystyle \| u \|_{L^{\Phi}} \leq 1 \iff \int_{\Omega} \Phi(u) \, \mathrm{d} \mu \leq 1.$

Orlicz spaces satisfy the following Hölder-type inequality:

$\displaystyle \| u v \|_{L^{1}} \leq 2 \| u \|_{L^{\Phi}} \| v \|_{L^{\Phi^{\ast}}},$

where Φ denotes the convex conjugate of Φ:

$\displaystyle \Phi^{\ast}(s) := \sup \{ st - \Phi(t) \mid t \geq 0 \},$

Another norm on LΦ is the Orlicz norm defined by

$\displaystyle | u |_{L^{\Phi}} := \sup \left\{ \int_{\Omega} u v \, \mathrm{d} \mu \,\middle|\, v \in L^{\Phi^{\ast}}, \| v \|_{L^{\Phi^{\ast}}} \leq 1 \right\}$

or, which is the same thing

$\displaystyle | u |_{L^{\Phi}} := \sup \left\{ \int_{\Omega} u v \, \mathrm{d} \mu \,\middle|\, v \in L^{\Phi^{\ast}}, \int_{\Omega} \Phi^{\ast} \circ v \, \mathrm{d} \mu \leq 1 \right\}.$

The Luxemburg and Orlicz norms are equivalent: they satisfy

$\displaystyle \| u \|_{L^{\Phi}} \leq | u |_{L^{\Phi}} \leq 2 \| u \|_{L^{\Phi}}.$