I have recently developed an interest in interpolation inequalities and the formal structure of interpolation spaces. Interpolation inequalities arise in mathematics when one controls the norm of a function *u* in one norm by the product of two other norms of the same function (or closely related functions like the derivatives of *u*). A classic example is the following interpolation inequality for *u*: ℝ^{n} → ℝ:

where the *L*^{r} norm is, as usual,

for finite *r*, and

and 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

A natural question to ask is whether we could in fact *characterize* *L*^{2} as being *precisely* the space of all *u* that are “between” the spaces *L*^{1} and *L*^{3} in the sense that

or at least remains bounded along some approximating sequence *u*_{n} → *u*. More exotically, could we construct a space of functions that are “half-differentiable” by finding a space somewhere between the space *C*^{0} of continuous functions and the space *C*^{1} of continuously differentiable functions? A natural candidate for such a space is the space of Hölder continuous functions, but this is little more than an educated guess — can we put it on a sounder footing?

The brief notes that follow owe a lot to some more detailed notes by Alessandra Lunardi.

**General Theory**

Consider two (real or complex) Banach spaces *X*_{0} and *X*_{1}, both of which embed continuously into a common Hausdorff topological vector space *Z*. (That is, the inclusion maps *ι*_{θ}: *X*_{θ} → *Z* for *θ* = 0, 1 are continuous linear maps.) For brevity, call such a pair (*X*_{0}, *X*_{1}) a **compatible pair**. The aim of interpolation is to construct, for each *θ* in the interval 0 ≤ *θ* ≤ 1, a space *X*_{θ} that somehow “lies between” their intersection *X*_{0} ∩ *X*_{1}, which is equipped with the norm

and their direct sum *X*_{0} + *X*_{1}, which is equipped with the norm

There are several methods of interpolation, but this post will concentrate on the **real K-method**. For

*u*∈

*X*

_{0}+

*X*

_{1}and

*t*∈ ℝ, define

Note that *K*(*u*, 1) = ‖*u*‖_{X0+X1}. Now, for 0 < *θ* < 1 and 0 < *q* ≤ ∞, define a “norm” ‖ · ‖_{θ,q}: *Z* → ℝ by

(to get some more motivation for this perhaps weird-looking norm, see the definition of Lorentz spaces in this earlier post) and define the **interpolation space** (*X*_{0}, *X*_{1})_{θ,q} to be the vector space of all *u* ∈ *X*_{0} + *X*_{1} for which ‖*u*‖_{θ,q} is finite. We also define the **continuous interpolation space** (*X*_{0}, *X*_{1})_{θ} to be the vector space of all *u* ∈ *X*_{0} + *X*_{1} for which

For all 0 < *θ* < 1 and 1 < *q* ≤ ∞, the interpolation space (*X*_{0}, *X*_{1})_{θ,q} is a Banach space with norm‖ · ‖_{θ,q}; the continuous interpolation space (*X*_{0}, *X*_{1})_{θ} is a Banach space with norm ‖ · ‖_{θ,∞}.

Having defined these spaces, it’s nice to know that they do indeed lie in between the spaces that we started off with: whenever for 0 < *θ* < 1 and 1 < *q*_{1} ≤ *q*_{2} ≤ ∞,

Note also that the order of the two spaces *X*_{0} and *X*_{1} is important:

and similarly for the continuous interpolation spaces.

Furthermore, the norms on the intermediate spaces satisfy the **interpolation inequality**

for some constant *C*_{θ,q} depending only on *θ* and *q*.

An important special case, relevant to our search for a space of half-differentiable functions, is that in which *X*_{1} ⊆ *X*_{0}. In this case, whenever 0 < *θ*_{1} < *θ*_{2} < 1, it follows that

and hence that, for all *p*, *q* ∈ [1, ∞],

**Spaces of Fractionally-Differentiable Functions**

**Strong Derivatives.** Fix a compact set *K* ⊆ ℝ^{n} and consider the following subspaces of *B*(*K*), the Banach space of bounded functions *u*: *K* → ℝ equipped with the supremum norm ‖ · ‖_{∞}. First, consider the space *C*^{0}(*K*) of continuous functions on *K*, again with the supremum norm. Secondly, for 0 < *α* ≤ 1, consider the space *C*^{α}(*K*) of *α*-Hölder functions on *K*, equipped with the norm

The space *C*^{1}(*K*) is the space of continuously differentiable functions on *K*. It seems intuitively reasonable to say that *C*^{1⁄2}(*K*) is a space of half-differentiable functions. This statement can be made rigorous by the following result: for every 0 < *θ* < 1,

and the norms on the respective spaces are equivalent.

**Weak Derivatives.** There is a very similar result for interpolation between the Lebesgue space *L*^{p}(ℝ^{n}) and the Sobolev space *W*^{1,p}(ℝ^{n}) of functions that are both in *L*^{p} and have a weak derivative in *L*^{p}. For 0 < *α* < 1, define *W*^{α,p}(ℝ^{n}) to be the vector space of functions *u* for which the norm

is finite. Then, for every 0 < *θ* < 1,

and the norms on the respective spaces are equivalent. Note also that these spaces are Hilbert spaces when *p* = 2.

**Lorentz Spaces and Reiteration**

Now consider a measure space (Ω, ℱ, *μ*). For 0 < *p* < ∞ and 0 < *q* ≤ ∞, the **Lorentz space** *L*^{p,q}(Ω, *μ*) is defined to be the vector space of all measurable functions *u* for which the “norm”

is finite. The Lorentz spaces are quasi-Banach spaces (the triangle inequality only holds up to a non-unit multiplicative constant) that generalize the Lebesgue *L*^{p} spaces in the sense that *L*^{p,p} = *L*^{p}.

When (Ω, ℱ, *μ*) is a *σ*-finite measure space, the pair (*L*^{1}(Ω), *L*^{∞}(Ω)) is a compatible pair and the interpolation spaces between *L*^{1} and *L*^{∞} are Lorentz spaces:

The natural next question is: what are the interpolation spaces between Lorentz spaces? More generally: what are the interpolation spaces of interpolation spaces? The answer is given by the following theorem:

**Reiteration Theorem.** Let (*X*_{0}, *X*_{1}) be a compatible pair of Banach spaces, 0 ≤ *θ*_{0} < *θ*_{1} ≤ 1, and 1 ≤ *q*_{0}, *q*_{1} ≤ ∞. Let

Then, for all 0 < *θ* < 1 and 1 ≤ *q* ≤ ∞,

It follows from the Reiteration Theorem that the interpolation spaces of Lorentz spaces are themselves Lorentz spaces:

where

Coming full circle, by choosing the right exponents (remember that *L*^{p,p} = *L*^{p}), it follows that the Lebesgue *L*^{p} spaces, for 1 < *p* < ∞, are also each other’s interpolation spaces, as suggested by the inequality

at the top of the page.

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