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: ℝn → ℝ:
where the Lr norm is, as usual,
for finite r, and
and the exponent α satisfies
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
A natural question to ask is whether we could in fact characterize L2 as being precisely the space of all u that are “between” the spaces L1 and L3 in the sense that
or at least remains bounded along some approximating sequence un → u. More exotically, could we construct a space of functions that are “half-differentiable” by finding a space somewhere between the space C0 of continuous functions and the space C1 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.
Consider two (real or complex) Banach spaces X0 and X1, both of which embed continuously into a common Hausdorff topological vector space Z. (That is, the inclusion maps ιθ: Xθ → Z for θ = 0, 1 are continuous linear maps.) For brevity, call such a pair (X0, X1) a compatible pair. The aim of interpolation is to construct, for each θ in the interval 0 ≤ θ ≤ 1, a space Xθ that somehow “lies between” their intersection X0 ∩ X1, which is equipped with the norm
and their direct sum X0 + X1, which is equipped with the norm
There are several methods of interpolation, but this post will concentrate on the real K-method. For u ∈ X0 + X1 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 (X0, X1)θ,q to be the vector space of all u ∈ X0 + X1 for which ‖u‖θ,q is finite. We also define the continuous interpolation space (X0, X1)θ to be the vector space of all u ∈ X0 + X1 for which
For all 0 < θ < 1 and 1 < q ≤ ∞, the interpolation space (X0, X1)θ,q is a Banach space with norm‖ · ‖θ,q; the continuous interpolation space (X0, X1)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 < q1 ≤ q2 ≤ ∞,
Note also that the order of the two spaces X0 and X1 is important:
and similarly for the continuous interpolation spaces.
Furthermore, the norms on the intermediate spaces satisfy the interpolation inequality
for some constant Cθ,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 X1 ⊆ X0. In this case, whenever 0 < θ1 < θ2 < 1, it follows that
and hence that, for all p, q ∈ [1, ∞],
Spaces of Fractionally-Differentiable Functions
Strong Derivatives. Fix a compact set K ⊆ ℝn 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 C0(K) of continuous functions on K, again with the supremum norm. Secondly, for 0 < α ≤ 1, consider the space Cα(K) of α-Hölder functions on K, equipped with the norm
The space C1(K) is the space of continuously differentiable functions on K. It seems intuitively reasonable to say that C1⁄2(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 Lp(ℝn) and the Sobolev space W1,p(ℝn) of functions that are both in Lp and have a weak derivative in Lp. For 0 < α < 1, define Wα,p(ℝn) 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 Lp,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 Lp spaces in the sense that Lp,p = Lp.
When (Ω, ℱ, μ) is a σ-finite measure space, the pair (L1(Ω), L∞(Ω)) is a compatible pair and the interpolation spaces between L1 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 (X0, X1) be a compatible pair of Banach spaces, 0 ≤ θ0 < θ1 ≤ 1, and 1 ≤ q0, q1 ≤ ∞. 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:
Coming full circle, by choosing the right exponents (remember that Lp,p = Lp), it follows that the Lebesgue Lp spaces, for 1 < p < ∞, are also each other’s interpolation spaces, as suggested by the inequality
at the top of the page.