Interpolation inequalities, interpolation spaces and fractional differentiability

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:

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

where the Lr norm is, as usual,

\displaystyle \| u \|_{L^{r}} := \left( \int_{\mathbb{R}^{n}} |u(x)|^{r} \, \mathrm{d} x \right)^{1/r}

for finite r, and

\displaystyle \| u \|_{L^{\infty}} := \inf \bigl\{ B > 0 \,\big|\, \mu \{ x \mid | u(x) | > B \} \bigr\}

and 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}.

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

\displaystyle \| u \|_{L^{1}}^{1/4} \| u \|_{L^{3}}^{3/4} < \infty,

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.

General Theory

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 ιθXθ → 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

\displaystyle \| u \|_{X_{0} \cap X_{1}} := \max \{ \| u \|_{X_{0}}, \| u \|_{X_{1}} \}

and their direct sum X0 + X1, which is equipped with the norm

\displaystyle \| u \|_{X_{0} + X_{1}} := \inf \left\{ \| u_{0} \|_{X_{0}} + \| u_{1} \|_{X_{1}} \,\middle|\, \begin{array}{c} u = u_{0} + u_{1} \\ u_{0} \in X_{0}, u_{1} \in X_{1} \end{array} \right\}.

There are several methods of interpolation, but this post will concentrate on the real K-method. For u ∈ X0 + X1 and t ∈ ℝ, define

\displaystyle K(u, t) := \inf \left\{ \| u_{0} \|_{X_{0}} + t \| u_{1} \|_{X_{1}} \,\middle|\, \begin{array}{c} u = u_{0} + u_{1} \\ u_{0} \in X_{0}, u_{1} \in X_{1} \end{array} \right\}

Note that K(u, 1) = ‖uX0+X1. Now, for 0 < θ < 1 and 0 < q ≤ ∞, define a “norm” ‖ · ‖θ,qZ → ℝ by

\displaystyle \| u \|_{\theta, q} := \bigl\| t^{-\theta} K(u, t) \bigr\|_{L^{q}(\mathbb{R}_{+}, t^{-1} \, \mathrm{d} t)}

(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 (X0X1)θ,q to be the vector space of all u ∈ X0 + X1 for which ‖uθ,q is finite. We also define the continuous interpolation space (X0X1)θ to be the vector space of all u ∈ X0 + X1 for which

\displaystyle \lim_{t \searrow 0} t^{-\theta} K(u, t) = \lim_{t \nearrow \infty} t^{-\theta} K(u, t) = 0.

For all 0 < θ < 1 and 1 < q ≤ ∞, the interpolation space (X0X1)θ,q is a Banach space with norm‖ · ‖θ,q; the continuous interpolation space (X0X1)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 ≤ ∞,

\displaystyle X_{0} \cap X_{1} \subseteq (X_{0}, X_{1})_{\theta, q_{1}} \subseteq (X_{0}, X_{1})_{\theta, q_{2}} \subseteq (X_{0}, X_{1})_{\theta} \subseteq (X_{0}, X_{1})_{\theta, \infty} \subseteq X_{0} + X_{1}.

Note also that the order of the two spaces X0 and X1 is important:

\displaystyle (X_{0}, X_{1})_{\theta, q} = (X_{1}, X_{0})_{1 - \theta, q}

and similarly for the continuous interpolation spaces.

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

\displaystyle \| u \|_{\theta, q} \leq C_{\theta, q} \| u \|_{X_{0}}^{1 - \theta} \| u \|_{X_{1}}^{\theta}

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

\displaystyle (X_{0}, X_{1})_{\theta_{2}, \infty} \subseteq (X_{0}, X_{1})_{\theta_{1}, 1}

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

\displaystyle (X_{0}, X_{1})_{\theta_{2}, p} \subseteq (X_{0}, X_{1})_{\theta_{1}, q}.

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 uK → 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

\displaystyle \| u \|_{C^{\alpha}} := \| u \|_{\infty} + \sup_{\substack{ x, y \in K \\ x \neq y }}  \frac{| u(x) - u(y) |}{| x - y |^{\alpha}}.

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,

\displaystyle \bigl( C^{0}(K), C^{1}(K) \bigr)_{\theta, \infty} = C^{\theta}(K),

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

\displaystyle \left( \| u \|_{L^{p}}^{p} + \iint_{\mathbb{R}^{n}} \frac{| u(x) - u(y) |^{p}}{| x - y |^{\alpha p + n}} \, \mathrm{d}x \mathrm{d}y \right)^{1/p}

is finite. Then, for every 0 < θ < 1,

\displaystyle \bigl( L^{p}(\mathbb{R}^{n}), W^{1,p}(\mathbb{R}^{n}) \bigr)_{\theta, p} = W^{\theta,p}(\mathbb{R}^{n}),

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”

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

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:

\displaystyle \bigl( L^{1}(\Omega, \mu), L^{\infty}(\Omega, \mu) \bigr)_{\theta, p} = L^{1/(1-\theta),p}(\Omega, \mu).

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

\displaystyle Y_{0} := (X_{0}, X_{1})_{\theta_{0}, q_{0}},
\displaystyle Y_{1} := (X_{0}, X_{1})_{\theta_{1}, q_{1}}.

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

\displaystyle (Y_{0}, Y_{1})_{\theta, q} = (X_{0}, X_{1})_{(1 - \theta) \theta_{0} + \theta \theta_{1}, q}.

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

\displaystyle (L^{p_{0},q_{0}}, L^{p_{1},q_{1}})_{\theta, q} = L^{p,q},

where

\displaystyle p = \frac{p_{0} p_{1}}{\theta p_{0} + (1 - \theta) p_{1}}.

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

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

at the top of the page.

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