Interpolation and fractional differentiability revisited

In this earlier post on interpolation spaces, part of the motivation for studying interpolation spaces was the search for a reasonable space of functions with a non-integer order of differentiability 0 < α < 1. In the case of strong (classical) derivatives, a suitable such space was the vector space Cα(K) of α-Hölder functions on a compact set K ⊆ ℝn with interior, i.e. the set of functions uK → ℝ for which the norm

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

is finite. In the case of weak derivatives, a suitable such space was the vector space Wα,p(K) of functions u for which the norm

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

is finite. These spaces are all Banach spaces, and interpolate in the sense of real K-interpolation between the spaces C0(K) of continuous functions and C1(K) of continuously differentiable functions (respectively the Lebesgue space Lp(K) and the Sobolev space W1,p(K). This post grew out of my noticing one simple omission in the previous post, now corrected: for p = 2, the spaces Wα,2 are Hilbert spaces under the inner product

\displaystyle (u, v)_{W^{\alpha, 2}(K)} := \int_{K} u(x) v(x) \, \mathrm{d} x + \iint_{K} \frac{( u(x) - u(y)) (v(x) - v(y))}{| x - y |^{2 \alpha + n}} \, \mathrm{d}x \mathrm{d}y .

On realizing this omission, I started to think more deeply about other notions of fractional differentiability. In particular, I wondered how the above Wα,p spaces are related to other fractional-order Sobolev spaces defined using Fourier transforms. So, the rest of this post is devoted to surveying the two main methods of constructing fractional-order Sobolev spaces and the relationships between them.

For neatly self-contained proofs of the assertions in this post, I recommend this set of notes by Eleonora Di Nezza, Giampiero Palatucci and Enrico Valdinoci.

The Aronszajn–Gagliardo–Slobodeckij Construction

Start by fixing a measurable domain Ω ⊆ ℝn; regularity of the domain is not needed for the definitions, although it will come in handy later on when proving embedding theorems. It helps to define s-times differentiable functions first for 0 < s < 1, and extend to s > 1 later. Given 0 < s < 1 and 1 ≤ p < ∞, define the Gagliardo semi-norm of a function u: Ω → ℝ by

\displaystyle [ u ]_{s, p} := \left( \iint_{\Omega} \frac{|u(x) - u(y)|^{p}}{|x+y|^{n + sp}} \, \mathrm{d}x\mathrm{d}y \right)^{1/p}.

Now define a norm by

\displaystyle \| u \|_{W^{s, p}(\Omega)} := \left( \| u \|_{L^{p}(\Omega)}^{p} + [u]_{s,p}^{p} \right)^{1/p}

and define the (Aronszajn–Gagliardo–Slobodeckij) fractional Sobolev space Ws,p(Ω) to be

\displaystyle W^{s, p}(\Omega) := \bigl\{ u \colon \Omega \to \mathbb{R} \,\big|\, \| u \|_{W^{s, p}(\Omega)} < \infty \bigr\},

with the usual identification of functions that agree almost everywhere. The spaces Ws,p(Ω) are Banach spaces, and the spaces Hs(Ω) ≔ Ws,2(Ω) are Hilbert spaces when given the inner product

\displaystyle (u, v)_{W^{s, 2}(\Omega)} := \int_{\Omega} u(x) v(x) \, \mathrm{d} x + \iint_{\Omega} \frac{( u(x) - u(y)) (v(x) - v(y))}{| x - y |^{2 \alpha + n}} \, \mathrm{d}x \mathrm{d}y .

Hopefully needless to say, when s is an integer, this coincides (up to equivalence of norms) with the usual definition of the Sobolev space Ws,p(Ω) as the space of all functions u having weak derivatives of all orders ≤ s, with all of them lying in Lp(Ω).

When the domain Ω is an open set, the “obvious” continuous embeddings with respect to s hold true: whenever 1 ≤ p < ∞ and 0 < s0s1 < 1, there is a constant C ≥ 1 such that

\displaystyle \| u \|_{W^{s_{0}, p}(\Omega)} \leq C \| u \|_{W^{s_{1}, p}(\Omega)},

so that Ws1,p(Ω) embeds continuously into Ws0,p(Ω). The same holds true in the limiting case s1 = 1 provided that Ω is a bounded domain with Lipschitz boundary.

To extend this discussion to orders of differentiability s > 1, it is not enough to simply take s > 1 in the above formulae: it turns out that if s > 1 and [u]s,p is finite, then u is constant on the connected components of Ω, i.e. u is boring. To get around this, we employ the simple expedient of differentiating in the usual way as many times as we can, and then applying the previous idea.

For s > 1, let m ≔ ⌊s⌋ (the greatest integer less than or equal to s) and σsm, and define

\displaystyle \| u \|_{W^{s, p}(\Omega)} := \left( \| u \|_{W^{m,p}(\Omega)}^{p} + \sum_{|\alpha| = m} \| \mathrm{D}^{\alpha} u \|_{W^{\sigma,p}(\Omega)}^{p} \right)^{1/p}

and define the (Aronszajn–Gagliardo–Slobodeckij) fractional Sobolev space Ws,p(Ω) to be

\displaystyle W^{s, p}(\Omega) := \bigl\{ u \colon \Omega \to \mathbb{R} \,\big|\, \| u \|_{W^{s, p}(\Omega)} < \infty \bigr\},

with the usual identification of functions that agree almost everywhere. As before, these are Banach spaces, and the spaces Hs(Ω) ≔ Ws,2(Ω) are Hilbert spaces.

When the domain Ω is an open set with Lipschitz boundary, the “obvious” continuous embeddings with respect to s hold true: whenever 1 ≤ p < ∞ and 0 < s0s1 < ∞, there is a constant C ≥ 1 such that

\displaystyle \| u \|_{W^{s_{0}, p}(\Omega)} \leq C \| u \|_{W^{s_{1}, p}(\Omega)},

so that Ws1,p(Ω) embeds continuously into Ws0,p(Ω).

As would be expected from the integral s case the smooth functions of compact support are dense in Ws,p(ℝn) with respect to its norm. For a domain Ω, the closure of the set of smooth functions of compact support in Ω is in general not the whole of Ws,p(Ω) — instead, that closure is denoted W0s,p(Ω), and it can be thought of as those functions in Ws,p(Ω) that satisfy zero boundary conditions on the boundary ∂Ω.

The spaces Ws,p(Ω) are (up to equivalence of norms) exactly the same spaces as one obtains by interpolating between the spaces Wk,p(Ω), where k is an integer, using the real K-method of interpolation:

\displaystyle W^{s,p}(\Omega) = \bigl( W^{k,p}(\Omega), W^{k+1,p}(\Omega) \bigr)_{\theta, p},

where k < s < k + 1 and θ = s − ⌊s⌋.

Before moving on to the alternative Bessel potential construction, it’s worth noting the relationship between fractional Sobolev spaces and traces — if only because, historically speaking, the weird differentiability properties of traces are what motivated the study of interpolation theory and fractional Sobolev spaces. Simply put, traces are the proper way to restrict Sobolev functions to lower-dimensional sets, most notably the boundary ∂Ω of the domain Ω. As it turns out, the trace Tu on ∂Ω of u ∈ W1,p(Ω) lies in W1−1⁄p,p(∂Ω), and in fact the trace operator TW1,p(Ω) → W1−1⁄p,p(∂Ω) is surjective. So, if you want to understand the degree of differentiability that a Sobolev function on Ω has on the boundary ∂Ω, then there really is no option but to study fractional Sobolev spaces!

The Bessel Potential Construction

An alternative construction of non-integer order Sobolev spaces is based on the following observation: for any smooth enough function u: ℝn → ℝ, its Fourier transform ℱu and the Fourier transform of its αth derivative (where α is a multi-index) are related by

\displaystyle \bigl( \mathcal{F} (\mathrm{D}^{\alpha} u) \bigr) (\xi) = (i \xi)^{\alpha} (\mathcal{F} u)(\xi).

Hence, if u is k times differentiable and all its kth order derivatives are in L2 (and so can have the Fourier transform applied to them), then

\displaystyle \int_{\mathbb{R}^{n}} |\xi|^{2k} |(\mathcal{F} u) (\xi)|^{2} \, \mathrm{d} \xi < \infty.

This suggests defining, for s ≥ 0,

\displaystyle \widehat{H}^{s}(\mathbb{R}^{n}) := \left\{ u \in L^{2}(\mathbb{R}^{n}) \,\middle|\, \int_{\mathbb{R}^{n}} (1 + |\xi|^{2s}) | (\mathcal{F}u)(\xi) |^{2} \, \mathrm{d} x < \infty \right\}.

More generally, the (Bessel potential) fractional Sobolev space is defined by

\displaystyle \widehat{W}^{s,p}(\mathbb{R}^n) := \left \{ u \in L^{p}(\mathbb{R}^n) \,\middle|\, \mathcal{F}^{-1} (1+|\xi|^2)^{s/2} \mathcal{F} u \in L^{p}(\mathbb{R}^n) \right\}

with the Banach norm

\displaystyle \| u \|_{\widehat{W}^{s,p}(\mathbb{R}^n)} := \bigl\| \mathcal{F}^{-1} (1+|\xi|^2)^{s/2} \mathcal{F} u \bigr\|_{L^{p}(\mathbb{R}^n)}.

As usual, when p = 2, these are Hilbert spaces.

The spaces \widehat{W}^{s,p}(\mathbb{R}^n) are (up to equivalence of norms) exactly the same spaces as one obtains by interpolating between the spaces Wk,p(Ω), where k is an integer, using complex interpolation:

\displaystyle \widehat{W}^{s,p}(\Omega) = \bigl[ W^{k,p}(\Omega), W^{k+1,p}(\Omega) \bigr]_{\theta},

where 0 < θ < 1 and

\displaystyle s = (1-\theta)k + \theta(k + 1).

At this point a rather awkward technicality rears its ugly head. Up to equivalence of norms, it is true that

W^{s,2}(\Omega) = \widehat{W}^{s,2}(\Omega).

However, for p ≠ 2, we have

W^{s,p}(\Omega) \neq \widehat{W}^{s,p}(\Omega).

The reason for this essentially lies in fundamental properties of the Fourier transform. The equivalence in the case p = 2 uses the Plancherel formula. However, except for the case p = q = 2, the Fourier transform does not take one back and forth between the spaces in a dual pair Lp and Lq, and this is enough to make the two approaches to fractional Sobolev spaces yield distinct results for p ≠ 2.

Sobolev Inequalities and Embeddings

Aside from the fairly obvious monotonicity relationships with respect to s, it is of interest to know how the spaces Ws,p(Ω) are related to other function spaces. An important notion in the results is that of an extension domain: a domain Ω ⊆ ℝn is called an extension domain if there is a constant C for which every u ∈ Ws,p(Ω) admits an extension to Eu ∈ Ws,p(ℝn) such that

\displaystyle (Eu)(x) = u(x) \quad \mbox{for all } x \in \Omega,
\displaystyle \| Eu \|_{W^{s,p}(\mathbb{R}^{n})} \leq C \| u \|_{W^{s, p}(\Omega)} .

Intuitively speaking, smooth domains Ω are likely to be extension domains with a moderate value of C. If Ω is highly irregular, then C is likely to be large, or Ω may fail to be an extension domain at all.

Sub-Critical Embedding. Let 0 < s < 1 and 1 ≤ p < ∞ be such that sp < n. Then there is a constant C such that, for every compactly supported and measurable function u,

\displaystyle \| u \|_{L^{p^{\ast}}(\mathbb{R}^{n})} \leq C \| u \|_{W^{s, p}(\mathbb{R}^{n})},
\displaystyle p^{\ast} := \dfrac{np}{n - sp}.

Hence, the Sobolev space Ws,p(ℝn) is continuously embedded in Lq(ℝn) for every pqp. The same holds true with any extension domain Ω in place of ℝn; furthermore, for a bounded extension domain Ω, we get continuous embedding of Ws,p(Ω) into Lq(Ω) for every 1 ≤ qp.

We also have a compact embedding of Ws,p(Ω) into Lq(Ω) for every 1 ≤ q < p (N.B. that this fails for qp) when Ω is a bounded extension domain. That is, every sequence that is bounded in Ws,p(Ω) with its norm has a subsequence that converges in Lq(Ω) with its norm.

Critical Embedding. Let 0 < s < 1 and 1 ≤ p < ∞ be such that sp = n. Let pq < ∞. Then there is a constant C such that, for every compactly supported and measurable function u,

\displaystyle \| u \|_{L^{q}(\mathbb{R}^{n})} \leq C \| u \|_{W^{s, p}(\mathbb{R}^{n})}.

Hence, the Sobolev space Ws,p(ℝn) is continuously embedded in Lq(ℝn) for every pq < ∞. The same holds true with any extension domain Ω in place of ℝn; furthermore, for a bounded extension domain Ω, we get continuous embedding of Ws,p(Ω) into Lq(Ω) for every 1 ≤ q < ∞.

Super-Critical Embedding. Let 0 < s < 1 and 1 ≤ p < ∞ be such that sp > n. If Ω is an extension domain, then there is a constant C such that, for every compactly supported and measurable function u,

\displaystyle \| u \|_{C^{\alpha}(\Omega)} \leq C \| u \|_{W^{s, p}(\Omega)},
\displaystyle \alpha := \dfrac{sp-n}{p}.

Advertisements

3 thoughts on “Interpolation and fractional differentiability revisited”

  1. Very nice post!
    Do you have a source for the claim that the Slobodeckij-spaces are the same as the interpolation spaces?
    For 0 < s 1 (I tought I had found a proof, but it turns out that it is incomplete). I know that the book by Triebel contains quite a long proof of a more general result involving Besov spaces, but I do not think I really want to get into the details of that.

    The book by Luc Tartar “An Introduction to Sobolev Spaces and Interpolation Spaces” just says that there were two ways to define the fractional Sobolev spaces for s>1, either like the Slobodeckij spaces or as interpolation spaces (between L^p and W^(m,p), but that does not matter due to reiteration) and there were a few technicalities to check in order to show that these notions coincide. He then proceeds to check that the reiteration theorem is applicable, but I do not see how that gives the result already.

    Maybe you can help me with that?

    Thank you very much!

  2. Thanks, Daniel. I think that Theorem 4 of [http://www.numa.uni-linz.ac.at/Teaching/LVA/2010w/SemNum/Topics/MutimbuVlcek_Report.pdf] answers your question.

    1. Unfortunately, Theorem 4 just extends the result from R^n to “regular” domains Omega and the result for R^n is only proven for the special case p=2, k=0.

      I have looked at it again and I think the proof in Lunardi’s book for the case k=0 extends to arbitrary k, not sure though, since it’s getting kind of late and I must just be doing stupid stuff ^^

      Thank you very much nontheless!

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s