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 *C*^{α}(*K*) of *α*-Hölder functions on a compact set *K* ⊆ ℝ^{n} with interior, i.e. the set of functions *u*: *K* → ℝ for which the norm

is finite. In the case of weak derivatives, a suitable such space was the vector space *W*^{α,p}(*K*) of functions *u* for which the norm

is finite. These spaces are all Banach spaces, and interpolate in the sense of real *K*-interpolation between the spaces *C*^{0}(*K*) of continuous functions and *C*^{1}(*K*) of continuously differentiable functions (respectively the Lebesgue space *L*^{p}(*K*) and the Sobolev space *W*^{1,p}(*K*). This post grew out of my noticing one simple omission in the previous post, now corrected: for *p* = 2, the spaces *W*^{α,2} are Hilbert spaces under the inner product

On realizing this omission, I started to think more deeply about other notions of fractional differentiability. In particular, I wondered how the above *W*^{α,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 ⊆ ℝ^{n}; 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

Now define a norm by

and define the **(Aronszajn–Gagliardo–Slobodeckij) fractional Sobolev space** *W*^{s,p}(Ω) to be

with the usual identification of functions that agree almost everywhere. The spaces *W*^{s,p}(Ω) are Banach spaces, and the spaces *H*^{s}(Ω) ≔ *W*^{s,2}(Ω) are Hilbert spaces when given the inner product

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

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

so that *W*^{s1,p}(Ω) embeds continuously into *W*^{s0,p}(Ω). The same holds true in the limiting case *s*_{1} = 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 *σ* ≔ *s* − *m*, and define

and define the **(Aronszajn–Gagliardo–Slobodeckij) fractional Sobolev space** *W*^{s,p}(Ω) to be

with the usual identification of functions that agree almost everywhere. As before, these are Banach spaces, and the spaces *H*^{s}(Ω) ≔ *W*^{s,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 < *s*_{0} ≤ *s*_{1} < ∞, there is a constant *C* ≥ 1 such that

so that *W*^{s1,p}(Ω) embeds continuously into *W*^{s0,p}(Ω).

As would be expected from the integral *s* case the smooth functions of compact support are dense in *W*^{s,p}(ℝ^{n}) 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 *W*^{s,p}(Ω) — instead, that closure is denoted *W*_{0}^{s,p}(Ω), and it can be thought of as those functions in *W*^{s,p}(Ω) that satisfy zero boundary conditions on the boundary ∂Ω.

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

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* ∈ *W*^{1,p}(Ω) lies in *W*^{1−1⁄p,p}(∂Ω), and in fact the trace operator *T*: *W*^{1,p}(Ω) → *W*^{1−1⁄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: ℝ^{n} → ℝ, its Fourier transform ℱ*u* and the Fourier transform of its *α*^{th} derivative (where *α* is a multi-index) are related by

Hence, if *u* is *k* times differentiable and all its *k*^{th} order derivatives are in *L*^{2} (and so can have the Fourier transform applied to them), then

This suggests defining, for *s* ≥ 0,

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

with the Banach norm

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

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

where 0 < θ < 1 and

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

However, for *p* ≠ 2, we have

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 *L*^{p} and *L*^{q}, 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 *W*^{s,p}(Ω) are related to other function spaces. An important notion in the results is that of an extension domain: a domain Ω ⊆ ℝ^{n} is called an **extension domain** if there is a constant *C* for which every *u* ∈ *W*^{s,p}(Ω) admits an extension to *E**u* ∈ *W*^{s,p}(ℝ^{n}) such that

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*,

Hence, the Sobolev space *W*^{s,p}(ℝ^{n}) is continuously embedded in *L*^{q}(ℝ^{n}) for every *p* ≤ *q* ≤ *p*^{∗}. The same holds true with any extension domain Ω in place of ℝ^{n}; furthermore, for a bounded extension domain Ω, we get continuous embedding of *W*^{s,p}(Ω) into *L*^{q}(Ω) for every 1 ≤ *q* ≤ *p*^{∗}.

We also have a **compact embedding** of *W*^{s,p}(Ω) into *L*^{q}(Ω) for every 1 ≤ *q* < *p*^{∗} (N.B. that this fails for *q* ≠ *p*^{∗}) when Ω is a bounded extension domain. That is, every sequence that is bounded in *W*^{s,p}(Ω) with its norm has a subsequence that converges in *L*^{q}(Ω) with its norm.

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

Hence, the Sobolev space *W*^{s,p}(ℝ^{n}) is continuously embedded in *L*^{q}(ℝ^{n}) for every *p* ≤ *q* < ∞. The same holds true with any extension domain Ω in place of ℝ^{n}; furthermore, for a bounded extension domain Ω, we get continuous embedding of *W*^{s,p}(Ω) into *L*^{q}(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*,

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!

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.

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!