# What does the metric $\dot{H}^{1/2}$ refer to?

I have a source that talks about the metric $$\dot{H}^{1/2}$$ on the space Diff($$S^1$$) (the space of $$C^\infty$$-diffeomorphisms from $$S^1$$ to $$S^1$$).

I know that in general the metric $$\dot{H}^k$$ refers to a degenerate version of the $$H^k$$ metric although I am not sure whether this degeneracy means to drop only the first term or to drop all but the highest-order term. In any case, I know that for an integer $$k$$, the metric $$\dot{H}^k$$ should be either $$\left(\sum_{i=1}^k \int \left|f^{(i)}\right|^2\right)^{1/2}\quad\text{or}\quad\left(\int \left|f^{(k)}\right|^2\right)^{1/2}.$$

For the first equation, I can see no immediate way to generalize to a non-integer parameter. For the second, it would seem that I will need to learn about fractional derivatives.

Can someone please tell me what the metric $$\dot{H}^{1/2}$$ refers to?

NB: The source in question is Recent Developments in Topological Hydrodynamics by Boris Khesin.

• As I checked, he has no such book! Jun 8 at 3:53
• It can be found as an appendix in Topological Methods in Hydrodynamics. I considered it a separate book because the book and Appendix can be read indepdenently Jun 8 at 11:10

Homogeneous Sobolev spaces are just concerned by the exact amount of regularity you are actually concerned with.

$$\bullet$$ For instance, homogeneous ($$\mathrm{L}^2$$-)Sobolev (semi-)norm of order $$k\in\mathbb{N}$$ over $$\mathbb{R}^n$$ is given by

$$\lVert u \rVert_{\dot{\mathrm{H}}^k(\mathbb{R}^n)} := \lVert \nabla^k u \rVert_{{\mathrm{L}}^2(\mathbb{R}^n)} (= \lVert (-\Delta)^{\frac{k}{2}} u \rVert_{{\mathrm{L}}^2(\mathbb{R}^n)}).$$

It can be easily generalized to $$s\in\mathbb{R}$$, setting the homogeneous Sobolev norms to be $$\lVert u \rVert_{\dot{\mathrm{H}}^s(\mathbb{R}^n)} :=\lVert (-\Delta)^{\frac{s}{2}} u \rVert_{{\mathrm{L}}^2(\mathbb{R}^n)}.$$

(Not really relevant here, but over the whole space and unbounded domains more generally there are few issues of definitions I wont discuss that much here : we need those quantities to be norms, hence we have to get rid of polynomials and to not work with $$\mathcal{S}'(\mathbb{R}^n)$$ as an ambient space).

It can be shown, [1, Proposition 1.37, p.28], that for all $$s\in(0,1)$$, all $$u\in\dot{\mathrm{H}}^s(\mathbb{R}^n)$$, one has \begin{align} \lVert u \rVert_{\dot{\mathrm{H}}^s(\mathbb{R}^n)}^2\sim_{s,n} \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\mathrm{d}x\mathrm{d}y. \end{align}

The previous equivalence of norms is often used as a definition for fractional Sobolev spaces, since we also have \begin{align} \lVert u \rVert_{{\mathrm{H}}^s(\mathbb{R}^n)}^2 \sim_{s,n} \lVert u \rVert_{{\mathrm{L}}^2(\mathbb{R}^n)}^2 + \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\mathrm{d}x\mathrm{d}y. \end{align}

$$\bullet$$ Getting back to your question about the definition over $$S^1$$, the idea has been mentionned in [2, Definition 2.2.2, p.21] to define the homogeneous Sobolev norms for $$s\in (0,1)$$ to be \begin{align} \lVert u \rVert_{\dot{\mathrm{H}}^s(S^1)}^2\sim_{s,n} \int_{S^1}\int_{S^1} \frac{|u(x)-u(y)|^2}{|x-y|^{1+2s}}\mathrm{d}\sigma_{x}\mathrm{d}\sigma_{y}. \end{align} where $$|\cdot|$$ is still understood as the standard euclidean metric via the imbedding $$S^1\hookrightarrow \mathbb{C}$$, $$\sigma$$ is the Lebesgue measure over $$S^1$$.

Notice that in [2, Definition 2.2.2, p.21] we are mainly concerned with Besov norms $$\lVert\cdot\rVert_{\dot{\mathrm{B}}^s_{2,2}(S)}$$. However it is expected, and usually true, in all the common cases that it should agree, up to equivalence, with $$\lVert\cdot\rVert_{\dot{\mathrm{H}}^s(S)}$$.

On may also try to show that one can recover those spaces via real or complex interpolation of normed vector spaces: $$\dot{\mathrm{H}}^s(S^1) = [{\mathrm{L}}^2(S^1),\dot{\mathrm{H}}^1(S^1)]_{s} = ({\mathrm{L}}^2(S^1),\dot{\mathrm{H}}^1(S^1))_{s,2}.$$ Here for $$\theta\in(0,1)$$, $$p\in[1,+\infty]$$, $$[\cdot,\cdot]_{\theta}$$ and $$(\cdot,\cdot)_{\theta,p}$$ are respectively the complex and real interpolation functors.

The fact that those definitions are all equivalent is pretty well known and folklore for the community I guess, but I don't have any clear references in mind for that. Any help will be welcome.

Let me know if some of you, readers, think my answer contains wrong or uncomplete statements. Feel free to comment.

1. Bahouri, Hajer; Chemin, Jean-Yves; Danchin, Raphaël, Fourier analysis and nonlinear partial differential equations, Grundlehren der Mathematischen Wissenschaften 343. Berlin: Heidelberg (ISBN 978-3-642-16829-1/hbk; 978-3-642-16830-7/ebook). xvi, 523 p. (2011). ZBL1227.35004.
2. Danchin, Raphaël; Mucha, Piotr Bogusław, Critical functional framework and maximal regularity in action on systems of incompressible flows, Mém. Soc. Math. Fr., Nouv. Sér. 143, 1-151 (2015). ZBL1335.35179.