0.1 Towards fractional smoothness

All the above spaces share the common feature that the regularity index is an integer. In many applications, one is interested to allowfractional order of smoothness, in order to describe the regularity of a function in a more precise way. The question thus arises of how to fill the gaps between integer smoothness classes . There are at least two instances where such a generalization is very natural:

• In the case of $L^2$ -Sobolev spaces $H^m:=W^{m,2}$ and when $\Omega ={\mathbb{R}}^d$ , we can define an equivalent norm based on the Fourier transform, since byParseval's formula we have the norm equivalence
  $${\parallel f\parallel }_{H^m}^2\sim {\int }_{{\mathbb{R}}^d}{(1+|\omega \left|\right)}^{2m}{\left|\widehat{f}\left(\omega \right)\right|}^{2}d\omega .$$
  For a non-integer $s\ge 0$ , it is thus natural to define the space $H^s$ as the set of all $L^2$ functions such that
  $${\parallel f\parallel }_{H^s}^2:={\int }_{{\mathbb{R}}^d}{(1+|\omega \left|\right)}^{2s}{\left|\widehat{f}\left(\omega \right)\right|}^{2}d\omega ,$$
  is finite.
• In the case of ${\mathcal{C}}^m$ spaces, we note that ${sup}_{x\in \Omega }|f\left(x\right)-f(x-h)|\le C\left|h\right|$ if $f\in {\mathcal{C}}^1$ for any $h\in {\mathbb{R}}^d$ whereas for an arbitrary function $f\in {\mathcal{C}}^0$ , ${sup}_{x\in \Omega }|f\left(x\right)-f(x-h)|$ might go to zero arbitrarily slow as $\left|h\right|\to 0$ . This motivates the definition of the Hölder space ${\mathcal{C}}^s$ , $0<s<1$ consisting of those $f\in {\mathcal{C}}^0$ such that
  $$\underset{x\in \Omega }{sup}|f\left(x\right)-f(x-h)|\le {C\left|h\right|}^s.$$
  If $m<s<m+1$ , a natural definition of ${\mathcal{C}}^s$ is given by $f\in {\mathcal{C}}^m$ and ${\partial }^{\alpha }f\in {\mathcal{C}}^{s-m}$ , $\left|\alpha \right|=m$ . It can be proved that this property can also be expressed by
  $$\underset{x\in \Omega }{sup}|{\Delta }_h^n{f\left(x\right)|\le C|h|}^s,$$
  where $n>s$ and ${\Delta }_h^n$ is the $n$ -th order finite difference operator defined recursively by ${\Delta }_h^{1}f\left(x\right)=f\left(x\right)-f(x-h)$ and ${\Delta }_h^{n}f\left(x\right)={\Delta }_h^1\left({\Delta }_h^{n-1}\right)f\left(x\right)$ (for example ${\Delta }_h^{2}f\left(x\right)=f\left(x\right)-2f(x-h)+f(x-2h)$ ). When $s$ is not an integer, the spaces ${\mathcal{C}}^s$ that we have defined are also denoted as $W^{s,\infty }$ . The space ${\mathcal{C}}^{\mathcal{s}}$ can be equiped with the norm
  $${\parallel f\parallel }_{{\mathcal{C}}^s\left(\Omega \right)}:={\parallel f\parallel }_{L^{\infty }\left(\Omega \right)}+\underset{h\in {\mathbb{R}}^d}{sup}{\left|h\right|}^{-s}{\parallel {\Delta }_h^{n}f\parallel }_{L^{\infty }\left(\Omega \right)}.$$

Let us give two important instances in which the above spaces appear in a natural way. The first is the study of the restriction of a function $f(x_1,\cdots ,x_d)$ to a manifold of lower dimension, for example the hyperplane defined by $x_d=0$ . If $g(x_1,\cdots ,x_{d-1})=f(x_1,\cdots ,x_{d-1},0)$ is such a restriction, then it is known that $f\in H^s({\mathbb{R}}^d$ for $s>\frac{1}{2}$ implies that $g\in H^{s-\frac{1}{2}}\left({\mathbb{R}}^{d-1}\right)$ . The second one is the study of theBrownian motion $W\left(t\right)$ on an interval $I$ , for which it is known that $W\left(t\right)$ is almost surely in $C^{\frac{1}{2}-\epsilon }$ for all $\epsilon >0$ .