Fractional Sobolev Spaces

Recall for a function f\in L^1(\Omega) for \Omega\subseteq\mathbb{R}^n we define the Fourier transform of f by

\displaystyle\hat{f}(x)=F(f)(x)=\int_\Omega e^{-2\pi ix\cdot\xi}f(\xi)\,d\xi

and the inverse Fourier transform by

\displaystyle \check{f}(x)=F^{-1}(f)(x)=\int_\Omega e^{2\pi i x\cdot\xi}f(\xi)\,d\xi.

If f,\hat{f}\in L^1(\Omega), then \check{\hat{f}}=\hat{\check{f}}=f a.e..

Now if f\in L^1 and D^\alpha (f)\in L^1 with \alpha\in\mathbb{Z}^n, we say D^\alpha f is the weak derivative of f provided

\displaystyle\int_\Omega f D^\alpha\phi\,dx=(-1)^{|\alpha|}\int_\Omega D^\alpha(f)\phi\,dx

with \phi\in C_C^\infty(\Omega), |\alpha|=\sum_i\alpha_i, and

\displaystyle D^\alpha\phi=\frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}}\phi.

The Sobolev space W^{k,p}(\Omega) is then defined as the set of all functions f\in L^p(\Omega) such that D^\alpha(f)\in L^p(\Omega) for all |\alpha|\leq k.  These become Banach spaces under the norm

\displaystyle\|f\|_{k,p}=\left(\sum_{|\alpha|\leq k}\int_\Omega \left|D^\alpha(f)\right|^p\,dx\right)^{1/p}

and Hilbert when p=2, whence we denote H^k(\Omega)=W^{k,2}(\Omega).

For 1<p<\infty we can say that W^{k,p}(\mathbb{R}^n) consists of f\in L^p(\mathbb{R}^n) such that

\displaystyle F^{-1}\circ(1+|2\pi x|^2)^{k/2}F\circ f=(1-\Delta)^{k/2}f(x)\in L^p(\mathbb{R}^n)

(using Fourier multipliers).

Bessel Potential Spaces:

We can generalize with r\in\mathbb{R} and 1<p<\infty and define the Bessel potential space W^{r,p}(\mathbb{R}^n) as all f\in L^p such that \|f\|_{r,p}:=\|(1-\Delta)^{r/2}f\|_p<\infty.  Note the fractional power of the Laplacian is defined by virtue of the fact that L^p\cap L^1 is dense in L^p, so we say

\displaystyle(1-\Delta)^{r/2}(f):=(F^{-1}\circ (1+|2\pi x|^2)\circ F)(f).

 These spaces satisfy our desire of being Banach (and Hilbert when p=2).

Sobolev-Slobodeckij Spaces:

There is an alternative approach.  Recall that the Holder space C^{k,\gamma}(\Omega) is defined as all functions f\in C^k(\Omega) such that

\displaystyle\|f\|_{C^{k,\gamma}}=\sum_{|\alpha|\leq k}\sup_{x\in\Omega}|D^\alpha|+\sum_{|\alpha|=k}\sup_{x,y\in\Omega}\frac{|D^\alpha f(x)-D^\alpha f(y)|}{|x-y|^\gamma}<\infty.

That is, it is the set of functions on \Omega which are C^k and whose k-th partial derivatives are bounded and Holder continuous of degree \gamma.  These spaces are Banach under the above norm.  We can generalize the Sobolev spaces to incorporate similar properties.  Let us define the Slobodeckij norm for f\in L^p(\Omega) with 1\leq p<\infty and \theta\in (0,1) by

\displaystyle [f]_{\theta,p}=\int_\Omega\int_\Omega\frac{|f(x)-f(y)|^p}{|x-y|^{\theta p+n}}\,dxdy.

The corresponding Sobolev-Slobodeckij space W^{s,p}(\Omega) is defined as all functions f\in W^{\lfloor s\rfloor,p}(\Omega) such that

\displaystyle\sup_{|\alpha|=\lfloor s\rfloor}[D^\alpha f]_{\theta,p}<\infty

where \theta=s-\lfloor s\rfloor\in(0,1).  This becomes a Banach space under the norm

\displaystyle\|f\|_{s,p}=\|f\|_{W^{\lfloor s\rfloor,p}}+\sup_{|\alpha|=\lfloor s\rfloor}[D^\alpha f]_{\theta,p}.

[1] (unclear text references)

[2]  Lieb, Elliot and Michael Loss.  Analysis.  2nd Edition.  Graduate Studies in Mathematics.  Vol. 14.  American Mathematical Society.  2001.

[3]  Evans, Lawrence.  Partial Differential Equations.  Graduate Studies in Mathematics.  Vol. 19.  American Mathematical Society.  1998.


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