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## Characterizations of Sobolev Spaces

I will present here a few useful characterizations of Sobolev spaces. The source is Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis. Consider ${u \in L^p(\Omega)}$ with ${1 < p \leq \infty}$. We denote by ${p'}$ the conjugate of ${p}$, i.e. ${\displaystyle \frac{1}{p}+\frac{1}{p'}=1}$. Then the following properties are equivalent:

• (i) ${u \in W^{1,p}(\Omega)}$;
• (ii) there exists a constant ${C}$ such that

$\displaystyle \left| \int_\Omega u \frac{\partial \varphi}{\partial x_i} \right| \leq C \| \varphi\|_{p'} ,\ \forall \varphi \in C_c^\infty(\Omega),\ \forall i=1..N;$;

• (iii) there exists a constant ${C}$ such that for all ${\omega \subset \subset \Omega}$ and all ${h \in \Bbb{R}^N}$ with ${|h|<\text{dist}(\omega,\partial \Omega)}$ we have

$\displaystyle \| \tau_hu-u\|_{L^p(\omega)}\leq C|h|.$

(note that ${\tau_hu(x)=u(x+h)}$ makes sense for ${x \in \omega}$ and ${|h|<\text{dist}(\omega,\partial \Omega)}$. Furthermore, we can take ${C=\|\nabla u\|_{L^p(\Omega)}}$ in (ii) and (iii). If ${\Omega=\Bbb{R}^N}$ we have

$\displaystyle \|\tau_hu-u\|_{L^p(\Bbb{R}^N)} \leq |h|\|\nabla u\|_{L^p(\Bbb{R}^N)}$.

Proof: (i) ${\Rightarrow}$ (ii). Obvious. Write the definition and apply Holder’s inequality.

(ii) ${\Rightarrow}$ (i). The functional

$\displaystyle \varphi \mapsto \int_\Omega u \frac{\partial \varphi}{\partial x_i}$

is a continuous linear functional on a dense subspace of ${L^{p'}(\Omega)}$, and therefore can be extended to a continuous linear functional on the whole ${L^{p'}(\Omega)}$. By Riesz’s representation theorem there exists a function ${g_i \in L^p(\Omega)}$ such that

$\displaystyle \int_\Omega u \frac{\partial \varphi}{\partial x_i}=-\int_\Omega g_i\varphi.$

Since this happens for every ${i=1..N}$ we find that ${u \in W^{1,p}(\Omega)}$.

(i) ${\Rightarrow}$ (iii). Assume first that ${ u \in C_c^\infty(\Bbb{R}^N)}$. Let ${h \in \Bbb{R}^N}$ and set

$\displaystyle v(t)=u(x+th),\ t \in \Bbb{R}.$

Then we find that ${v'(t)=h \cdot \nabla u(x+th)}$ and therefore

$\displaystyle u(x+h)-u(x)=v(1)-v(0)=\int_0^1 v'(t)dt =\int_0^1 h \cdot \nabla u(x+th)dt.$

For ${1 < p< \leq \infty}$ we have

$\displaystyle |\tau_hu(x)-x)|^p\leq |h|^p \int_0^1 |\nabla u (x+th)|^p dt$

and

$\displaystyle \int_\omega |\tau_h u(x) -u(x)|^pdx \leq |h|^p \int_\omega\int_0^1 |\nabla u(x+th)|^pdtdx$

$\displaystyle = |h|^p \int_0^1 \int_{\omega+th} |\nabla u(y)|^pdydt.$

If ${|h|}$ is small enough then there exists an open set ${\omega' \subset \subset \Omega}$ such that ${\omega +th \subset \omega'}$ for all ${t \in [0,1]}$ and therefore

$\displaystyle \|\tau_hu-u\|_{L^p(\omega)} \leq |h| \|\nabla u\|_{L^p(\omega')},$

which concludes the proof for ${u \in C_c(\Bbb{R}^N)}$ and ${1 \leq p <\infty}$. But we know that ${C_c(\Bbb{R}^N)}$ is dense in ${W^{1,p}}$ in the sense that there exists ${u_n \in C_c(\Bbb{R}^N)}$ such that ${u_n \rightarrow u}$ in ${L^p(\Omega)}$ and ${\nabla u_n \rightarrow \nabla u}$ in ${L^p(\omega)}$ for every ${\omega \subset \subset \Omega}$. This finishes the argument. For ${p=\infty}$ apply the above for ${p<\infty}$ and make ${p \rightarrow \infty}$.

(iii) ${\Rightarrow}$ (ii). Let ${\varphi \in C_c^\infty(\Omega)}$ and consider ${\omega \subset \subset \Omega}$ such that ${\text{supp} \varphi \subset \omega}$. Take ${h}$ with ${|h|}$ small enough such that ${\omega+h\subset \subset \Omega}$. Because of (iii) we have

$\displaystyle \left| \int_\Omega (\tau_hu-u)\varphi \right| \leq C |h| \|\varphi\|_{L^{p'}(\Omega)}.$

which translates to

$\displaystyle \left|\int_\Omega u(y) \frac{\varphi(y-h)-\varphi(y)}{|h|}dy \right| \leq C \|\varphi\|_{L^{p'}(\Omega)}.$

If we choose ${h=te_i}$ in the last inequality and make ${ t \rightarrow 0}$ then we obtain (i).

In the case ${p=1}$ the following implications still hold:

$\displaystyle \text{(i)} \Rightarrow \text{(ii)} \Leftrightarrow \text{(iii)}.$

The functions which satisfy (ii) and (iii) in the case when ${p=1}$ are the functions of bounded variation.