## Notes of Nicola Garofalo’s lecture nr 4

1. The isoperimetric problem

I want to show how PDE results can be used to solve geometric problems.

1.1. The isoperimetric inequality

I will prove the isoperimetric inequality in Carnot groups, $\displaystyle \begin{array}{rcl} |E|^{\frac{Q-1}{Q}}\leq\mathrm{const.}\,|\partial E|. \end{array}$

It has lots of applications, see the conference in Paris at the end of september.

1.2. Doubling metric spaces

A metric space ${S}$ is doubling if it admits a Borel measure ${\nu}$ such that for all balls, ${\nu(B(x,2r))\leq C_1\,\nu(B(x,r))}$. On can define a dimension by ${Q=\log_2(C_1)}$.

Exercise: Prove that this implies ${\nu(B(x,tr))\geq \frac{1}{C_1}\,t\,\nu(B(x,r))}$ for all ${t>1}$.

1.3. Weak ${L^p}$ spaces

The weak (Marcinkiewicz) ${L^p}$ space, denoted by ${L^{p,\infty}}$, is the set of functions ${f}$ such that $\displaystyle \begin{array}{rcl} \sup_{t>0}t|\{x\,;\,|f(x)|>t\}^{1/p}<\infty. \end{array}$

It contains ${L^p}$ (Cavalieri’s principle) strictly. For instance, ${f(x)=\frac{1}{|x|^2}}$ belongs to ${L^{n/2,\infty}({\mathbb R}^n)}$ but not to ${L^{n/2}({\mathbb R}^n)}$. The standard operators of analysis often fail to send ${L^p}$ to ${L^q}$, but send ${L^p}$ to weak ${L^q}$. The loss is not so serious since Marcinkiewicz’ interpolation theorem tells us that interpolating ${L^p}$ and weak ${L^p}$ spaces gives ${L^p}$ spaces.

1.4. Fractional integration

The Riesz fractional integration operator ${I_\alpha}$ is $\displaystyle \begin{array}{rcl} I_\alpha f(x)=\int_{B}f(y)\frac{d(x,y)^\alpha}{\nu((B(x,d(x,y))))}\,dy. \end{array}$

Theorem 1 If ${0<\alpha, then ${I_\alpha}$ is bounded ${L^1(B)\rightarrow L^{q,\infty}(B)}$, provided ${q=\frac{Q}{Q-\alpha}}$. Morover, its norm is at most $\displaystyle \begin{array}{rcl} C_2\frac{R}{|B|^{1/Q}}. \end{array}$

In fact, the theorem holds for doubling metric spaces.

Theorem 2 (Nagel-Stein-Wainger 1984) Carnot manifolds are locally doubling.

1.5. Fundamental solutions, again

In this section, we deal with a bracket-generating family of vectorfields ${X_j}$, the corresponding sub-Laplacian ${L=\sum X_j^*X_j}$, and the correponding gradient $\displaystyle \begin{array}{rcl} |\nabla u|=(\sum |X_i u|^2)^{1/2}. \end{array}$

Everything is local.

Theorem 3 (NSW, Sanchez-Calle 1984) There exists a fundamental solution ${\Gamma}$ of ${L}$, it satisfies $\displaystyle \begin{array}{rcl} 0\leq \Gamma(x,y)\leq C\,\frac{d(x,y)^2}{|B(x,d(x,y))}. \end{array}$

Furthermore, $\displaystyle \begin{array}{rcl} |\nabla\Gamma(x,y)|\leq C\,\frac{d(x,y)}{|B(x,d(x,y))}. \end{array}$

An integration by parts gives

Corollary 4 (Citti-Garofalo-Lanconelli) For compactly supported functions ${u}$, $\displaystyle \begin{array}{rcl} |u(x)|\leq C\,I_1(|\nabla u|)(x). \end{array}$

Indeed, $\displaystyle \begin{array}{rcl} |u(x)|\leq \int|\nabla u(y)||\nabla\Gamma(x,y)|dy\leq C\,\int|\nabla u(y)|\frac{d(x,y)}{|B(x,d(x,y))}\,dy. \end{array}$

Corollary 5 For compactly supported functions ${u}$, $\displaystyle \begin{array}{rcl} \|I_1(|\nabla u|)\|_{L^{q,\infty}}\leq C\,\|\nabla u\|_{L^1}. \end{array}$

This easily follows from previous results. Combining the last two corollaries yields

Theorem 6 For ${q=\frac{Q}{Q-1}}$, for compactly supported functions ${u}$, $\displaystyle \begin{array}{rcl} \|u\|_{L^{q,\infty}}\leq C\,\frac{R}{|B|^{1/Q}}\|\nabla u\|_{L^1}. \end{array}$

1.6. From weak to strong Sobolev inequality

Fleming and Richel observed in 1971 that, thanks to coarea formula, the weak Sobolev inequality implies the strong one. This works only for ${p=1}$, the geometric case, which is equivalent to the isoperimetric inequality, since one uses this equivalence.

1.7. Perimeter

To give a precise statement of the isoperimetric inequality, we need to define perimeter. The following definition, in case ${X_j=\partial_j}$, is due to de Giorgi.

The norm of a vectorfield ${\xi}$ is ${(\sum a_i^2)^{1/2}}$ if ${\xi=\sum a_i X_i}$, and ${+\infty}$ if ${\xi\notin\mathrm{span}(X_1,\ldots,X_m)}$.

The total variation of an ${L^1}$ function ${u}$ is $\displaystyle \begin{array}{rcl} Var(u,\Omega):=\sup\{\int_{\Omega}u\,div(\xi)\,;\,\xi\textrm{ vector field },\|\xi\|_{L^{\infty}}\leq 1\}. \end{array}$

The space of functions of bounded variation ${BV(\Omega)}$ has norm $\displaystyle \begin{array}{rcl} \|u\|_{BV(\Omega)}:=\|u\|_1 + Var(u,\Omega). \end{array}$

Note that ${W^{1,1}}$ ( ${L^1}$ functions with ${X_i u\in L^1}$) is strictly contained in ${BV}$. It does not contain indicators ${1_E}$ of sets ${E}$, for instance, although they are often in ${BV}$.

Definition 7 $\displaystyle P(E,\Omega)=Var(1_E;\Omega).$

In ${{\mathbb R}^n}$, for smooth sets, one gets back the surface measure.

1.8. Proof of the isoperimetric inequality

Theorem 8 $\displaystyle \begin{array}{rcl} |E|^{\frac{Q-1}{Q}}\leq C\,|B|^{-1/Q}P(E,B). \end{array}$

\proof

Let ${E}$ be a smooth domain. The idea is to apply the weak Sobolev inequality to the indicator ${u=1_E}$. ${P(E,\Omega)}$ plays the role of ${\|\nabla u\|_{L^1}}$ on the right hand side. On the left hand side, $\displaystyle \begin{array}{rcl} |\{x\,;\,|u(x)|>t\}=|E| \textrm{ iff }0\leq t<1, \end{array}$

hence $\displaystyle \begin{array}{rcl} \|u\|_{L^{q,\infty}}=|E|^{1/q}. \end{array}$

To justify replacement of perimeter with ${\|\nabla u\|_{L^1}}$, approximate ${1_E}$ with smooth functions ${u}$ and apply the coarea formula as in next subsection.

1.9. Proof of the strong Sobolev inequality

Theorem 9 $\displaystyle \begin{array}{rcl} \|u\|_{L^{\frac{Q-1}{Q}}(B)}\leq C\,|B|^{-1/Q}\|\nabla u\|_{L^1(B)}. \end{array}$

\proof

Assume ${u}$ is smooth and compactly supported. By Sard’s theorem, for a.e. ${t}$, ${E_t=\{u>t\}}$ is a smooth manifold. In general, Federer’s coarea formula states that, for ${g}$ a Lipschitz function, $\displaystyle \begin{array}{rcl} \int_{{\mathbb R}^n}f|D g|=\int_{{\mathbb R}}(\int_{\{g=t\}}f\,d\mathcal{H}^{n-1})\,dt \end{array}$

We apply it to ${g=u}$ and ${f=\frac{|\nabla u|}{|Du|}\geq 1}$. $\displaystyle \begin{array}{rcl} \int_{B}|\nabla u|\geq\int_{{\mathbb R}}(\int_{\partial E_t}\,d\mathcal{H}^{n-1})\,dt=\int_{{\mathbb R}}P(E_t,B)\,dt \end{array}$

Finally, express the ${L^{\frac{Q}{Q-1}}}$-norm of ${u}$ as an integral, $\displaystyle \begin{array}{rcl} (\int_{B}|u|^\frac{Q}{Q-1})^{\frac{Q-1}{Q}}&=&(\frac{Q}{Q-1}\int_{0}^{\infty}t^{\frac{1}{Q-1}}|E_t|\,dt)^{\frac{Q-1}{Q}}\\ &\leq& C\,(\int_{0}^{\infty}|E_t|^\frac{Q}{Q-1}\,dt)^{\frac{Q-1}{Q}}, \end{array}$

which concludes the proof. We have used the easy fact that, for every nondecreasing function ${V(t)}$ and ${a>1}$, $\displaystyle \begin{array}{rcl} F(x)=a\int_{0}^{x}t^{a-1}V(t)\,dt-(\int_{0}^{x}V(t)^{1/a}\,dt)^a \end{array}$

is a non decreasing function of ${x}$ (differentiate !) and thus nonnegative.

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