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<Q}, 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.

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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