Notes of Andrea Bonfiglioli’s lecture

Maximal principles and Harnack inequalities for PDO’s in divergence form

1. Motivation

CR geometry (sub-Laplacians), stochastic PDE’s.

2. Introduction

2.1. Standing assumptions

  1. Total nondegeneracy.
  2. Smooth hypoellipticity.

Sometimes, we require that {L-\epsilon} is hypoelliptic as well. Or even existence of a global, positive fundamental solution (unfortunately, this is known only for special classes, like homogeneous operators on nilpotent groups, Nagel Stein 1990.

2.2. Earlier work

Theorem 1 (Bony 1969) Maximum principle and Harnack inequality for a class of degenerate elliptic operators (sums of squares of Hörmander vectorfields).

Bony uses a Hopf-type lemma and maximum propagation to get maximum principle. Then is used to get Harnack inequality.

Huge litterature in the 1980’s : Fabes, Jerison, Serapioni Franchi, Lanconelli, Chanillo, Wheeden, Sanchez-Calle. All assume hypo-ellipticity.

Nowadays, the framework has been enlarged : doubling metric spaces satisfying Poincaré inequality.

2.3. Examples

Sub-Laplacians.

Fedii 1971 : sum of squares of non Hörmander vectorfields (a constant basis, whose vectors are multiplied with flat functions). This can be hypo-elliptic but not sub-elliptic (Fefferman-Phong 1981).

3. Results

3.1. Hopf Lemma

Let {F} be the set where {u} achieves its maximum. Let {y\in F} and {\nu} be orthogonal to {F} (meaning that the interior of some ball centered at {y+\epsilon \nu} and passing through {y} is disjoint from {F}). Then…

3.2. From Hopf lemma to maximum principle

Theorem 2 Non total degeneracy and hypoellipticity imply strong maximum principle.

\proof

Principal vectorfields {X} have to be tangent to {F}. This implies {F} has to be invariant under {X}. How can one build them ? Use columns of the matrix defining the operator. Note that Hörmander’s condition need not hold for these vectorfields.

Amano 1979 observed that non total degeneracy and hypoellipticity imply connectivity of {{\mathbb R}^n} with respect to such vectorfields plus a drift vectorfields. Thus maximum principle follows.

3.3. Harnack inequality

Theorem 3 Non total degeneracy and hypoellipticity of {L-\epsilon} imply strong a Harnack inequality where, however, the constant depends on the shape of the domain and of the considered subdomain.

\proof

Follows Bony’s approach. Solve the Dirichlet for {L-\epsilon} (based on maximal principle). Prove existence of the Green kernel of {L-\epsilon}. Get a weak Harnack inequality. Use potential theory to get Harnack from weak Harnack.

By maximum principle, the Green kernel {k_\epsilon} of {L-\epsilon} is positive. Then for {u\geq 0} such that {Lu=0}, Bony proves that

\displaystyle  \begin{array}{rcl}  u(x)\geq \epsilon\int u(y)k_\epsilon(x,y)\,d\nu(y). \end{array}

Since {k_\epsilon>0}, this allows to locally bound {u(x)} from below with the {L^1_{loc}}-norm of {u}. On the space of {L}-harmonic functions, the {L^1_{loc}} and {C^\infty} topologies coincide. This way, we get the weak Harnack inequality

\displaystyle  \begin{array}{rcl}  \sup_K u \leq C(x_0)u(x_0). \end{array}

3.4. Role of potential theory

Theorem 4 (Mokobodzki-Brelot 1964) Very abstract setting. Assume weak Harnack inequality holds and that Dirichlet problem on small open sets has a solution, then strong Harnack inequality holds.

4. More on potential theory

How can one characterize {L}-subharmonic functions ?

Use balls defined by Green’s function ({\Gamma}-balls) to define inradius of a domain. Then a representation formula follows, based on the divergence theorem, with kernel expressible in terms of Green’s function. A mean value formula holds for {L}-harmonic functions on {\Gamma}-balls, with a correction term. The corresponding inequality characterizes sub-harmonicity. So does monotonicity of mean values on {\Gamma}-balls.

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Pierre Pansu’s slides on Differential Forms and the Hölder Equivalence Problem

Here is the completed set of slides

CIRMsep14_beamer

If you want to know more about the construction of horizontal submanifolds and how Gromov uses it to bound Hausdorff dimensions from below, see Pansu’s Trento notes (2005).

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Notes of Anton Thalmaiers’s lecture nr 4

1. Probabilistic content of Hörmander’s condition

1.1. Statement

Theorem 1 Suppose that the Lie algebra generated by {A_1,\ldots,A_r} and brackets {[A_0,A_i]} fills {T_xM}. Then the bilinear form {C_t(x)} on {T_xM} is non-degenerate

1.2. Proof

Let

\displaystyle  \begin{array}{rcl}  G_s=span\{{X_s^{-1}}_* A_i\textrm{ at }x\,;\,i=1,\ldots,r\}\subset T_x M,\quad U^+_t=span\bigcup_{s\leq t}G_s. \end{array}

By Blumenthal’s 0/1-law, {U^+_t} is not random. We prove by contradiction that {U_0^+=T_x M} (this will suffice to prove the theorem). Introduce

\displaystyle  \begin{array}{rcl}  \sigma=\inf\{t>0\,;\,U_0^+\not=U_t^+\} \end{array}

Let {\xi\in T_x^*M} be orthogonal to {U_0^+} (and thus to {U_t^+} for {t<\sigma}). Since {\xi} is orthogonal to all {{X_s^{-1}}_* A_i}, {s<\sigma}. But for all vectorfields {V}, {{X_s^{-1}}_* V} satisfies (first line is Stratonovich, the second is Ito)

\displaystyle  \begin{array}{rcl}  d({X_s^{-1}}_* V)&=&({X_s^{-1}}_* [A_0,V])_X \,dt+\sum ({X_s^{-1}}_* [A_i,V])_X \cdot dB_s^i\\ &=&({X_s^{-1}}_* [A_0,V])_X \,dt+\sum ({X_s^{-1}}_* [A_i,V])_X \,dB_s^i+\sum_j ({X_s^{-1}}_*[A_j, [A_j,V]])_X\,ds\\ \end{array}

thus for all {t<\sigma},

\displaystyle  \begin{array}{rcl}  \langle\xi,({X_s^{-1}}_*A_i)_X)\rangle&=&\langle\xi,A_i(X)\rangle\\ &&+\int_{0}^{t}\langle\xi,({X_s^{-1}}_* [A_0,A_i])_X \,ds\rangle\\ &&+\int_{0}^{t}\sum_j\langle\xi,({X_s^{-1}}_* [A_j,A_i])_X \rangle dB_s^i+\int_{0}^{t}\sum_j\langle\xi,({X_s^{-1}}_*[A_j, [A_j,A_i]])_X\rangle\,ds \end{array}

By uniqueness of the solution of an SDE, this implies that {\langle\xi,({X_s^{-1}}_* [A_j,A_i])_X\rangle=0} for all {i,j\geq 1} and {s<\sigma}. Replacing {[A_i]} with {[A_i,Aj]} shows that

\displaystyle  \begin{array}{rcl}  \langle\xi,({X_s^{-1}}_* [A_j,[A_j,A_i]])_X\rangle=0, \end{array}

and

\displaystyle  \begin{array}{rcl}  \langle\xi,({X_s^{-1}}_* [A_0,A_i])_X\rangle=0 \end{array}

Iterating the procedure shows orthogonality of {\xi} with all iterated brackets, and thus {\xi=0}.

2. Probabilistic proof of hypoellipticity

Theorem 2 Assume that {A_i} and there derivatives satisfy suitable growth conditions. Assume that the bilinear form {C_t(x)} is non-degenerate and

\displaystyle  \begin{array}{rcl}  |C_t(x)|^{-1}\in L^p \end{array}

for all {p\geq 1}. Then {P_t(x,dy)=p_t(x,y)\,dy} with a smooth density {p_t(x,y)}.

The proof we are about to give is due to a large extent to Bismut, although many details are skipped in Bismut’s original paper. We use more elementary tools. We shall rely on the following standard fact.

2.1. Girsanov’s theorem

Let {B} a Brownian motion on Euclidean space. Add an absolutely continuous process, i.e. {d\hat{B}_t=dB_t+u_t\,dt} such that

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb E}(\exp(\frac{1}{2}\int_{0}^{t}|u(s)|^2\,ds))<\infty. \end{array}

{\hat{B_t}} is not a martingale any more, but this can be recovered by changing the probability measure.

Theorem 3 (Girsanov) {\hat{B}_t} is a Brownian measure with respect to the mesure {\hat{P}} whose density with respect to {P} is

\displaystyle  \begin{array}{rcl}  G_t:=\frac{d\hat{P}}{dP}_{|\mathcal{F}_t}=\exp(-\int_{0}^{t}u_s\,dB_s-\frac{1}{2}\int_{0}^{t}|u(s)|^2\,ds). \end{array}

In other words, if {F} is a functional on the space of Brownian motions, then

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb E}_{P}(F(B_.))=\mathop{\mathbb E}_{\hat{P}}((F\hat{B}_.)). \end{array}

2.2. A criterion for a measure to have a smooth density

We want to prove that {P_t(x,dy)=p_t(x,\cdot)\,dvol} for {t>0}. We use the following criterion.

Lemma 4 Let {\mu} be a probability measure on some manifold, viewed as a distribution. Assume that for all {\alpha\in{\mathbb N}} and all test functions {f},

\displaystyle  \begin{array}{rcl}  |\langle f,D^\alpha \mu\rangle|\leq C_\alpha\,\|f\|_\infty. \end{array}

Then {\mu} has a smooth density.

2.3. Proof of Theorem 2

Fix {x}. Identify {T_xM} with {{\mathbb R}^n}. We apply Girsanov’s theorem to {u_s=a_s\cdot\lambda} where {a_s} takes values in {T_xM\otimes{\mathbb R}^r} and {\lambda\in T_x^*}. The modified flow is denoted by {X^\lambda_t(x)}. Let {g} be a function to be specified later. Up to introducing the density {G_t^\lambda}, nothing changes, and

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb E}(f(X^\lambda_t(x))g(B^\lambda_\cdot )G_t^\lambda) \end{array}

does not depend on {\lambda}. Let us differentiate with respect to {\lambda} at {\lambda=0}.

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb E}((D_i f)(X_t(x))(\frac{\partial}{\partial \lambda_k}_{|\lambda=0}X_t^\lambda(x))^i g(B_.))=-\mathop{\mathbb E}(f(X_t(x))\frac{\partial}{\partial \lambda_k}_{|\lambda=0}(g(B^\lambda_.)G_t^\lambda))) \end{array}

Remember that SDE can be formally differentiated with respect to a parameter. Notation: {\frac{\partial}{\partial \lambda_k}_{|\lambda=0}X_t^\lambda(x))^i =(\partial X_t(x))_{ik}}. Get

\displaystyle  \begin{array}{rcl}  \partial X_t(x)={(X_t)}_*\int_{0}^{t}(X_s^{-1}A)_X u_s\,ds. \end{array}

This suggests choosing

\displaystyle  \begin{array}{rcl}  u_s=(X_s^{-1})_* A)_X^*:T_x^*M\rightarrow{\mathbb R}^r. \end{array}

With this choice,

\displaystyle  \begin{array}{rcl}  \partial X_t(x)={(X_t)}_*C_t(x). \end{array}

By assumption, {C_t(x)} is invertible, so we take

\displaystyle  \begin{array}{rcl}  g(B^*_.)=(C_t(x)^{-1}({X_t^{-1}}_*)^{-1})_{kj}\gamma(B^\lambda_.), \end{array}

where {\gamma} is to be specified later. This yields

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb E}((D_j f)(X_t(x))\gamma(B_.))=-\mathop{\mathbb E}(f(X_t(x))H_j(\gamma)), \end{array}

for some rather complicated expression {H_j(\gamma)}. Iteration gives

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb E}((D_i D_j D_k f)(X_t(x)))=-\mathop{\mathbb E}(f(X_t(x))H_k(H_j(H_i(1)))))), \end{array}

from which we get the estimate

\displaystyle  \begin{array}{rcl}  |\mathop{\mathbb E}((D_i D_j D_k f)(X_t(x)))|\leq\|f\|_{\infty}\|\cdots H_k(H_j(H_i(1)))\|_{L^1} \end{array}

The right hand side involves only polynomial expressions, except {C_t(x)^{-1}} and its derivatives with respect to {\lambda}. These have to be computed and estimated too. Then the Lemma applies, it shows that the distribution of {X_t(x)} has a smooth density.

3. Subjects I could not cover

There was no time to treat

  1. the short time asymptotics of the heat kernel,
  2. bounds on the lifetime of Brownian motion (differentiating {d(x,X_t(x))}, leads to the Laplacian of the distance function, and to Ricci curvature).
  3. Bismut’s interpolation between the geodesic flow and an hypoelliptic diffusion.

There is more that probability theory can do for sub-Riemannian geometry and hypoelliptic PDE’s.

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Notes of Anton Thalmaier’s lecture nr 3

1. Stochastic flows of diffeomorphisms

We continue our study of SDE {dX=A_0(X)dt+\sum A_i(X)\cdot dB^i}. Up to now, the starting point {x} was fixed. Now we exploit the dependance on {x}.

1.1. Random continuous paths of diffeomorphisms

Let us introduce the random set {M_t(\omega)=\{x\in M\,;\,\zeta(x)(\omega)>t\}} of starting points whose trajectory is still alive at time {t}. Then

  • {M_t(\omega)} is open (in fact, the lifetime {\zeta(\cdot)(\omega)} is lower semi-continuous in {x}).
  • {X_t(\cdot)(\omega):M_t(\omega)\rightarrow R_t(\omega)} is a diffeomorphism onto an open subset of {M}.
  • {s\mapsto X_s(\cdot)(\omega)} is continuous: {[0,t]\rightarrow C^{\infty}(M_t(\omega),M)}.

Furthermore, under mild growth conditions on vectorfields and their derivatives (for instance, if {M} is compact), {X_t(\cdot)(\omega)\in Diffeo(M)} for all {t}.

Consider the tangent flow {U={X_t}_*} on {TM}. It solves the formally differentiated SDE

\displaystyle  \begin{array}{rcl}  dU=\sum (DA_i)_X U\cdot dZ^i. \end{array}

1.2. Crucial observation

Let us transport a vectorfield {V} under our stochastic flow. We get a random vectorfield {{X_t}^{-1}_{*}V}. This means that, for a test function {f},

\displaystyle  \begin{array}{rcl}  ({X_t}^{-1}_{*}V)(f)=(V(f\circ X_t^{-1}))\circ X_t. \end{array}

Maillavin’s covariance matrix is defined as follows. For {t>0},

\displaystyle  \begin{array}{rcl}  C_t(x)=\sum_{i=1}^r\int_{0}^{t}({X_s}^{-1}_{*}A_i)\otimes({X_s}^{-1}_{*}A_i)_X\, dt. \end{array}

This is a random smooth section of {TM\otimes TM} over {M_t}. We shall see later that the condition we need to make this nondegenerate is Hörmander’s condition.

On may view

\displaystyle  \begin{array}{rcl}  ({X_s}^{-1}_{*}A)_X:{\mathbb R}^r \rightarrow T_X M \end{array}

as a linear map from {{\mathbb R}^r} to {T_X M}. Its adjoint is a linear map from {T_X^* M} to {{\mathbb R}^r}. Then {C_t(x)} may be viewed as an endomorphism of {T_x M}.

Lemma 1 The SDE satisfied by {{X_t}^{-1}_{*}V} is

\displaystyle  \begin{array}{rcl}  d({X_t}^{-1}_{*}V)=\sum_{i=0}^{r}({X_t}^{-1}_{*}[A_i,V])_X \cdot dZ^i. \end{array}

In particular, if {V} commutes with vectorfields {A_i}, {{X_t}^{-1}_{*}V=V}.

2. Stochastic flows and hypoellipticity

We assume that all constant coefficient combinations of the {A_i} are complete. The flow defines two canonical measures,

  • The distribution of {X_t(x)}, {P_t(x,dy)=P\{X_t(x)\in dy\}},
  • Green’s measure {G_\lambda(x,dy)=\int_{0}^{\infty}P_t(x,dy)\,dt}.

Let us study the following Dirichlet boundary problem

\displaystyle  \begin{array}{rcl}  -Lu+ku&=&f \textrm{ on }D,\\ u_{\partial D}&=&\phi. \end{array}

The solution takes the following form (Feyman-Kac formula).

\displaystyle  \begin{array}{rcl}  u(x)=\mathop{\mathbb E}(\phi(X_{\tau_D})\exp(-\int_{0}^{\tau_D}k(X_s)\,ds)+\int_{0}^{\tau_D}f(X_s)\exp(\int_{0}^{s}k(X_r)\,dr)\,ds). \end{array}

2.1. Hörmander’s condition

Question: When do {P_t(x,dy)} and {G_\lambda(x,dy)} have a density ?

Let

  • {\mathcal{L}} denote the Lie algebra generated by the vectorfield {A_i},
  • {\mathcal{B}} the Lie algebra generated by {A_1,\ldots,A_r} only,
  • {\mathcal{J}} by {A_1,\ldots,A_r} and brackets {[A_0,A_i]},
  • {\hat{\mathcal{L}}} by {A_0+\partial_t} and {A_1,\ldots,A_r} on {M\times{\mathbb R}}.

Hörmander’s theorem states

  • hypoellipticity of {L} under {\mathcal{L}(x)=T_xM},
  • hypoellipticity of {L+\partial_t} under {\hat{\mathcal{L}}(x)=T_{x,t}M\times{\mathbb R}}.

It follows that

  • Under {\mathcal{L}(x)=T_xM}, {G_\lambda(x,dy)=g_\lambda(x,y)\,dy},
  • Under {\hat{\mathcal{L}}(x)=T_{x,t}M\times{\mathbb R}}, {P_t(x,dy)=p_t(x,y)\,dy},

where the densities are smooth.

2.2. A probabilistic proof of hypoellipticity ?

In 1970, in his Kyoto lectures, Paul Maillavin proposed a toolbox to prove this, called Maillavin Calculus. This calculus deals with infinite dimensional path spaces.

Instead, I will describe a more direct root. The existence of smooth densities {g_\lambda} and {p_t} in turn imply hypoellipticity, so it suffices to prove this.

Theorem 2 Assume that {A_i} and there derivatives satisfy suitable growth conditions. Assume that the bilinear form {C_t(x)} is non-degenerate and

\displaystyle  \begin{array}{rcl}  |C_t(x)|^{-1}\in L^p \end{array}

for all {p\geq 1}. Then {P_t(x,dy)=p_t(x,y)\,dy} with a smooth density {p_t(x,y)}.

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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.

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Notes of Ludovic Rifford’s lecture nr 4

Open problems

  1. The Sard conjecture
  2. Regularity of geodesics
  3. Small balls

1. The Sard conjecture

1.1. Statement

Theorem 1 (Morse 1939 for {p=1}, Sard 1942) If {f:{\mathbb R}^d\rightarrow{\mathbb R}^p} is of class {C^k},

\displaystyle  \begin{array}{rcl}  k\geq\max\{1,d-p+1\}\quad \Rightarrow\quad \mathcal{L}^p(\textrm{critical values})=0, \end{array}

and this is sharp (Whitney).

Does this theorem generalize to the endpoint map of a smooth control system ?

Conjecture. The set of all positions at time {t} of singular paths starting at {x} has measure zero.

Remark. There are examples of smooth (even polynomial) functions on {L^2} which do not satisfy Sard’s theorem. The only infinitesimal version of Sard’s theorem is Smale’s for Fredholm maps.

Conjecture is open for Carot groups (which may be harder).

1.2. Positive cases

Fat distributions have no singular curves but constants.

For rank two distributions in dimension 3, singular curves are contained in the Martinet surface which is known to be countably 2-rectifiable. Conjecturally, the singular values of the endpoint map have Hausdorff dimension {\leq 1}. Generically, the horizontal curves on the Martinet surface form a foliation whose singularity are either saddles or foci. At foci, the length of leaves is infinite, so one can ignore them.

1.3. The minimizing Sard conjecture

Let {S} denote the set of points joined to {x} by a minimizing geodesic which is singular. Let {S_s\subset S} denote the set of points joined to {x} by a minimizing geodesic which is singular and not the projection of a normal extremal.

The following partial result turns out to be rather easy.

Proposition 2 (Rifford-Trélat, Agrachev) {S} has empty interior.

Lemma 3 Assume that there is a function {\phi:M\rightarrow{\mathbb R}} such that

  1. {\phi} is differentiable at {y},
  2. {\phi(y)=d(x,y)^2} and {d(x,y)^2>\phi(z)} for all neigboring {z\not=y}.

Then there is a unique minimizing geodesic between {x} and {y}, which is the projection of a normal extremal {\psi} such that {\psi(1)=(y,D_y\psi}.

\proof

Let {v} be the control of some minimizing geodesic. For {u\in L^2} close to {v},

\displaystyle  \begin{array}{rcl}  \|u\|_{L^2}^2 =C(u)\geq e(x,E^x(u)), \end{array}

with equality at {u=v}. By assumption, {e(x,E^x(u))\geq\phi(E^x(u))}, with equality at {u=v}. Therefore {v} minimizes {C(u)-\phi(E^x(u))} in a neighborhood of {v}, and it is locally unique. So there is {p\in T^*_y M} such that {p\cdot D_vE^x=D_vC}, {v} is normal, q.e.d.

\proof

of Proposition. Any continuous function has a smooth (even constant) support function at a dense set of points, q.e.d.

Question. Can one improve this to full measure ?

2. Regularity of minimizers

Projections of normal extremals are smooth.

Question. Are abnormal minimizing geodesics of class {C^1} ?

2.1. Partial results

Theorem 4 (Monti-Leonardi) Consider an equiregular ({Lie^k} all have constant dimension) distribution. Assume that {[Lie^k,Lie^\ell]\subset Lie^{k+\ell+1}}. Then curves with a corner cannot be minimizing.

Theorem 5 (Süssmann) If data are real analytic, singular controls are real analytic on an open dense subset of their interval f definition.

This comes from sub-analytic geometry.

3. Small balls

Question. Are small spheres homeomorphic to spheres ?

It is true in Carnot groups.

Yuri Baryshnikov claims that the answer is yes in the contact case, but the proof does not seem to be correct.

In the absence of abnormal geodesics, then almost every sphere at {x} is a Lipschitz submanifold.

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Notes of Nicola Garofalo’s lecture nr 3

1. Fundamental solutions

Exercise (related to the Hopf-Rinow): compute the sub-Riemannan metric associated to vectorfield {X=(1+x^2)\partial_x}. Observe that balls are non compact, i.e. metric is not complete.

2. Existence

Theorem 1 (Folland) On a Carnot group, all sub-Laplacians {\Delta_H} have a unique fundamental solution, i.e. a smooth fonction {\Gamma} on {G\setminus\{e\}} such that

  1. {\Delta_H \Gamma=\delta}, Dirac distribution at the origin,
  2. {\lim_{|g|\rightarrow\infty} \Gamma(g)=0}.

It is homogeneous of degree {2-Q} under dilations.

\proof of homogeneity. Consider {v=\Gamma\circ\delta_\lambda-\lambda^{2-Q}}. Then {\Delta_H v=0}. By hypoellipticity, {v} is smooth and classically harmonic.

By Bony’s maximal principle, since {v} teds to 0 at infinity, {v=0}. Alternatively, use Liouville’s theorem.

2.1. The case of groups of Heisenberg type

Charles Feffermann, studying several complex variables, suggested the form that the fundamental solution should take in the Heisenberg group. This was implemented by Folland and Kaplan.

Theorem 2 (Folland 1972, Kaplan 1981) Let {G} be of Heiseberg type. The function

\displaystyle  \begin{array}{rcl}  \Gamma(g)=\frac{C}{(|z|^4+16|t|^2)^{\frac{Q-2}{4}}} \end{array}

is a fundamental solution of {-\Delta_H}. Here, {C} is a suitable constant,

\displaystyle  \begin{array}{rcl}  C^{-1}=m(Q-2)\int_{G}\frac{1}{((|z|^2+1)^2+16|t|^2)^{\frac{Q+2}{4}}}. \end{array}

2.2. A Lemma

Lemma 3 If {G} is of Heisenberg type,

  1. {\Delta_H(|t|^2)=\frac{k}{2}|z|^2},
  2. {|\nabla_H(|t|^2)|^2=|z|^2|t|^2},
  3. {\langle\nabla_H(|z|^2),\nabla_H(|t|^2)\rangle=0}.

For this, use Baker-Campbell-Hausdorff to compute

\displaystyle  \begin{array}{rcl}  z_j(g\exp(se_i))&=&z_j(g)+s\delta_{ij},\\ t_\ell(g\exp(se_i))&=&t_\ell(g)+\frac{s}{2}\langle[z,e_i],\epsilon_\ell\rangle. \end{array}

Differentiating with respect to {s} at {s=0}, this gives

\displaystyle  \begin{array}{rcl}  X_i(z_j)(g)&=&\delta_{ij},\\ X_i(t_\ell)(g)&=&\frac{1}{2}\langle[z,e_i],\epsilon_\ell\rangle=\frac{1}{2}\langle J(\epsilon_\ell)z,e_i\rangle. \end{array}

This leads rather easily to all 3 formulae.

2.3. Proof of the Folland-Kaplan Theorem

We see that {\Gamma(g)=C\,\rho^{2-Q}} where {\rho(g)=(|z|^4+16|t|^2)^{1/4}} is a gauge. Let us regularize it,

\displaystyle  \begin{array}{rcl}  \rho_\epsilon(g)=((|z|^2+\epsilon^2)^2+16|t|^2)^{1/4}. \end{array}

Then

\displaystyle  \begin{array}{rcl}  |\nabla_H \rho_\epsilon|^2&=&\frac{|z|^2}{\rho_\epsilon^2},\\ \Delta_H\rho_\epsilon&=&\frac{Q-1}{\rho_\epsilon}|\nabla_H \rho_\epsilon|^2+\frac{m\epsilon^2}{\rho_\epsilon^3}. \end{array}

Given an arbitrary function {h:{\mathbb R}\rightarrow{\mathbb R}}, differentiate {v=h\circ\rho_\epsilon}. Then apply it to {h(t)=t^{2-Q}} and observe that this kills a term, yielding

\displaystyle  \begin{array}{rcl}  \Delta_H v&=&\frac{m\epsilon^2}{\rho_\epsilon^3}h'(\rho_\epsilon)\\ &=&m(2-Q)\epsilon^2\rho_\epsilon^{-2-Q}\\ &=&-m(Q-2)\epsilon^2 v^{\frac{Q+2}{Q-2}}. \end{array}

This equation is known as the CR Yamabe equation. This is the conformally invariant form of the sub-Laplacian. It indicates that {v} is critical for the sub-Riemannian Sobolev inequality.

Observe that

\displaystyle  \begin{array}{rcl}  \rho_\epsilon=\epsilon\delta_{\epsilon^{-1}}\circ\rho_1. \end{array}

Thus

\displaystyle  \begin{array}{rcl}  \Delta_H v&=&\epsilon^{-Q}\delta_{\epsilon^{-1}}\circ\Delta_H(\rho_1^{2-Q})\\ &=&-m(Q-2)\epsilon^{-Q}\delta_{\epsilon^{-1}}\circ v_1^{\frac{Q+2}{Q-2}}. \end{array}

It turns out that {v_1^{\frac{Q+2}{Q-2}}\in L^1(G)}. So up to a multiplicative constant, {\epsilon^{-Q}\delta_{\epsilon^{-1}}\circ v_1^{\frac{Q+2}{Q-2}}} converges to the Dirac distribution as {\epsilon\rightarrow 0}. Indeed, given a test function {\phi},

\displaystyle  \begin{array}{rcl}  \langle \rho^{2-Q},\Delta_H\phi\rangle&=&\langle v,\Delta_H\phi\rangle\\ &=&\lim_{\epsilon\rightarrow 0} \langle v,\Delta_H\phi\rangle\\ &=&\lim_{\epsilon\rightarrow 0} \langle\Delta_H v,\phi\rangle\\ &=&-m(Q-2)\lim_{\epsilon\rightarrow 0} \epsilon^{-Q}\langle\delta_{\epsilon^{-1}}\circ v_1^{\frac{Q+2}{Q-2}},\phi\rangle\\ &=&-m(Q-2)\lim_{\epsilon\rightarrow 0} \epsilon^{-Q}\langle v_1^{\frac{Q+2}{Q-2}},\phi\circ \delta_{\epsilon^{-1}}\rangle\\ &=&-m(Q-2)\phi(e)\int_G v_1^{\frac{Q+2}{Q-2}}. \end{array}

2.4. The CR Yamabe problem

The problem: let {M} be a compact strictly pseudoconvex CR manifold, find a choice of the contact form {\theta}, for which the Tanaka-Webster scalar curvature is constant.

This is a sub-Riemannian analogue of a problem posed in 1959 by Yamabe, and which has been solved (Yamabe, Trudinger, Aubin, Schoen).

Theorem 4 (Jerison-Lee 1990) The CR Yamabe problem is solvable when dim{(M)\geq 5} and {M} is not locally CR equivalent to the round CR sphere.

The CR version

After a decade, Gamara and Yaccoub, two students of Abbas Bahri, solved the problem when {M} is CR equivalent to the CR round sphere. The 3-dimensional case was later completed by Gamara.

These cases non treated by Jerison and Lee are analogues of the Riemannian cases where the positive mass conjecture in general relativity plays a role. There have been recent progress along similar lines in CR geometry recently. Attend the relevant workshop this fall!

2.5. The sub-Riemannian Sobolev embedding theorem

Observe that

\displaystyle  \begin{array}{rcl}  \int_{G}|\nabla_H v|^2=-\int_{G}v\Delta_H v=\int_{G}v^{\frac{2Q}{Q-2}}. \end{array}

This is an equality case in a Sobolev type inequality. The Euclidean Sobolev inequality reads

\displaystyle  \begin{array}{rcl}  (\int_{{\mathbb R}^n}|u|^{q})^{1/q}\leq S(\int_{{\mathbb R}^n}|\nabla u|^p)^{1/p}. \end{array}

The numerology {\frac{1}{p}-\frac{1}{q}=\frac{1}{n}} is forced by dilaton invariance.

Theorem 5 (Folland-Stein 1975) In a Carnot group, let {1<p<Q}. There exists a constant {S_q(G)} such that, for all smooth compactly supported functions {u},

\displaystyle  \begin{array}{rcl}  (\int_{G}|u|^{q})^{1/q}\leq S_q(G)(\int_{G}|\nabla u|^p)^{1/p}, \end{array}

provided {\frac{1}{p}-\frac{1}{q}=\frac{1}{Q}}.

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