Propagation of Singularities for Subelliptic Wave Equations

Abstract

Hörmander’s propagation of singularities theorem does not fully describe the propagation of singularities in subelliptic wave equations, due to the existence of doubly characteristic points. In the present work, building upon a visionary conference paper by Melrose (in: Hyperbolic equations and related topics, Academic Press, pp 181–192, 1986), we prove that singularities of subelliptic wave equations only propagate along null-bicharacteristics and abnormal extremals, which are well-known curves in optimal control theory. As a consequence, we characterize the singular support of subelliptic wave kernels outside the diagonal. These results show that abnormal extremals play an important role in the classical-quantum correspondence between sub-Riemannian geometry and sub-Laplacians.

Appendix

1.1 Sign conventions in symplectic geometry

In the present work, we take the following conventions (the same as [18], see Chapter 21.1): on a symplectic manifold with canonical coordinates $(x,\xi )$, the symplectic form is $\omega =d\xi \wedge dx$, and the Hamiltonian vector field $H_f$ of a smooth function f is defined by the relation $\omega (H_f,\cdot )=-df(\cdot )$. In coordinates, it reads

$$\begin{aligned} H_f=\sum _j (\partial _{\xi _j}f) \partial _{x_j} -(\partial _{x_j} f)\partial _{\xi _j}. \end{aligned}$$

In these coordinates, the Poisson bracket is

$$\begin{aligned} \{f,g\}=\omega (H_f,H_g)=\sum _j (\partial _{\xi _j}f) (\partial _{x_j}g) -(\partial _{x_j} f)(\partial _{\xi _j}g), \end{aligned}$$

which is also equal to $H_fg$ and $-H_gf$.

1.2 Pseudodifferential operators

This appendix is a short reminder on basic properties of pseudodifferential operators. Most proofs can be found in [18]. In this paper, we work with the class of polyhomogeneous symbols (defined below), which is slightly smaller than the usual class of symbols but has the advantage that the subprincipal symbol can be read easily when using the Weyl quantization (see [18], the paragraph before Section 18.6).

We consider $\Omega$ an open set of a d-dimensional manifold, and $\mu$ a smooth volume on $\Omega$. The variable in $\Omega$ is denoted by q. Let $\pi :T^*\Omega \rightarrow \Omega$ be the canonical projection.

$S_{\text {hom}}^n(T^*\Omega )$ stands for the set of homogeneous symbols of degree n with compact support in $\Omega$. We also denote by $S_{\mathrm{phg}}^n(T^*\Omega )$ the set of polyhomogeneous symbols of degree n with compact support in $\Omega$. Hence, $a\in S_{\mathrm{phg}}^n(T^*\Omega )$ if $a\in C^\infty (T^*\Omega )$, the projection $\pi (\text {supp}(a))$ is a compact of $\Omega$, and there exist $a_j\in S^{n-j}_{\text {hom}}(T^*\Omega )$ such that for any $N\in {\mathbb {N}}$, $a-\sum _{j=0}^N a_j\in S_{\mathrm{phg}}^{n-N-1}(T^*\Omega )$. We denote by $\Psi ^n_\mathrm{phg}(\Omega )$ the space of polyhomogeneous pseudodifferential operators of order n on $\Omega$, with a compactly supported kernel in $\Omega \times \Omega$.

We use the Weyl quantization denoted by $\mathrm{Op}:S^n_\mathrm{phg}(T^*\Omega )\rightarrow \Psi ^n_\mathrm{phg}(\Omega )$. It is obtained by using partitions of unity and the formula in local coordinates

$$\begin{aligned} \mathrm{Op}(a)f(q)=\frac{1}{(2\pi )^d}\int _{{\mathbb {R}}^d_{q'}\times {\mathbb {R}}^d_{p}}e^{i\langle q-q',p\rangle }a\left( \frac{q+q'}{2},p\right) f(q')dq'dp. \end{aligned}$$

If a is real-valued, then $\mathrm{Op}(a)^*=\mathrm{Op}(a)$. Moreover, with this quantization, the principal and subprincipal symbols of $A=\mathrm{Op}(a)$ with $a\sim \sum _{j\leqslant n} a_j$ are simply $\sigma _p(A)=a_n$ and $\sigma _{\text {sub}}(A)=a_{n-1}$ (usually, the subprincipal symbol is defined for operators acting on half-densities, but we make here the identification $f\leftrightarrow fd\nu ^{1/2}$).

We also have the following properties:

• 1. If $A\in \Psi ^l_\mathrm{phg}(\Omega )$ and $B\in \Psi ^n_\mathrm{phg}(\Omega )$, then $[A,B]\in \Psi ^{l+n-1}_\mathrm{phg}(\Omega )$. Moreover, $\sigma _p([A,B])=\frac{1}{i}\{\sigma _p(a),\sigma _p(b)\}$ where the Poisson bracket is taken with respect to the canonical symplectic structure of $T^*\Omega$.

• 2. If X is a vector field on $\Omega$ and $X^*$ is its formal adjoint in $L^2(\Omega ,\mu )$, then $X^*X$ is a second order pseudodifferential operator, with $\sigma _p(X^*X)=h_X^2$ and $\sigma _{\text {sub}}(X^*X)=0$. Here, for X a vector field, we denoted by $h_{X}$ the momentum map given in canonical coordinates $(x,\xi )$ by $h_{X}(x,\xi )=\xi (X(x))$.

• 3. If $A\in \Psi _\mathrm{phg}^n(\Omega )$, then A maps continuously the space $H^s(\Omega )$ to the space $H^{s-n}(\Omega )$.

Finally, we define the essential support of A, denoted by $\mathrm{essupp}(A)$, as the complement in $T^*\Omega$ of the points (q, p) which have a conic-neighborhood W so that A is of order $-\infty$ in W.

1.3 The cones $\Gamma _m$ as generalized Hamiltonians

In this section, we interpret the set $B=B(m)$ which appears in the formula (15), namely

$$\begin{aligned} B(m)=\left\{ b \in \mathrm{ker}(a_m)^{\perp _{\omega _{X}}}, \ a_m^*({\mathcal {I}}(b))\leqslant 1\right\} , \end{aligned}$$

as a generalized Hamiltonian, just adapting the notion of Clarke generalized gradient (see [6, Chapter 1.2]) to the “Hamiltonian” framework.

Definition 32

Let f be an almost everywhere differentiable function on $T^*X$ and let $\Omega _f$ be the set of points where it is not differentiable. Its generalized Clarke Hamiltonian ${\mathcal {H}}f(x)$ at $x\in \Omega _f$ is the set

$$\begin{aligned} {\mathcal {H}}f(x)=\mathrm{cxhl}\left\{ \underset{j\rightarrow +\infty }{\lim } H_f(x_j), \ x_j\rightarrow x, \ x_j\notin \Omega _f\right\} \subset T_x(T^*X) \end{aligned}$$

where $\mathrm{cxhl}$ denotes the convex hull.

The main result of this section is the following:

Proposition 33

For any $m\in \Sigma _{(2)}$, $B(m)={\mathcal {H}}\sqrt{a}(m)$.

This proposition, beside giving an alternative proof of Lemma 7, draws a link between our computations and the Pontryagin maximum principle in the Clarke formulation, which asserts that any sub-Riemannian geodesic (see Sect. 5.1) is a solution of the differential inclusion

$$\begin{aligned} {\dot{\gamma }}(s)\in {\mathcal {H}}\sqrt{a}(\gamma (s)). \end{aligned}$$

The projection of a null-ray in $T^*X$ is also by Definition 9 a solution of this differential inclusion, and this “explains” why abnormal extremals appear naturally in Corollary 3.

Before proving Proposition 33, we introduce the “fundamental matrix” F (see [18, Section 21.5]) defined as follows:

$$\begin{aligned} \forall Y,Z\in T_{m}(T^*X), \qquad \omega _{X}(Y,FZ)=a_m(Y,Z). \end{aligned}$$

(64)

Here $a_m(Y,Z)=\frac{1}{2} (\mathrm{Hess\;} a)(m)(Y,Z)$. Then, $\omega _{X}(FY,Z)=-\omega _{X}(Y,FZ)$. As already explained in Sect. 2.2, there is here a slight abuse of notations since $T_m(T^*X)$ stands for $T_{\pi _2(m)}(T^*X)$ where $\pi _2:M\rightarrow T^*X$ is the canonical projection on the second factor.

Lemma 34

The fundamental matrix induces an isomorphism

$$\begin{aligned} F:T_{m}(T^*X)/\mathrm{ker}(a_m)\rightarrow \mathrm{ker}(a_m)^{\perp _{\omega _{X}}} \end{aligned}$$

Proof

F clearly passes to the quotient by $\mathrm{ker}(a_m)$ by (64). Let $b \in \mathrm{ker}(a_m)^{\perp _{\omega _{X}}}$. We set $b_0=-{\mathcal {I}}(b)\in \mathrm{ker}(a_m)^\perp$. The bilinear form $a_m$ is continuous and coercive on $T_m(T^*X)/\mathrm{ker}(a_m)$, and $b_0$ is a linear form on this space, thus by Lax-Milgram’s lemma we get the existence of $Z\in T_m(T^*X)/\mathrm{ker}(a_m)$ such that $b_0=a_m(Z,\cdot )$. Finally, we have

$$\begin{aligned} -{\mathcal {I}}(b)=b_0=a_m(\cdot ,Z)=\omega (\cdot ,FZ)=-{\mathcal {I}}(FZ) \end{aligned}$$

according to (16), which means that $b=FZ$. $\square$

Now we derive a formula for B(m) in terms of the fundamental matrix (see formula (2.6) in [29]):

Lemma 35

There holds

$$\begin{aligned} B(m)=\mathrm{cxhl}\left\{ \frac{FZ}{a_m(Z)^{\frac{1}{2}}}, \ Z\in T_{m}(T^*X)/\mathrm{ker}(a_m)\right\} . \end{aligned}$$

(65)

Proof

We have to compare (15) with (65).

First, let $b \in \mathrm{ker}(a_m)^{\perp _{\omega _{X}}}$ with $a_m^{*}({\mathcal {I}}(b ))\leqslant 1$. By the proof of Lemma 34, there exists $Z\in T_{m}(T^*X)/\mathrm{ker}(a_m)$ such that $-{\mathcal {I}}(b)=b_0=a_m(Z,\cdot )$. Using that $a_m^{*}(b_0)\leqslant 1$, we obtain $a_m(Z)\leqslant 1$, hence $b_0=\lambda a_m(Z,\cdot )/a_m(Z)^{\frac{1}{2}}$ where $|\lambda |\leqslant 1$. It follows that $b =-{\mathcal {I}}^{-1}(b_0)=\lambda FZ/a_m(Z)^{\frac{1}{2}}$. This proves that the cones given by (15) are included in those given by (65).

For the converse, we first notice that $FZ/a_m(Z)^{\frac{1}{2}}\in \mathrm{ker}(a_m)^{\perp _{\omega _{X}}}$, and thus it is also the case for any convex combination. Also, it follows from the definitions of ${\mathcal {I}}$, F, $a_m^*$ and the Cauchy-Schwarz inequality that

$$\begin{aligned} \forall Z\in T_{m}(T^*X)/\mathrm{ker}(a_m), \quad a_m^*({\mathcal {I}}(FZ)/a_m(Z)^{\frac{1}{2}})\leqslant 1. \end{aligned}$$

By convexity of $a_m^*$, we obtain that any convex combination b of elements of the form $FZ/a_m(Z)^{\frac{1}{2}}$ satisfies $a_m^*({\mathcal {I}}(b))\leqslant 1$. This concludes the proof. $\square$

Proof of Proposition 33

As in Sect. 2.3, we work in a chart near m. Following the computations of Lemma 8, we have for any sequence of points $(m_j)_{j\in {\mathbb {N}}}$ such that $m_j-m\notin \mathrm{ker}(a_m)$,

$$\begin{aligned} \frac{1}{2}\omega _{X}(H_a(m_j),w)&=-\frac{1}{2}da(m_j)(w)=-a_m(m_j-m,w)+o(m_j-m)\\&=\omega _{X}(F(m_j-m),w)+o(m_j-m), \end{aligned}$$

which implies

$$\begin{aligned} H_{\sqrt{a}}(m_j)=\frac{1}{2}\frac{H_a(m_j)}{a(m_j)^{\frac{1}{2}}}=\frac{F(m_j-m)}{a_m(m_j-m)^{\frac{1}{2}}}+o(1). \end{aligned}$$

(66)

Choosing $m_j=m+\varepsilon _j Z$ with $\varepsilon _j\rightarrow 0$, we obtain

$$\begin{aligned} H_{\sqrt{a}}(m_j)\underset{j\rightarrow +\infty }{\longrightarrow }\frac{FZ}{a_m(Z)^{\frac{1}{2}}} \end{aligned}$$

which proves that $B(m)\subset {\mathcal {H}}\sqrt{a}(m)$ according to Lemma 35.

Conversely, since F is a linear isomorphism (see Lemma 34), it is not difficult to see that any limit of $\frac{F(m_j-m)}{a_m(m_j-m)^{\frac{1}{2}}}$ is of the form $\frac{FZ}{a_m(Z)^{\frac{1}{2}}}$. Using (66) and taking convex hulls, this proves that ${\mathcal {H}}\sqrt{a}(m)\subset B(m)$. $\square$