On evolution equations for Lie groupoids

The main motivation of this paper is the construction of an approximate solution to the problem

in the framework of a Lie groupoid $G\rightrightarrows M$. This means that $P$ here is a suitable order $1$ pseudodifferential $G$-operator, that $f,g$ live in suitable spaces of distributions and that the approximate solution will be seeked among Fourier integral $G$-operators. The present article can be considered as a continuation of [17], where properties of distributions on Lie groupoids, and convolution of them, are studied in a certain generality, and of [18], where Hörmander’s notion and calculus of Fourier integral operators on manifolds [11, 12] are exported to the framework of Lie groupoids. We will frequently refer to the results of these papers, and one of their cornerstones, namely the symplectic groupoid structure of $T^{*}G$ [6]: $s_{{}_{\Gamma}},r_{{}_{\Gamma}}:\Gamma=T^{*}G\rightrightarrows A^{*}G$, will be of great importance here again.

The Cauchy problem (1) has been and can be of course investigated in many situations under many different assumptions. We refer more precisely to [12, Theorem 29.1.1] to illustrate the kind of results that we want to achieve on Lie groupoids. This can be summarized by the following problem:

$(\mathcal{P})$ Under an ellipticity assumption on $P$, the fundamental solution of (1) should have, up to suitable regularizing error terms, an explicit approximation by Fourier integral $G$-operators that describes in a simple and geometric way how the singularities of the initial data $g$ propagate at time $t$ under the action of the principal symbol of $P$.

To set the problem on firm foundations, we first study in Section 3 existence and unicity conditions for (1) in the general framework of $C^{*}$-algebras and Hilbertian modules, and we require there that $P$ is an unbounded self-adjoint regular operator on a $C^{*}$-algebra $A$ [2, 3, 33, 13, 32, 30]. Then the fundamental solution of (1) denoted by $E(t)=e^{-itP}$ is obtained by continuous functional calculus, which yields the existence of solutions, while easy computations identical to those for Hilbert spaces show the uniqueness. We get in particular:

This preliminary result allows us to speak about the fundamental solution of (1) in the case of a Lie groupoid $G$ with compact units space $M$ and of a first order elliptic symmetric and compactly supported, polyhomogeneous pseudodifferential $G$-operator $P$. Indeed, we then know by [32] that (the closure of) $P$ is selfadjoint and regular on, for instance, the reduced $C^{*}$-algebra of $G$, denoted by $C^{*}_{r}(G)$. In particular, the theorem above applies and the task to find a nice approximation to $E(t)$ among Fourier integral $G$-operators is meaningful. Note that, because of (2), the error term will automatically belong to the space $\mathcal{H}^{\infty}=H^{\infty}\cap(H^{\infty})^{*}$. Our answer to the problem $(\mathcal{P})$ is the main result of the paper:

Let us now explain in some details the ingredients and the intermediate results, some of them being interesting on their owns, required in the proof of the main theorem.

First of all, Theorem 2 immediately rises the preliminary question of the regularity of elements in ${\mathcal{H}^{\infty}}$. Strictly speaking, elements of ${\mathcal{H}^{\infty}}$ live in a noncommutative $C^{*}$-algebra so dealing with their local properties makes a priori no sense. We manage on the one hand to prove that elements of the reduced $C^{*}$-algebra $C^{*}_{r}(G)$ of a Lie groupoid $G$ are distributions on $G$, in a canonical way, and on the other hand, to precise the regularity of elements in ${\mathcal{H}^{\infty}}$. These intermediate tasks are the subject of Sections 4 and 5 and the details can be summarized as follows.

The space of distributions we deal with, denoted by $\mathcal{D}^{\prime}(G)$, is the one of distributions on $G$ with values in the density bundle $\Omega^{1/2}:=\Omega^{1/2}(r^{*}AG)\otimes\Omega^{1/2}(s^{*}AG)$ and thus the space of test functions we use, denoted by $\mathcal{D}(G)$, is the one of compactly supported $C^{\infty}$ sections of the density bundle $\Omega^{1/2}_{0}:=\Omega^{-1/2}\otimes\Omega^{1}_{G}$. Thus $\mathcal{D}^{\prime}(G)=(\mathcal{D}(G))^{\prime}$ and the choice of $\Omega^{1/2}$ is relevant because $C^{\infty}_{c}(G):=C^{\infty}_{c}(G,\Omega^{1/2})\subset\mathcal{D}^{\prime}(G)$ is in a canonical way an involutive algebra, whose a certain completion is precisely the algebra $C^{*}_{r}(G)$. Also, we have proved in [17] that the product $\star$ in $C^{\infty}_{c}(G)$ (called the convolution product for obvious reasons) widely generalizes, by continuity, to distributions in $\mathcal{D}^{\prime}(G)$. For instance the space $\mathcal{E}^{\prime}_{r,s}(G)$ of compactly supported distributions on $G$ whose pushforwards by the source and range maps are $C^{\infty}$ on $M$ (transversal distributions) forms a unital involutive algebra for the convolution product. Another justification for these choices of densities comes from the present work, indeed we prove that transversal distributions also act by convolution on $\mathcal{D}(G)$ in a nice way, and that weak factorizations in the sense of [10] are available:

This material allows us in Section 5 to answer to the question about the local nature of elements in ${\mathcal{H}^{\infty}}$, and along the way, that of elements in $C^{*}_{r}(G)$:

Next, we explain how the principal symbol $p$ of $P$ gives rise to the family of Lagrangian submanifolds $\Lambda_{t}$, $t\in{\mathbb{R}}$, that will describe the propagation of singularities as expected in Problem $(\mathcal{P})$. By definition, $P\in I^{1+(n^{(1)}-n^{(0)})/4}(G,M,\Omega^{1/2})$ is a polyhomogeneous conormal distribution, thus posseses a homogeneous principal symbol $p^{0}_{0}\in C^{\infty}(A^{*}G\setminus 0)$. If one considers the family $(P_{x})_{x\in M}$ of ordinary pseudodifferential operators in the fibers of $s$ and collects their principal symbols into a homogeneous function $p^{0}\in C^{\infty}(T_{s}^{*}G\setminus 0)$, $T^{*}_{s}G=(\ker ds)^{*}$, that will be called the principal $G$-symbol of $P$. After lifting $p^{0}$ to a function on $T^{*}G\setminus\ker r_{{}_{\Gamma}}$, one gets the following identity:

Here $r_{{}_{\Gamma}}$ is the range map of the symplectic groupoid $\Gamma=T^{*}G$. The computations also give a local expression for the sub-principal $G$-symbol of $P$, that is, for the collection of the sub-principal symbols of the operators $P_{x}$. Now it turns out that the Hamiltonian flow $\chi$ of the principal $G$-symbol $p$ is complete and right invariant, and we get the required Lagrangian submanifolds that will describe the evolution of singularities:

This already produces a $C^{\infty}$ family of homogeneous Lagrangian submanifolds of $T^{*}G\setminus 0$ that satisfies the group relation, with respect to the product in $T^{*}G$:

Moreover $\Lambda_{t}\subset\accentset{\mbox{\large.}}{T}^{*}G:=T^{*}G\setminus(\ker r_{{}_{\Gamma}}\cup\ker s_{{}_{\Gamma}})$, that is, in the vocabulary of [18], every $\Lambda_{t}$ is a $G$-relation, while the global object coming with the family $(\Lambda_{t})_{t}$:

is a family ${\mathbb{R}}\times G$-relation. As in [18], this construction highlights the important role of the symplectic groupoid structure of $T^{*}G$ in analysis.

There is a last result, of technical nature, that intervenes in the proof of Theorem 2. Indeed, assuming that the Lagrangian submanifolds $\Lambda_{t}$ provide the good candidate for Theorem 2, we are led to search a first order parametrix $U_{0}$ for $\partial_{t}+iP$ among the Fourier integral $G$-operators associated with $(\Lambda_{t})_{t}$. This amounts to solve the transport equation for principal symbols:

and thus it requires to express the principal symbol of the convolution product $PU_{0}$ of the lagrangian distributions $P$ and $U_{0}$. Since by construction and on purpose, the principal symbol $p_{0}$ vanishes on $r_{{}_{\Gamma}}\Lambda$, we need to look for the next term in the asymptotic expansion of the total symbol of $PU_{0}$. This is what is achieved, modulo some technical details, by using the following result:

Many interesting situations produce non compactly supported operators $P$: for instance, if $\Delta$ is a Laplacian on $G$ then $\sqrt{\Delta}=P+S$ with $P$ as above and $S\in\mathcal{H}^{\infty}$ [32]. The main theorem trivially extends to such non compactly supported operators: one just needs to replace $C^{\infty}_{c}(G)$ by ${\mathcal{H}^{\infty}}$ in (3). We describe at the end of the paper several situations where Theorem 2 applies:

1. (1)

   The usual pseudodifferential calculus on a compact manifold without boundary $X$. We use the pair groupoid $G=X\times X\rightrightarrows X$. Since $X$ itself is an orbit, we have $C^{\infty,0}_{\mathop{\mathrm{orb}}\nolimits}(G,E)=C^{\infty}(X\times X,E)$ and we just recover the classical result on manifolds.

2. (2)

   The longitudinal calculus on foliations [4]. We use the holonomy groupoid. We recover a construction in [15] by a quite different approach.

3. (3)

   Right invariant calculus on a Lie group $G$. We use $G$ as a groupoid with units space $\{e\}$.

4. (4)

   The $b$-calculus on manifolds with corners [23]. We use the $b$-groupoid [25].

5. (5)

   The calculus on manifolds with fibred boundary or with iterated fibred corners [20, 8]. We use the groupoid of [8].

As far as we know, the results obtained for cases (3), (4), (5) above are new.

The next section contains the basic definitions and notation necessary for the sequel and can be considered as an extension of the introduction for the unfamiliar reader.

A fundamental point is that compactly supported one densities on $X$ can be integrated over $X$. More precisely, there is a unique linear form $\int_{X}:C^{\infty}_{c}(X,\Omega^{1}_{X})\longrightarrow{\mathbb{R}}$ such that if $f=\mathsf{f}(x)|dx|$ is compactly supported in a local chart $U$ with local coordinates $x=(x_{1},\ldots,x_{n})$, then

Above, $|dx|$ is the one density defined by $|dx|=|dx_{1}\wedge\cdots\wedge dx_{n}|$. Diffeomorphisms $\phi:X\to Y$ provide isomorphisms $\phi^{*}:\Omega^{\alpha}_{Y}\to\Omega^{\alpha}_{X}$ given by $\phi^{*}\omega(V)=\omega(\phi_{*}V)$. By construction, the integral of one densities is invariant under the action of diffeomorphisms. Densities are usually handled with the following canonical isomorphisms:

1. -

   $\Omega^{\alpha}(E)\otimes\Omega^{\beta}(E)\simeq\Omega^{\alpha+\beta}(E)$

2. -

   $\Omega^{\alpha}(E\oplus F)\simeq\Omega^{\alpha}(E)\otimes\Omega^{\alpha}(F)$,

3. -

   $\Omega^{\alpha}(E^{*})\simeq\Omega^{-\alpha}(E)$

4. -

   if $0\to F\to E\to G\to 0$ is exact, then $\Omega^{\alpha}(E)\simeq\Omega^{\alpha}(F)\otimes\Omega^{\alpha}(G)$.

We will consider in this paper homogeneous lagrangian submanifods of $T^{*}G\setminus 0$ that avoids the kernel of $r_{{}_{\Gamma}}$ and $s_{{}_{\Gamma}}$. We call them $G$-relations, in reference to the term canonical relations often employed for (product) manifolds. Under mild assumptions, $G$-relations compose well in the groupoid $T^{*}G$ [18]. $G$-relations $\Lambda$ such that $s_{{}_{\Gamma}}|_{\Lambda}$ and $r_{{}_{\Gamma}}|_{\Lambda}$ are diffeomorphisms onto their ranges are called invertible. We will sometimes use densities along the $s_{{}_{\Gamma}}$ and $r_{{}_{\Gamma}}$-fibers of the cotangent groupoid $T^{*}G$. Both are naturally isomorphic and:

where $\hat{E}$ denotes the pull back to $T^{*}G$ of the bundle $E\to G$. Also, we note that $\Omega^{1}_{s_{{}_{\Gamma}}}|_{A^{*}G}=\Omega^{1}(AT^{*}G)=(\mathcal{D}_{AG}^{\mathop{\mathrm{tr}}\nolimits})^{-1}$ where $\mathcal{D}_{AG}^{\mathop{\mathrm{tr}}\nolimits}$ is the transverse density bundle of $AG$ [7].

The convolution algebra structure on $C^{\infty}_{c}(G)$ refers to the product $\star$ canonically defined from any of the following three intuitive formulas:

This is justified as follows. Write $f=\mathsf{f}(\mu_{s}\mu_{r})^{1/2}$, $g=\mathsf{g}(\mu_{s}\mu_{r})^{1/2}$ with $\mathsf{f},\mathsf{g}\in C^{\infty}_{c}(G,{\mathbb{C}})$, $\mu_{s}=r^{*}\mu\in C^{\infty}(G,\Omega^{1}_{s})$, $\mu_{r}=s^{*}\mu\in C^{\infty}(G,\Omega^{1}_{r})$ for some positive $\mu\in C^{\infty}(M,\Omega^{1}(AG))$. Then, whenever $\gamma_{1}\gamma_{2}=\gamma$:

We now may set rigorously:

This gives consistance to the first formula in (6). The second and third formulas are obtained from the first one using the diffeormorphisms $L_{\gamma}\circ\iota:G_{s(\gamma)}\to G^{r(\gamma)}$ and $(L_{\gamma}\circ\iota,\operatorname{Id}):G_{s(\gamma)}\to m^{-1}(\gamma)$. Equivalently, one can directly define them as we did for the first one using the suitable structural isomorphisms to create the appropriate one densities on $G^{r(\gamma)}$ and $m^{-1}(\gamma)$. With the notation above, this concretely means:

with, for the last line:

By $C^{\infty,0}_{\pi}(G)$, we denote the space of elements in $C_{c}(G)$ that belong to $C(U_{(0)},C^{\infty}(U_{(1)}))$ over any local trivializations $\kappa:U\overset{\simeq}{\to}U_{(0)}\times U_{(1)}$ of $\pi$ (here $\pi=\mathop{\mathrm{pr}}\nolimits_{1}\circ\kappa$). The topology is modeled on that of $C(U_{(0)},C^{\infty}(U_{(1)}))$ and is Fréchet. We write $C^{\infty,0}_{\pi,c}$ for $C^{\infty,0}_{\pi}\cap C_{c}$, and equip it with the corresponding LF-topology.