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Exponential coordinates and regularity of groupoid heat kernels

We prove that on an asymptotically Euclidean boundary groupoid, the heat kernel of the Laplacian is a smooth groupoid pseudo-differential operator.


Introduction
In this notes we study the regularity of certain groupoid heat kernels.
1.1. The problem. We begin with stating our problem, recalling some basic notations along the way. Our motivation will be explained in the next section.
First of all, we briefly recall the notion of geoupoid pseudo-differential operators and heat kernels. The theory is developed by Nistor, Weinstein and Xu [8] and is considered classical.
Let G ⇒ M be a Lie groupoid. We shall assume G is Hausdorff. Denote the source and the target maps by s and t respectively. We shall use the convention s(ab) = s(b), ∀a, b ∈ G, t(b) = s(a). For each x ∈ M, write G x := s −1 (x) for the s-fiber over x, and T ⊥ G := {X ∈ T G : ds(X) = 0}. Note that T ⊥ G is just the tangent space of all the G x . Definition 1.1. A pseudo-differential operator Ψ on a groupoid G of order ≤ m is a smooth family of pseudo-differential operators {Ψ x } x∈M , where Ψ x ∈ Ψ m (G x ), and satisfies the right invariance property Ψ s(a) (r * a f ) = r * g Ψ t(a) (f ), ∀a ∈ G, f ∈ C ∞ c (G s(a) ). If, in addition, all Ψ x are classical of order m, then we say that Ψ is classical of order m. One particularly interesting groupoid differential operator is the Laplacian. Let A → M be the Lie algebroid of G with anchor map ν : A → T M, and E be a vector bundle over M. Let g A be a Riemannian metric on A and ∇ E be an A-connection on E (for the definition of an A-connection and other details, see [3]). By right invariance, one obtains a (family of) Riemannian metric and connection, which we still denote by G A and ∇ E respectively, on G x . Motivations and known results. The most notable partial result for Conjecture 1.6 is probably that of [4], where the author proves that the heat kernel on the holonomy groupoid is smooth. His proof uses heavily the constant rank of the (in general singular) foliation, which in turn enables one to naturally lift the orthogonal complement of the foliation on M to G.
Not much is known about groupoids with non-constant rank. The case of manifolds with boundary was proven by Melrose [6], using explicit construction with boundary defining functions. Essentially the same argument was used by Albin [1], generalizing the result to edge calculus. However these techniques do not generalize easily to other groupoids, because there is no obvious definition of boundary function.
For groupoids not of the edge type, the only example we know is that of [9]. There, the author gives a criterion for the derivatives of the groupoid multiplication, and then verify that the symplectic groupoid of the Bruhat Poisson structure satisfies these properties. Our argument here will essentially follow similar lines.
Our main motivation of establishing the regularity is in line of the authors quoted above. For example, the renormalized integral considered in [1] and [9] requires the integrand to be differentiable. That in turn means it is necessary to establish Conjecture 1.6 in order to obtain a renormalized index formula by the heat kernel method.
Another potentially interesting application of the results in this notes is on the polyhedral domain technique developed by Nistor et. al. The arguments developed in here may result in novel a-prior estimates for their numerical calculation.
Last but not the least, this notes can been seen as a sequel of [10], where the author attempts to generalize the classical theory of singular pseudo-differential calculus, using groupoids only. It appears the exponential coordinates defined here is central to many results (for example, smooth extension property) for boundary groupoids.

1.
3. An overview of the argument. As far as we know, the exponential coordinates [7] is the only way to construct local complements of the s-foliation. In general, exponential coordinates are very arbitrary and therefore its use in estimating derivatives is limited. In the case of boundary groupoid with exponential isotropy subgroups (which implies these Lie groups are solvable and simply connected), the situation is easier as one only needs a finite cover of the singular invariant submanifolds.
In Section 2, we shall give a technical introduction of these exponential coordinates. Then we study the change in coordinates formulas, as well as groupoid multiplication under these coordinates.
In Section 3, we first recall the construction of the heat kernel by the method of Levi parametix. Then we directly show that on each exponential coordinate patch, derivatives of the Levi paramterix converges uniformly, thanks to the estimation done in Section 2. Hence we conclude that the groupoid heat kernel is smooth.

Asymptotically Euclidean boundary groupoids
for some Lie groups G k ; (iv) For each k, there exists (unique) sub-bundlesĀ k ⊂ A|M k such thatĀ| M k = ker(ν| M k ).
For simplicity, we shall also assume that G k and M k are connected, hence all s-fibers are connected.
On any boundary groupoids we have the following fundamental estimates [10]. Fix any Riemannian metricḡ on M. For each k ≥ 1, let d(·,M k ) be the distance function defined by g. For each k ≥ 0, fix a function ρ k ∈ C ∞ (M) such that ρ k > 0 on M \M k and ρ k = d(·,M k ) on some open set containingM k .
Lemma 2.3. For each k, let ω k be defined in the previous Lemma 2.2. Suppose further that |dρ k • ν(X)| ≤ ω k ρ k (x)|X| g A for any X ∈ A. Then for any x ∈ M, a, b ∈ G x , ) .
In the same vein, we recall the definitions of uniformly degenerate and nondegenerate in [10].
for any x ∈M k−1 , X ∈ A x , and X⊥Ā k ; The groupoid G is said to be uniformly degenerate if there exist constants ω 1 , ω 2 , · · · , ω r , ω ′ 1 , · · · ω ′ r > 0 and exponents λ 1 , · · · λ r , Given any smooth section X ∈ Γ ∞ (A), denote by X r the right invariant vector field on G with s * X r = 0 and X r | M = X. Since M is compact, it is standard that X r is a complete vector field on G, hence one has a well defined map exp X : M → G, given by the flow of X r form each x ∈ M ⊂ G. It is a well known fact that t • exp X equals the flow of ν(X) on M and hence is a exp X is an admissible section. Define We list some basic properties of the exponential map [7], [5]:

Notation 2.5. For any collection of sections
We specialize the construction of exponential coordinates in [7] to our case. Fix an orthonormal bases {Y 1 , · · · , Y q } of R q . We regard it as a basis of T M 1 × R q and extend to an orthonormal set of sections on some neighborhood of M 1 , which we still denote the extension by {Y 1 , · · · , Y q }. LetṼ α ⊂ M ⊂ G be a coordinate patch on M, V α M 1 = ∅. We may assume that A|Ṽ α is trivial. Fix a orthonormal basis and extend X (α) to smooth compactly supported sections.
Notation 2.6. For any τ ∈ R q , denote by q } as above, and a arbitrary collec- Proof. By the inverse function theorem it suffices to prove that the differential of x is invertible. It is straightforward to compute for any a = x(x, µ, τ ), the 'coordinate vector fields' are given by In view that ds(∂ Observe that at x ∈ M 1 , w ji = 1 if i = j, w j ′ i ′ = 1, and zero otherwise. Moreover, it is clear that the x-derivatives of w are bounded by C 1 e M 1 τ . It follows that for any (x, τ ) ∈ T (r, M 1 ) with r > 0 sufficiently small, we can write with the last term on the right hand side uniformly small, hence invertible, and the assertion follows.
In the rest of this notes we shall make the following additional assumption: Assumption 2.9. For some T (r 0 , M ), the map x (α) are diffeomorphisms. In other words, x (α) defines a local coordinate patch.
2.2. Groupoid multiplication. The main advantage of the exponential coordinates we constructed, is that the groupoid multiplication can be described rather explicitly.
Integrating with respect to t, we get Using the same arguments as above, we get w ji ′ (x,φ,ψ) = 0 for any x ∈ M 1 and Remark 2.12. Observe that one can also write The techniques leading to Equation (6) can be further refined to give an estimate of the derivatives of the multiplication map. First consider d(exp tY i )∂ x (x (α) (x, τ, µ)). Clearly, one has To estimate ∂ x ψ(x, µ, τ, t) and ∂ x ϕ(x, µ, τ, t), differentiate Equation (5) with respect to x to get ∂ψ k ∂x ∂v ji ∂τ k (x, ϕ(x, µ, τ, t), ψ(x, µ, τ, t)).
The case for ∂ τ k and ∂ µ k ′ is much simpler. Observe that both vector lies in T ⊥ G, therefore ). Since the metric g on T ⊥ Gis right invariant, we have for some constants C 4 , M 4 , and clearly similar estimate holds for ∂ µ k ′ . Here, the important point to note is that the estimate does not depend on (x, µ, τ ).

Change of coordinates.
In this section we turn to the change of coordinate formulas. Observe that suppose one has x (β) (x, µ, τ ) = x (α) (x, ϕ(x, µ, τ ), ψ(x, µ, τ )), then Then, by possibly by restricting T (r, K), ϕ(x, µ, τ ) and ψ(x, µ, τ ) can be computed using Equation (6). In particular, the second equation of (6) implies ϕ(x, µ, τ ) still lies in the domain of x (α) . Moreover, repeated use of the first equation of (6) now implies that Here, note the assumption (x, τ ) ∈ T (r, K), K ≥ C 2 , implies all term on the right hand side are bounded. On the other hand, Lemma 2.13 implies that one has estimations for the change of coordinate vector fields: Fix a collection of coordinate charts and ε > 0 such that for all x, ∈ V α , µ (α) ∈ (−ε (α) , ε (α) ), there exists β (possibly same as α) such that Since the product is by definition just an iteration of left multiplication by admissible sections of the form exp Z then followed by exp τ 2 · Y . In particular, we may change coordinates so that Then by Equation (6) we have Obviously the same arguments can be iterated to compute (11).
At the same time, one can estimate of the differential of each successive left multiplication by Lemma 2.13 and Equation (10). Hence we conclude that Theorem 2.14. There exist some r, H > 0 such that whenever x ∈ B(re −kH ), τ i ≤ 1, i = 1, 2, · · · k, for some exponential coordinate chart x (β) . Moreover, one has where ∂ x on the right hand side is the coordinate vector field on x (β) , and V ∈ T ⊥ G (i.e. V is spanned by ∂ τ (β) j and ∂ µ (β) j ′ ), is such that for some constant C 7 , M 7 > 0.

The groupoid heat kernel
In terms of reduced kernel and convolution product, the heat kernel is defined as: such that Q(a, t), Q(a −1 , t) are smooth when restricted to all G x ×(0, ∞), and satisfies: (i) The heat equation (∂ t + ∆)Q(a, t) = 0; (ii) The initial condition where • denotes the convolution product 3.1. Construction of the heat kernel. In this section, we briefly review how one constructs the heat kernel of a generalized Laplacian operator [2]. For simplicity, we only consider the scalar case. By [2], there exists a formal power series Φ(a, t) := ∞ i=1 t i Φ i (a) that satisfies the equation For each N > n 2 , define the sections R where Σ k := {(s 1 , · · · , s k ) ∈ R k , s 1 , · · · , s k ≥ 0, s 1 + · · · + s k ≤ t}, and ds is the Lebesgue measure. Then the heat kernel is constructed by taking the sum Q : N (·, t). To simplify notation, we shall write R N := R (1) , and omit the reference to t.
In the construction of G N , we may assume that ̺ is sufficiently small, such that Fix another cutoff function χ ∈ C ∞ c (R) such that χ = 1 on [0, 1 2 ] and 0 on [1, ∞).
Given smooth compactly supported functions κ 0 , · · · , κ k on G, it is straightforward to write for any a ∈ G: Suppose that κ i is supported on U 0 for all i. Then one can write Consider differentiating the integrand with respect to x.
3.3. C 1 estimation away from G 1 . In the following, we make the additional assumption that G is non-degenerate or uniformly degenerate. Denote by ι : M 0 ×M 0 → G the embedding of the invariant sub-manifold. Since the target and source maps on M 0 × M 0 are respectively the projection onto the first and second factor, one has Using the fact that dt(Ṽ ) = ν(dr −1 a (Ṽ )), for anyṼ ∈ T a G, a ∈ G satisfying ds(Ṽ ) = 0, it follows that It is straightforward to compute the coordinate vector fields for the exponential coordinates: Similarly, one computes where ∂ x (x) on the right hand side is regarded as a tangent vector on M.

3.4.
Regularity of the heat kernel. Adding Equation (17) and Lemma 3.2 together, and using the fact that the volume of the set Σ ⊂ R k equals (k!) −1 t k , it follows that the sum ∞ k=0 (−1) k Q (k) N (·, t).
converges uniformly and absolutely in C 1 for N sufficiently large.
Clearly, similar arguments holds for all l = 1, 2, · · · (in particular Equation (5) can be differentiated repeatedly and estimates in the form of Lemma 2.13 still holds). Hence we conclude that Theorem 3.3. Let G be a boundary groupoid of the form G = M 0 × M 0 M 1 × M 1 ×R q , that is either non-degenerate or uniformly degenerate. Suppose furthermore that there exists a finite collection of exponential coordinates charts {x (α) } satisfying Assumption 2.9. Then the heat kernel Q lies in C ∞ (G × (0, ∞)).