Relative heat content asymptotics for sub-Riemannian manifolds
Abstract
The relative heat content associated with a subset of a sub-Riemannian manifold, is defined as the total amount of heat contained in at time , with uniform initial condition on , allowing the heat to flow outside the domain. In this work, we obtain a fourth-order asymptotic expansion in square root of of the relative heat content associated with relatively compact non-characteristic domains. Compared to the classical heat content that we studied in [RR21], several difficulties emerge due to the absence of Dirichlet conditions at the boundary of the domain. To overcome this lack of information, we combine a rough asymptotics for the temperature function at the boundary, coupled with stochastic completeness of the heat semi-group. Our technique applies to any (possibly rank-varying) sub-Riemannian manifold that is globally doubling and satisfies a global weak Poincaré inequality, including in particular sub-Riemannian structures on compact manifolds and Carnot groups.
1 Introduction
In this paper we study the asymptotics of the relative heat content in sub-Riemannian geometry. The latter is a vast generalization of Riemannian geometry, indeed a sub-Riemannian manifold is a smooth manifold where a metric is defined only on a subset of preferred directions at each point (called horizontal directions). For example, can be a sub-bundle of the tangent bundle, but we will consider the most general case of rank-varying distributions. Moreover, we assume that satisfies the so-called Hörmander condition, which ensures that is horizontally-path connected, and that the usual length-minimization procedure yields a well-defined metric.
Let be a sub-Riemannian manifold, equipped with a smooth measure , let be an open relatively compact subset of , with smooth boundary, and consider the Cauchy problem for the heat equation in this setting:
| (1) | ||||||
where is the indicator function of the set , and is the sub-Laplacian, defined with respect to . By classical spectral theory, there exists a unique solution to (1),
| (2) |
where denotes the heat semi-group in , associated with . The relative heat content is the function
| (3) |
This quantity has been studied in connection with geometric properties of subsets of , starting from the seminal work of De Giorgi [DG54], where he introduced the notion of perimeter of a set in and proved a characterization of sets of finite perimeter in terms of the heat kernel. His result was subsequently refined, using techniques of functions of bounded variation: it was proven in [Led94] for balls in , and in [MPPP07] for general subsets of , that a borel set with finite Lebesgue measure has finite perimeter à la De Giorgi if and only if
| (4) |
where is the Lebesgue measure and is the perimeter measure in . Notice that (4) is equivalent to a first-order111Here and throughout the paper, the notion of order is computed with respect to . asymptotic expansion of . A further development in this direction was then obtained in [AMM13], where the authors extended (4) to an asymptotic expansion of order in , assuming the boundary of to be a set. For simplicity, we state here the result of [AMM13, Thm. 1.1] assuming is smooth222The statement of Theorem 1.1 in [AMM13] differs from (5) by a sign in the third-order coefficient: the correct sign appears a few lines below the statement, in the expansion of the function . :
| (5) |
as , where is the Hausdorff measure and, denoting by the principal curvatures of at the point ,
| (6) |
In the Riemannian setting, Van den Berg and Gilkey in [vdBG15] proved the existence of a complete asympotic expansion for , generalizing (5), when is smooth. Moreover, they were able to compute explicitly the coefficients of the expansion up to order in . Their techniques are based on pseudo-differential calculus, and cannot be immediately adapted to the sub-Riemannian setting. In particular, what is missing is a global parametrix estimate for the heat kernel , cf. [vdBG15, Sec. 2.3]: for any , there exist such that
| (7) |
where are suitable smooth functions, given explicitly in terms of the Euclidean heat kernel and iterated convolutions. The closest estimate analogue to (7) in the sub-Riemannian setting is the one proved recently in [CdVHT21, Thm. A] (see Theorem 2.9 for the precise statement), where the authors show an asymptotic expansion of the heat kernel in an asymptotic neighborhood of the diagonal, which is not enough to reproduce (7) and thus the argument of Van den Berg and Gilkey. Moreover, in this case, is expressed in terms of the heat kernel of the nilpotent approximation and iterated convolutions, thus posing technical difficulties for the explicit computations of the coefficients (which would be no longer “simple” gaussian-type integrals).
In this paper, under the assumption of not having characteristic points, we prove the existence of the asymptotic expansion of , up to order in , as . We remark that we include also the rank-varying case. In order to state our main results, let us introduce the following operator, acting on smooth functions compactly supported close to ,
| (8) |
where denotes the sub-Riemannian signed distance function from , see Section 4 for precise definitions.
Theorem 1.1.
Let be a compact sub-Riemannian manifold, equipped with a smooth measure , and let be an open subset whose boundary is smooth and has no characteristic points. Then, as ,
| (9) |
where denotes the sub-Riemannian perimeter measure.
Remark 1.2.
The compactness assumption in Theorem 1.1 is technical and can be relaxed by requiring, instead, global doubling of the measure and a global Poincaré inequality, see section 7 and in particular Theorem 7.3. Some notable examples satisfying these assumptions are:
-
•
is a Lie group with polynomial volume growth, the distribution is generated by a family of left-invariant vector fields satisfying the Hörmander condition and is the Haar measure. This family includes also Carnot groups.
-
•
, equipped with a sub-Riemannian structure induced by a family of vector fields with bounded coefficients together with their derivatives, and satisfying the Hörmander condition.
-
•
is a complete Riemannian manifold, equipped with the Riemannian measure, and with non-negative Ricci curvature.
See Section 7.1 for further details. In all these examples, Theorem 1.1 holds.
The strategy of the proof of Theorem 1.1 follows a similar strategy of [RR21], inspired by the method introduced in [Sav98], used for the classical heat content (11). However, as we are going to explain in Section 1.1, new technical difficulties arise, the main one being related to the fact that now . At order zero, we obtain the following result, see Section 2 for precise definitions.
Theorem 1.3.
Let be a sub-Riemannian manifold, equipped with a smooth measure and let be an open relatively compact subset, whose boundary is smooth and has no characteristic points. Let and consider a chart of privileged coordinates centered at , such that . Then,
| (10) |
where denotes the nilpotentization of at and denotes the heat kernel associated with the nilpotent approximation of at and measure .
This result can be seen as a partial generalization of [CCSGM13, Prop. 3], where the authors proved an asymptotic expansion of up to order in for , for a special class of non-characteristic domains in Carnot groups.
Remark 1.4.
Remark 1.5.
When has no characteristic points, the conormal bundle does not intersect the characteristic set and, as a consequence, the principal symbol of the sub-Laplacian is elliptic near . Thus, it is likely that microlocal analysis techniques in the spirit of [CdVHT18] could yield the existence of a complete asymptotic expansion of the relative heat content (but not an explicit expression and geometric interpretation of the coefficients). We thank Yves Colin de Verdière and the anonymous referee for pointing out this fact.
1.1 Strategy of the proof of Theorem 1.1
To better understand the new technical difficulties in the study of the relative heat content , let us compare it with the classical heat content and illustrate the strategy of the proof of Theorem 1.1.
The classical heat content.
We highlight the differences between the relative heat content and the classical one : let an open set in , then for all , we have
| (11) |
where is the solution to (1) and is the solution to the Dirichlet problem for the heat equation, associated with , i.e.
| (12) | ||||||
The crucial difference is that , for any , whereas in general. Thus, there is no a priori relation between and : the only relevant information is given by domain monotonicity, which implies that:
| (13) |
and clearly this does not give the asymptotics of the latter. See also [vdB13] for other comparison results in the Euclidean setting.
Failure of Duhamel’s principle.
In [RR21], we established a complete asymptotic expansion of , as , provided that has no characteristic points. The proof of this result relied on an iterated application of the Duhamel’s principle and the fact that . Following the same strategy, we apply Duhamel’s principle to a localized version of : fix a function , compactly supported in a tubular neighborhood around and such that and is identically , close to . Then, using off-diagonal estimates for the heat kernel, one can prove that:
| (14) |
where is defined for and as
| (15) |
here , with denoting the distance function from the boundary. Hence, the small-time behavior of is captured by . By Duhamel’s principle and the sub-Riemannian mean value lemma, cf. Section 4 for details, we obtain the following:
| (16) |
For the classical heat content, satisfies Dirichlet boundary condition, cf. (12), hence (16) would give the first-order asymptotics (and then one could iterate). On the contrary, in this case, we do not have prior knowledge of as and . Thus, already for the first-order asymptotics, Duhamel’s principle alone is not enough, and we need some information on the asymptotic behavior of .
First-order asymptotics.
We study the asymptotics of . Using the notion of nilpotent approximation of a sub-Riemannian manifold, cf. Section 2.3, we deduce the zero-order asymptotic expansion of as , proving Theorem 1.3. This is enough to infer the first-order expansion of , by means of (16). At this point, we iterate the Duhamel’s principle to obtain the higher-order terms of the expansion of . However, already at the first iteration, we obtain the following formula for :
| (17) |
as . Therefore, the zero-order asymptotic expansion of no longer suffices for obtaining the second-order asymptotics of .
The outside contribution .
We mentioned that the crucial difference between and , defined in (11), is related to the fact that , whereas , for any . From a physical viewpoint, this distinction comes from the fact that, since the boundary is no longer insulated, the heat governed by the Cauchy problem , solution to (1), can flow also outside of , whereas , solution to the Dirichlet problem (12), is confined in , and the external temperature is . Hence, we can imagine that the asymptotic expansion of is affected by the boundary, both from the inside and from the outside of .
Interpreting as the inside contribution to the asymptotics of , we are going to formalize the physical intuition of having heat flowing outside of , defining an outside contribution, to the asymptotics333The notation “superscript ” stands for complement. Indeed the outside contribution is the inside contribution of the complement of , see Section 5.1.. The starting observation is the following simple relation: setting
| (18) |
we have, by divergence theorem,
| (19) |
Similarly to (15), for a suitable smooth function , one may define a localized version of , which we call , so that
| (20) |
see Section 5.1 for precise definitions. Using (14), (19) and (20), we show the following relation:
| (21) |
for a suitable smooth function . On the other hand, for the localized quantity we have a Duhamel’s principle, thanks to which we are able to study the asymptotic expansion, up to order , of the integral of over , cf. Theorem 5.4. The limitation to the order of the asymptotics is technical and seems difficult to overcome, cf. Remark 5.5. Inserting this asymptotics in (17), we obtain the asymptotics up to order of the expansion of , as .
Fourth-order asymptotics.
Since we have at disposal only the asymptotics of the integral of over , up to order , we need a finer argument to obtain the fourth-order asymptotics of . The simple but compelling relation is based once again on (14), (19) and (20), thanks to which we can write:
| (22) |
Now for the sum of the contributions , the Duhamel’s principle implies the following:
| (23) |
This time notice how the integral of over appears in a first-order term (as opposed to what happened in (16) or (17)), thus its asymptotic expansion up to order implies a fourth-order expansion for , concluding the proof of Theorem 1.1.
1.2 From the heat kernel asymptotics to the relative heat content asymptotics
In [CdVHT21, Thm. A], the authors proved the existence of small-time asymptotics of the hypoelliptic heat kernel, , see Theorem 2.9 below for the precise statement. In Theorem 1.3 we are able to exploit this result to obtain the zero-order asymptotics of the function
| (24) |
However, we are not able to extend Theorem 1.3 to higher-order asymptotics since, roughly speaking, the remainder terms in Theorem 2.9 are not uniform as . If we had a better control on the remainders, we could indeed integrate (in a suitable way) the small-time heat kernel asymptotics to obtain the corresponding expansion for . Finally, from such an expansion, the relative heat content asymptotics would follow from the localization principle (14) and the (iterated) Duhamel’s principle (16). This is done in Section 6.
1.3 Characteristic points
In order to prove our main results, we need the non-characteristic assumption on the domain . We recall that for a subset with smooth boundary, is a characteristic point if . As was the case for the classical heat content, cf. [RR21], the non-characteristic assumption is crucial to follow our strategy, since it guarantees the smoothness of the signed distance function close to , cf. Theorem 4.1. Nevertheless, one might ask whether Theorem 1.1 holds for domains with characteristic points, at least formally.
On the one hand, the coefficients, up to order , are well-defined even in presence of characteristic points, cf. [Bal03]. While, for what concerns the integrand of the third-order coefficient, its integrability, with respect to the sub-Riemannian induced measure , is related to integrability of the sub-Riemannian mean curvature , with respect to the Riemannian induced measure. The latter is a non-trivial property, which has been studied in [DGN12], and holds in the Heisenberg group, for surfaces with mildly-degenerate characteristic points in the sense of [Ros23].
On the other hand, differently from what happens in the case of the Dirichlet problem, the heat kernel associated with (1) is smooth at the boundary of , for positive times, even in presence of characteristic points. Thus, in principle, there is no obstacle in obtaining an asymptotic expansion of also in that case. Moreover, in Carnot groups of step , a similar result to (4) holds, cf. [BMP12, GT20]. In particular, the characterization of sets of finite horizontal perimeter in Carnot groups of step is independent of the presence of characteristic points, indicating that an asymptotic expansion such as (9) may still hold, dropping the non-characteristic assumption.
1.4 Notation
Throughout the article, for a set , we will use the notation , even in the compact case, so that all the statements need not be modified in the non-compact case, when the generalization is possible, cf. Theorem 7.3. Moreover, in the non-compact and complete case, the set is assumed to be open and bounded.
Acknowledgments.
This work was supported by the Grant ANR-18-CE40-0012 of the ANR, by the Project VINCI 2019 ref. c2-1212 and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 945655).
We thank Yves Colin de Verdière, Luc Hillairet and Emmanuel Trélat for stimulating discussion regarding the small-time asymptotics of hypoelliptic heat kernels. We also thank the anonymous referee for valuable comments and remarks.
2 Preliminaries
We recall some essential facts in sub-Riemannian geometry, following [ABB20].
2.1 Sub-Riemannian geometry
Let be a smooth, connected finite-dimensional manifold. A sub-Riemannian structure on is defined by a set of global smooth vector fields , called a generating frame. The generating frame defines a distribution of subspaces of the tangent spaces at each point , given by
| (25) |
We assume that the distribution satisfies the Hörmander condition, i.e. the Lie algebra of smooth vector fields generated by , evaluated at the point , coincides with , for all . The generating frame induces a norm on the distribution at , namely
| (26) |
which, in turn, defines an inner product on by polarization, which we denote by . Let . We say that is a horizontal curve, if it is absolutely continuous and
| (27) |
This implies that there exists , such that
| (28) |
Moreover, we require that . If is a horizontal curve, then the map is integrable on . We define the length of a horizontal curve as follows:
| (29) |
The sub-Riemannian distance is defined, for any , by
| (30) |
By the Chow-Rashevsky Theorem, the distance is finite and continuous. Furthermore it induces the same topology as the manifold one.
Remark 2.1.
The above definition includes all classical constant-rank sub-Riemannian structures as in [Mon02, Rif14] (where is a vector distribution and a symmetric and positive tensor on ), but also general rank-varying sub-Riemannian structures. Moreover, the same sub-Riemannian structure can arise from different generating families.
2.2 The relative heat content
Let be a sub-Riemannian manifold. Let be a smooth measure on , i.e. by a positive tensor density. The divergence of a smooth vector field is defined by
| (31) |
where denotes the Lie derivative in the direction of . The horizontal gradient of a function , denoted by , is defined as the horizontal vector field (i.e. tangent to the distribution at each point), such that
| (32) |
where acts as a derivation on . In terms of a generating frame as in (25), one has
| (33) |
We recall the divergence theorem (we stress that is not required to be orientable): let be open with smooth boundary, then
| (34) |
for any smooth function and vector field , such that the vector field is compactly supported. In (34), is the inward-pointing normal vector field to and is the induced sub-Riemannian measure on (i.e. the one whose density is ).
The sub-Laplacian is the operator , acting on . Again, we may write its expression with respect to a generating frame (25), obtaining
| (35) |
We denote by , or simply by , the space of real functions on which are square-integrable with respect to the measure . Let be an open relatively compact set with smooth boundary. This means that the closure is a compact manifold with smooth boundary. We consider the Cauchy problem for the heat equation on , that is we look for functions such that
| (36) | ||||||
where is a shorthand notation for the -limit of as . Notice that is symmetric with respect to the -scalar product and negative, moreover, if is complete as a metric space, it is essentially self-adjoint, see [Str86]. Thus, there exists a unique solution to (36), and it can be represented as
| (37) |
where denotes the heat semi-group, associated with . We remark that for all , the function is smooth for all , by hypoellipticity of the heat operator and there exists a heat kernel associated with (36), i.e. a positive function such that:
| (38) |
Definition 2.2 (Relative heat content).
Let be the solution to (36). We define the relative heat content, associated with , as
| (39) |
Remark 2.3.
If we consider, instead of , a set which is the closure of an open set, then the Cauchy problem (36) has a unique solution and relative heat content is still well-defined.
We recall here a property of the solution to (36): it satisfies a weak maximum principle, meaning that
| (40) |
This can be proven following the blueprint of the Riemannian proof (see [Gri09, Thm. 5.11]).
Definition 2.4 (Characteristic point).
We say that is a characteristic point, or tangency point, if the distribution is tangent to at , that is
| (41) |
We will assume that has no characteristic points. We say in this case that is a non-characteristic domain.
2.3 Nilpotent approximation of
We introduce the notion of nilpotent approximation of a sub-Riemannian manifold, see [Jea14, Bel96] for details. This will be used only in Sections 3 and 6.
Sub-Riemannian flag.
Let be an -dimensional sub-Riemannian manifold with distribution . We define the flag of as the sequence of subsheafs such that
| (42) |
with the convention that . Under the Hörmander condition, the flag of the distribution defines an exhaustion of , for any point , i.e. there exists such that:
| (43) |
The number is called degree of nonholonomy at . We set , for any , then the collection of integers
| (44) |
is called growth vector at , and we have . Associated with the growth vector, we can define the sub-Riemannian weights at , setting for any ,
| (45) |
A point is said to be regular if the growth vector is constant in a neighborhood of , and singular otherwise. The sub-Riemannian structure on is said to be equiregular if all points of are regular. In this case, the weights are constant as well on . Finally, given any , we define the homogeneous dimension of at as
| (46) |
We recall that, if is regular, then coincides with the Hausdorff dimension of at , cf. [Mit85]. Moreover, , for any such that .
Privileged coordinates.
Let be a sub-Riemannian manifold with generating frame (25) and be the germ of a smooth function at . We call nonholonomic derivative of order of , the quantity
| (47) |
for any family of indexes . Then, the nonholonomic order of at the point is
| (48) |
Definition 2.5 (Privileged coordinates).
Let be a -dimensional sub-Riemannian manifold and . A system of local coordinates centered at is said to be privileged at if
| (49) |
Notice that privileged coordinates at satisfy the following property
| (50) |
A local frame of consisting of vector fields and satisfying (50) is said to be adapted to the flag (43) at . Thus, privileged coordinates are always adapted to the flag. In addition, given a local frame adapted to the sub-Riemannian flag at , say , we can define a set of privileged coordinates at , starting from , i.e.
| (51) |
Moreover, in these coordinates, the vector field is exactly .
Nilpotent approximation.
Let be a sub-Riemannian manifold and let with weights as in (45). Consider a chart of privileged coordinates at , where is a relatively compact neighborhood of and is a neighborhood of . Then, for any , we can define the dilation at as
| (52) |
Using such dilations, we obtain the nilpotent (or first-order) approximation of the generating frame (25), indeed setting , for any , define
| (53) |
where the limit is taken in the -topology of . Notice that the vector field is defined on the whole , even though was defined only on .
Theorem 2.6.
Let be a -dimensional sub-Riemannian manifold with generating frame and consider its first-order approximation at as in (53). Then, the frame of vector fields on generates a nilpotent Lie algebra of step and satisfies the Hörmander condition.
The proof of this theorem can be found in [Jea14]. Recall that a Lie algebra is said to be nilpotent of step if is the smallest integer such that all the brackets of length greater than are zero.
Definition 2.7 (Nilpotent approximation).
Let be a sub-Riemannian manifold and let . Then, Theorem 2.6 implies that the frame is a generating frame for a sub-Riemannian structure on : we denote the resulting sub-Riemannian manifold . This is the so-called nilpotent approximation of at the point .
Notice that the sub-Riemannian distance of , denoted by , is -homogeneous with respect to the dilations (52).
Nilpotentized sub-Laplacian.
Let be a sub-Riemannian manifold, equipped with a smooth measure , and let be a set of privileged coordinates at . We will use the same symbol to denote measure in coordinates. The nilpotentization of at is defined as follows:
| (54) |
Notice that, denoting by the Lebesgue measure on , we have
| (55) |
thus, the limit in (54) exists. Finally, we can define the nilpotentized sub-Laplacian according to (35), acting on , i.e.
| (56) |
We remark that in (56) there is no divergence term, since
| (57) |
As in the general sub-Riemannian context, in the nilpotent approximation , we may consider the Cauchy heat problem (36) in . We will the denote the associated heat kernel as
| (58) |
Heat kernel asymptotics.
Let be a sub-Riemannian manifold, equipped with a smooth measure and denote by the heat kernel (38). We have the following result.
Theorem 2.9 ([CdVHT21, Thm. A]).
Let be a sub-Riemannian manifold and let be a chart of privileged coordinates at . Then, for any ,
| (59) |
in the -topology of , where ’s are smooth functions satisfying the following homogeneity property: for
| (60) |
where, for , we set . In (59), we are considering the heat kernel in coordinates, with a little abuse of notation.
Remark 2.10.
We will drop the dependence on the center of the privileged coordinates if there is no confusion.
3 Small-time asymptotics of at the boundary
We prove here Theorem 1.3, regarding the zero-order asymptotics of as .
Theorem 3.1.
Let be a compact sub-Riemannian manifold, equipped with a smooth measure and let be an open subset, whose boundary is smooth and has no characteristic points. Let and consider a chart of privileged coordinates centered at , such that . Then,
| (61) |
where denotes the nilpotentization of at and denotes the heat kernel associated with the nilpotent approximation of at and measure .
Remark 3.2.
A chart of privileged coordinates, such that always exists, provided that has no characteristic points. Indeed, in this case, there exists a tubular neighborhood of the boundary, cf. Theorem 4.1, which is built through the flow of , namely
| (62) |
is a diffeomorphism such that and . Here is the signed distance function444We warn the reader that without a subscript always denotes the signed distance function and should not be confused with dilations . from and , see Section 4.1 for precise definitions. Therefore, choosing an adapted frame for the distribution at , say where , we can define a set of privileged coordinates as in (51):
| (63) |
The resulting set of coordinates satisfies and, denoting by the neighborhood of in where is invertible, . Here, denotes the flow of the vector field , starting at , evaluated at time .
Proof of Theorem 3.1.
Let be the heat kernel of , then we may write
| (64) |
For a fixed , denoting by any relatively compact neighborhood of , we have
| (65) |
as . Indeed, since the heat kernel is exponentially decaying outside the diagonal, cf. [JSC86, Prop. 3],
| (66) |
as . Now, for , fix the set of privileged coordinates , defined as in the statement and assume without loss of generality that for any , where is the dilation (52) of the nilpotent approximation of . Also set
| (67) |
when the limits exist, we have:
| (68) |
where, in the last equation, we are considering the expression of the heat kernel and the measure in coordinates. We want to apply (59) at order in , so let us rephrase the statement as follows: for any compact set ,
| (69) |
where is a smooth function such that
| (70) |
with . Notice that (70) is not uniform in , in the sense that can explode as , in general. Moreover, without loss of generality and, up to restrictions of , we can assume that (70) holds globally on . For a fixed parameter , we set and in (69), obtaining:
| (71) |
where the remainder is bounded as on the compact sets of , but with a constant depending on . Inserting the above expansion in (68), and writing the measure in coordinates with , we have:
| (72) | ||||
| (73) | ||||
| (74) | ||||
| (75) | ||||
| (76) |
where in the third equality we performed the change of variable in the first integral. Let us discuss the terms appearing in (75) and (76). First of all, for any , by definition of the nilpotentization of given in (54), we get
| (77) |
Moreover, for a fixed , the integral of is bounded as , therefore, using (70), we have:
| (78) |
where is a constant depending on the fixed . Secondly, by an upper Gaussian bound for the heat kernel in compact sub-Riemannian manifold, [JSC86, Thm. 2], we obtain the following estimate for (76):
| (79) |
where are positive constants. Now, by the Ball-Box Theorem [Jea14, Thm. 2.1], the sub-Riemannian distance function at the origin is comparable with the sub-Riemannian distance of , denoted by . In particular, there exists a constant such that
| (80) |
Since in (79) we are integrating over the set and is 1-homogeneous with respect to , we conclude that
| (81) |
Therefore, using also (80), the term (79) can be estimated as follows:
| (82) |
where is independent of and . The last inequality in (82) follows from the fact that, after a change of variable , the integral
| (83) |
is uniformly bounded with respect to .
Therefore, for any , we obtain the following estimates for the limit of :
| (84) |
In order to evaluate the limits in (84), let us firstly notice that, since enjoys upper and lower Gaussian bounds (see for example [CdVHT21, App. C]), reasoning as we did for (82), we can prove the following:
| (85) |
Secondly, thanks to (60) for , we have the following parity property
| (86) |
and, by the choice of privileged coordinates, . Thus, using also the stochastic completeness of the nilpotent approximation, we obtain for any ,
| (87) | ||||
| (88) |
having performed the change of variables in the last equality. Hence, the integral in (85) is
| (89) |
Finally, we optimize the inequalities (84) with respect to , taking and concluding the proof. ∎
4 First-order asymptotic expansion of
In this section, we introduce the technical tools that allow us to prove the first-order asymptotic expansion of the relative heat content starting from Theorem 3.1. The new ingredient is a definition of an operator , which depends on the base set .
4.1 A mean value lemma
Define to be the signed distance function from , i.e.
| (90) |
where denotes the usual distance function from . Let us introduce the following notation: for any , with , we set
| (91) |
with the understanding that if (or ) is omitted, it is assumed to be (or ), for example555Notice that the set , thus omitting both indexes can create confusion. We will never do that and will always denote the starting subset of .
| (92) |
In the non-characteristic case, [FPR20, Prop. 3.1] can be extended without difficulties to the signed distance function.
Theorem 4.1 (Double-sided tubular neighborhood).
Let be a sub-Riemannian manifold and let be an open relatively compact subset of whose boundary is smooth and has no characteristic points. Let be the signed distance function from . Then, we have:
-
i)
is Lipschitz with respect to the sub-Riemannian distance and a.e.;
-
ii)
there exists such that is smooth;
-
iii)
there exists a smooth diffeomorphism , such that
(93) Moreover, on .
In particular, the following co-area formula for the signed distance function holds
| (94) |
where is the induced measure on , namely the positive measure with density . From (94), we deduce the sub-Riemannian mean value lemma, see [RR21, Thm. 4.1] for a proof.
Proposition 4.2.
Let be a compact sub-Riemannian manifold, equipped with a smooth measure , let be an open subset of with smooth boundary and no characteristic points and let be the signed distance function from . Fix a smooth function and define
| (95) |
Then there exists such that the function is smooth on and, for :
| (96) |
where is the inward-pointing unit normal to , is the induced measure on .
Remark 4.3.
If , then neither nor is required to be compact for Proposition 4.2 to be true, indeed its proof relies on (94), which continues to hold, and the divergence theorem (34), which applies if is compact. Moreover, we remark that is equal to up to sign. We prefer to keep in (95), since we are going to apply it when the integral is performed over or its complement.
If we choose the function in the definition of to be , where , then, satisfies a non-homogeneous one-dimensional heat equation.
Corollary 4.4.
Under the hypotheses of Proposition 4.2, the function
| (97) |
where , satisfies the following non-homogeneous one-dimensional heat equation:
| (98) |
Here is the inward-pointing unit normal to , is the induced measure on .
Corollary 4.4 holds only for , however we would like to extend it to the whole positive half-line, in order to apply a Duhamel’s principle. This can be done up to an error which is exponentially small.
4.2 Localization principle
Proposition 4.5.
Let be a compact sub-Riemannian manifold, equipped with a smooth measure , and let be an open subset of , with smooth boundary. Moreover, let be a closed set such that . Then
| (99) |
where .
Proof.
The statement is a direct consequence of the off-diagonal estimate for the heat kernel in compact sub-Riemannian manifold (see [JSC86, Prop. 3]):
| (100) |
for suitable constants , depending only on . Now, since , we can write as a disjoint union
| (101) |
At this point, for , set by hypothesis, and let . Then, using the stochastic completeness of , we have:
| (102) |
which is exponentially decaying, uniformly in . Analogously, if , we have
| (103) |
uniformly in . ∎
Remark 4.6.
Let be compact. Thanks to Proposition 4.5, we can extend the function defined in (97), to a solution to a non-homogeneous heat equation such as (98) on the whole half-line. More precisely, let such that
| (104) |
where is defined in Proposition 4.2. We have then, for ,
| (105) | ||||
| (106) | ||||
| (107) |
where we used Proposition 4.5 to deal with the second term, having set . For this reason, we may focus on the first term in (107).
Definition 4.7.
For all and , we define the operators , associated with , by
| (108) | ||||
| (109) |
for any , and where denotes the induced measure on and .
Remark 4.8.
We stress that, for every , , are indeed smooth in both variables thanks to the choice of the parameter as in Proposition 4.2, together with the smoothness of the solution to the heat equation. Moreover, is compactly supported in the -variable.
Thanks to the localization principle, we can improve Corollary 4.4, obtaining a better result for .
Lemma 4.9.
Let be the one-dimensional heat operator. Then, for any ,
| (110) |
where is the operator defined by:
| (111) |
and is the inward-pointing unit normal to .
4.3 Duhamel’s principle for
We recall for the convenience of the reader a one-dimensional version of the Duhamel’s principle, see [RR21, Lem. 5.4].
Lemma 4.10 (Duhamel’s principle).
Let , , such that and are compactly supported and assume that
| (112) |
Consider the non-homogeneous heat equation on the half-line:
| (113) | ||||||
where . Then, for and , the solution to (113) is given by
| (114) |
where is the Neumann heat kernel on the half-line, that is
| (115) |
Finally, we apply Lemma 4.10 to obtain an asymptotic equality for . The main difference with the result of [RR21, Thm. 5.6] is that the former will not be a true first-order asymptotic expansion.
Corollary 4.11.
Let be a compact sub-Riemannian manifold, equipped with a smooth measure , and let be an open subset whose boundary is smooth and has no characteristic points. Then, for any function ,
| (116) |
as , where .
Proof.
By Lemma 4.9, the function satisfies the following Neumann problem on the half-line:
| (117) | ||||||
where the source is given by . Thus, applying Duhamel’s formula (114), we have:
| (118) |
Since the source is uniformly bounded by the weak maximum principle (40), the first integral is a remainder of order , as , concluding the proof. ∎
Remark 4.12.
We mention that a relevant role in the sequel will be played by the operators , cf. Definition 4.7, associated with either or its complement .
4.4 First-order asymptotics
In this section we prove the first-order asymptotic expansion of , cf. Theorem 1.1 at order . We will use Corollary 4.11, for the inside contribution:
| (119) |
for any , and where denotes the induced measure on and is the solution to (36). The quantity (119) is just Definition 4.7, applied with base set .
Theorem 4.13.
Let be a compact sub-Riemannian manifold, equipped with a smooth measure , and let be an open subset whose boundary is smooth and has no characteristic points. Then,
| (120) |
Proof.
Let be as in (104), namely
| (121) |
Then, by the localization principle, cf. (107), we have that
| (122) |
Applying Corollary 4.11, we have:
| (123) |
Thus, to infer the first-order term of the asymptotic expansion we have to compute the following limit:
| (124) |
Firstly, by the change of variable in the integral , we rewrite the argument of the limit (124) as
| (125) |
Secondly, we apply the dominated convergence theorem. Indeed, on the one hand, by Theorem 3.1 we have point-wise convergence
| (126) |
and on the other hand, by the maximum principle
| (127) |
for any . Therefore, we finally obtain that:
| (128) |
Recalling that , we conclude the proof. ∎
5 Higher-order asymptotic expansion of
We iterate Duhamel’s formula (114) for the inside contribution to study the higher-order asymptotics of . We obtain the following expression for , at order :
| (129) |
where denotes the solution to (36) and is the operator defined by
| (130) |
with the signed distance function from . The computations for deriving (129) are technical. We refer to Appendix A for further details, and in particular to Lemma A.6. Motivated by (129), we introduce the following functional.
Definition 5.1.
Let be a sub-Riemannian manifold, equipped with a smooth measure , let be a relatively compact subset with smooth boundary and let . Define the functional , for any as:
| (131) |
where is the sub-Riemannian induced measure on .
Notice that the functional is linear with respect to the subscript function , by linearity of the integral. Moreover, when the function is chosen to be the solution to (36), we easily obtain the following corollary of Theorem 3.1, which is just a rewording of (124).
Corollary 5.2.
Let be a compact sub-Riemannian manifold, equipped with a smooth measure , and let be an open subset whose boundary is smooth and has no characteristic points. Let , then,
| (132) |
Then, we can rewrite (129) in a compact notation:
| (133) |
However, on the one hand, the application of Corollary 5.2 to (129) does not give any new information on the asymptotics of , as the first term produces an error of . On the other hand, it is clear that an asymptotic series of is enough to deduce the small-time expansion of .
5.1 The outside contribution and an asymptotic series for
In this section, we deduce an asymptotic series, at order , of as . This is done exploiting the fact that the diffusion of heat is not confined in , and as a result we can define an outside contribution, namely the quantity obtained from Definition 4.7, applied with base set :
| (134) |
for any , and where denotes the induced measure on the boundary of and . We remark that, since and its complement share the boundary, then . It is convenient to introduce (134), because it turns out that the quantity , where is the inside contribution (119), has an explicit asymptotic series in integer powers of .
Proposition 5.3.
Let be a compact sub-Riemannian manifold, equipped with a smooth measure , and let be an open subset with smooth boundary. Let , then, for any ,
| (135) |
where
| (136) |
Proof.
Recall that in the definition of the outside contribution (134), the integrand function involves . Since is compact, and hence stochastically complete, we have:
| (137) |
having used the point-wise equality in . Therefore, we can write the difference as follows:
| (138) | ||||
| (139) | ||||
| (140) |
Since is the solution to (36), the function (140) is smooth as . Indeed, the smoothness in the open interval is guaranteed by hypoellipticity of the sub-Laplacian. At , the divergence theorem, together with the fact that has compact support in , implies that
| (141) |
The previous limit shows that (140) is smooth at , and also that its asymptotic expansion at order , as , coincides with its -th Taylor polynomial at . Finally, we recover (135), applying once again the divergence theorem:
| (142) |
recalling that is the inward-pointing normal vector to at its boundary. ∎
Applying the (iterated) Duhamel’s principle (114) to the difference , we are able to obtain relevant information on the functional .
Theorem 5.4.
Let be a compact sub-Riemannian manifold, equipped with a smooth measure , and let be an open subset whose boundary is smooth and has no characteristic points. Then, for any ,
| (143) |
Proof.
Let us study the difference of the inside and outside contributions : on the one hand, we have an iterated Duhamel’s principle, cf. Lemma A.7, which we report here:
| (144) | ||||
| (145) | ||||
| (146) |
where we recall that is the operator acting on smooth functions compactly supported close to defined by
| (147) |
Using Corollary 5.2 and the linearity of with respect to , we know that
| (148) |
Therefore, applying (148) to the function , we obtain
| (149) |
In addition, an application of Theorem 3.1 and the dominated convergence theorem implies that
| (150) |
Thus, using (149) and (150), we can improve (144), obtaining
| (151) |
On the other hand, the quantity has a complete asymptotic series by Proposition 5.3, which at order becomes:
| (152) |
Comparing (151) and (152), we deduce that, as ,
| (153) | ||||
| (154) |
Finally, using the linearity of the functional with respect to , we conclude the proof. ∎
Remark 5.5.
The asymptotics (143) for the functional is the best result that we are able to achieve. In the expression (144), the problematic term is given by (150), i.e.
| (155) |
which can not be expressed in terms of , hence the only relevant information is given by Theorem 3.1. In conclusion, we can not repeat the strategy of the proof of Theorem 5.4, replacing the series of at order in (144) to deduce the higher-order terms.
5.2 Fourth-order asymptotics
In this section we prove Theorem 1.1. We recall here the statement.
Theorem 5.6.
Let be a compact sub-Riemannian manifold, equipped with a smooth measure , and let be an open subset whose boundary is smooth and has no characteristic points. Then, as ,
| (156) |
Before giving the proof of the theorem, let us comment on its strategy. Recall that, on the one hand, for a cutoff function which is identically close to , cf. (104), the localization principle (122) holds, namely
| (157) |
Moreover, by the iterated Duhamel’s principle for , cf. Lemma A.6, we can deduce expression (133), namely
| (158) |
On the other hand, we have an asymptotic series of the functional at order , cf. Theorem 3.1. Therefore, if we naively insert this series in (158), we can obtain, at most, a third-order asymptotic expansion of the relative heat content , whereas we are interested in the fourth-order expansion.
Using the outside contribution, we are able to overcome this difficulty. In particular, applying Proposition 5.3, for a function which is identically close to , we have the following asymptotic relation:
| (159) |
Notice that (159) is a direct consequence of Proposition 5.3 since all the coefficients of the expansion vanish. Therefore, thanks to (159), we can rephrase (157) as follows:
| (160) |
The advantage of (160) is that we can now apply the iterated Dirichlet principle for the sum , cf. Lemma A.8. Already at order , we obtain
| (161) |
where is the operator defined in (130). As we can see, in (161), the functional occurs for the first time in the second iteration of the Duhamel’s principle, as opposed to the expansion for , where it appeared already in the first application, cf. (158). Hence we gain an order with respect to the asymptotic series of . More generally, if we were able to develop the -th order asymptotics for , this would imply the -th order expansion for .
Proof of Theorem 5.6.
Following the discussion above, it is enough to expand the sum , with . For this quantity, Lemma A.8 holds, namely we have the following iterated version of Duhamel’s principle:
| (162) | ||||
where is defined in (130). Moreover, recall that by Theorem 5.4, the functional has the following expansion for any :
| (163) |
Thus, replacing the term in (162) with the expansion (163) for , we obtain the following asymptotic as :
| (164) |
for any . In particular, if we choose such that close to , then on the one hand, from (164), we obtain, as ,
| (165) |
On the other hand, the asymptotic relation (160) holds. This concludes the proof. ∎
Third-order vs fourth-order asymptotics.
We stress that we could have obtained the third-order asymptotic expansion of without introducing the sum of the inside and outside contributions , and only using the Duhamel’s principle for , cf. Lemma A.6, and the asymptotic series for , cf. Theorem 5.4. However, for the improvement to the fourth-order asymptotics, the argument of the sum of contributions seems necessary.
5.3 The weighted relative heat content
Adapting the proof of Theorem 5.6, one can prove a slightly more general result which we state here for completeness.
Proposition 5.7.
Let be a compact sub-Riemannian manifold, equipped with a smooth measure , and let be an open subset whose boundary is smooth and has no characteristic points. Let and define the weighted relative heat content
| (166) |
Then, as ,
| (167) |
Proof.
Let us consider a cutoff function as in (104). Then, applying the usual localization argument, cf. (107), we have:
| (168) |
where now the function and close to .
As we did in the proof of Theorem 5.6, we relate with the sum of contributions. Applying Proposition 5.3, we have the following asymptotic relation at order :
| (169) |
as , having used the fact that close to . Notice that this relation coincides with (159) when close to . Thus, we obtain
| (170) |
Finally, applying (164) for , we conclude. ∎
Remark 5.8.
We compare the coefficients of the expansions of and , defined in (11), respectively. On the one hand, by [RR21, Thm. 5.8], the -th coefficient of the expansion of is of the form
| (171) |
where is a differential operator acting on and belonging to as algebra of operators. On the other hand, Proposition 5.7 shows that this is no longer true for the third coefficient of the expansion of , as we need to add the operator multiplication by .
6 An alternative approach using the heat kernel asymptotics
As we can see by a first application of Duhamel’s principle, cf. (16), and its iterations, the small-time asymptotics of , together with uniform estimates on the remainder with respect to , would be enough to determine the asymptotic expansion of the relative heat content, at any order.
In Theorem 3.1, we studied the zero-order asymptotics of . The technique used for its proof does not work at higher-order, since the exponential remainder term in (84) would be unbounded as . In this section, we comment how such an higher-order asymptotics of can be obtained exploiting the asymptotic formula for the heat kernel proved in [CdVHT21, Thm. A].
Let be a compact sub-Riemannian manifold and an open subset with smooth boundary. For , let us consider a chart of privileged coordinates centered at , with a relatively compact set. Since the heat kernel is exponentially decaying outside the diagonal, cf. (66),
| (172) |
where , and we denote with the same symbols and the coordinate expression of the measure and heat kernel, respectively. For example, if is non-characteristic, we may choose as in (63), then . Recall the asymptotic expansion of the heat kernel of Theorem 2.9, evaluated in : for any and compact set ,
| (173) |
uniformly as , where , and ’s are defined in Section 2. We will omit the dependance on the center of the privileged coordinates , it being fixed for the moment. At this point, we would like to integrate (173) to get information of as . Proceeding formally, let us choose the parameters in (173) such that:
| (174) |
for some to be fixed. For convenience of notation, set
| (175) |
then, split the integral over in (172) in two, so that the first one is computed on and the second one is computed on its complement in , i.e. . Notice that, by usual off-diagonal estimates, see [JSC86, Prop. 3] and our choice of the parameter as in (174), the following is a remainder term, independently of the value of :
| (176) |
Thus, writing the measure in coordinates with , we have, as ,
| (177) | ||||
| (178) |
where is a smooth function on , such that
| (179) |
for any compact set , according to (173). Up to restricting the domain of privileged coordinates , we can assume that (179) holds on . By our choices (174), we would like the following term
| (180) |
to be an error term of order greater than , as . Thus, assume for the moment that compact and , and such that
| () |
Thanks to assumption (), choosing large enough, we see that (180) is a . Performing the change of variables in (178), and exploiting the homogeneity properties of and , namely (60), we finally obtain the following expression for as :
| (181) |
having set , for all . Therefore, we find an asymptotic expansion of under assumption (), which is crucial to overcome the fact that (173) is formulated on an asymptotic neighborhood of the diagonal, and not uniformly as . It is likely666Personal communication by Yves Colin de Verdière, Luc Hillairet and Emmanuel Trélat. that () can be proven in the nilpotent case, and more generally when the ambient manifold is and the generating family of the sub-Riemannian structure, satisfies the uniform Hörmander polynomial condition, see [CdVHT21, App. B] for details. Although this strategy could be used to prove the existence of an asymptotic expansion of , we refrain to go in this direction since two technical difficulties would arise nonetheless:
- •
-
•
Computations of the coefficients. The coefficients appearing in (181) depend on the nilpotent approximation at and are not clearly related to the invariants of .
Our strategy avoids almost completely the knowledge of the small-time asymptotics of , it being based on an asymptotic series of the auxiliary functional . Moreover, we stress that our method to prove the asymptotics of up to order , cf. Theorem 1.1, holds for any sub-Riemannian manifold, including also the non-equiregular ones.
Remark 6.1.
In order to pass from (181) to the asymptotic expansion of , we would use Duhamel’s formula, which holds under the non-characteristic assumption. This means that, even though (173) of course is true even in presence of characteristic points, we can’t say much about the asymptotics of in the general case.
7 The non-compact case
In the non-compact case, we have the following difficulties:
-
•
The localization principle, cf. Proposition 4.5, may fail.
-
•
Set and . If the manifold is not stochastically complete, the relation does not hold.
- •
Definition 7.1.
Let be a sub-Riemannian manifold, equipped with a smooth measure . We say that is (globally) doubling if there exist constants such that:
| (182) |
where . We say that satisfies a (global) weak Poincaré inequality, if there exist constants such that,
| (183) |
for any smooth function . Here . We refer to these properties as local whenever they hold for any .
Remark 7.2.
If is a sub-Riemannian manifold, equipped with a smooth globally doubling measure , then it is stochastically complete, namely
| (184) |
This is a straightforward consequence of the characterization given by [Stu94, Thm. 4] on the volume growth of balls.
Theorem 7.3.
Let be a complete sub-Riemannian manifold, equipped with a smooth measure . Assume that is globally doubling and satisfies a global weak Poincaré inequality. Then, there exist constants , for any integer , depending only on , such that, for any and ,
| (185) |
where we recall .
In addition, there exists constants , depending only on , such that, for any and ,
| (186) |
Proof.
Define the sub-Riemannian Hamiltonian as the smooth function ,
| (187) |
where is a generating family for the sub-Riemannian structure. Then, following the notations of [Stu96], one can easily verify that
| (188) |
where is the sub-Riemannian Hamiltonian viewed as a bilinear form on fibers, defines a strongly local Dirichlet form with domain . Notice that the Friedrichs extension of is exactly the sub-Laplacian, moreover, the intrinsic metric
| (189) |
coincides with the usual sub-Riemannian distance, as , cf.
[BBS16, Ch. 2, Prop. 12.4]. Thus, is also strongly regular and, by our assumptions on , [SC92, Thm. 4.3] holds true, proving (185).
For the Gaussian lower bound (186), it is enough to apply [Stu96, Cor. 4.10], cf. also [SC92, Thm. 4.2]. This concludes the proof.
∎
Remark 7.4.
We prove now the non-compact analogue of Proposition 4.5.
Corollary 7.5.
Proof.
Let us assume that such that . The other part of the statement can be done similarly.
Since is stochastically complete, cf. Remark 7.2, for any , we can write:
| (191) |
Thanks to Theorem 7.3, we can apply (185) for obtaining
| (192) |
for suitable constants not depending on , . Now, fix : since is closed with empty intersection with , and thus well-separated from , we deduce there exists such that for any , and so
| (193) | ||||
| (194) |
where . Thus, if we prove that the integral in (194) is finite, we conclude. Firstly, recall the Gaussian lower bound (186), which holds thanks to Theorem 7.3:
| (195) |
for suitable constants constants , not depending on , . Secondly, by the doubling property of , it is well-known that there exists depending only on such that
| (196) |
Therefore, choosing so big that and applying (196) for and , we have and
| (197) |
having denoted by . Finally, using (197) and the Gaussian lower bound (195), we can estimate the integral in (194) as follows:
| (198) |
where . Since the resulting constant does not depend on , we conclude the proof. ∎
Using Corollary 7.5 and adopting the same strategy of the compact case, one can finally prove the following result.
Theorem 7.6.
Let be a complete sub-Riemannian manifold, equipped with a smooth measure . Assume that is globally doubling and satisfies a global weak Poincaré inequality. Let be an open and bounded subset whose boundary is smooth and has no characteristic points. Then, as ,
| (199) |
Remark 7.7.
7.1 Notable examples
We list here some notable examples of sub-Riemannian manifolds satisfying the assumptions of Theorem 7.3. For these examples Theorem 7.6 is valid.
- •
-
•
, equipped with a sub-Riemannian structure induced by a family of vector fields with bounded coefficients together with their derivatives, and satisfying the Hörmander condition. Under these assumptions, the Lebesgue measure is doubling, cf. [NSW85, Thm. 1], and the Poincaré inequality is verified in [Jer86]. We remark that these works provide the local properties of Definition 7.1, with constants depending only on the -norms of the vector fields , for . Thus, if the -norms are globally bounded, we obtain the corresponding global properties.
-
•
is a complete Riemannian manifold with metric , equipped with the Riemannian measure, and with non-negative Ricci curvature.
We mention that a Riemannian manifold with Ricci curvature bounded below by a negative constant satisfies only locally Definition 7.1, i.e. for some , depending on the Ricci bound. Nevertheless, we can prove Corollary 7.5 in this case, as Li and Yau provides an upper Gaussian bound, see [LY86, Cor. 3.1], and a lower bound as (186) holds, cf. [BQ99, Cor. 2]. Thus, the first-order asymptotic expansion of , cf. Theorem 4.13, is valid in this setting.
Appendix A Iterated Duhamel’s principle for
In this section, we study the iterated Duhamel’s principle for the , cf. Definition 4.7. The main result is Lemma A.6, which will imply formulas (129), (144) and (162).
The next proposition is a version of the iterated Duhamel’s principle taken from [RR21, Prop. A.1], which we recall here.
Proposition A.1.
Let be a smooth function compactly supported in the second variable and let . Assume that the following conditions hold:
-
(i)
exists in the sense of distributions777Namely, for any , there exists finite . With a slight abuse of notation, we define . for any ;
-
(ii)
and converge to a finite limit as , for any .
Then, for all and , we have
| (200) |
where is the Neumann heat kernel on the half-line, cf. (115).
We want to apply Proposition A.1 to the function , thus, we study in detail the operators , for any . Define iteratively the family of matrices of operators, acting on smooth functions:
| (201) |
as follows. Set
| (202) |
and, for all and , set
| (203) |
while , for all other values of the indices, i.e. , or . Here is the operator defined in (111), namely
| (204) |
where is the inward-pointing normal from .
Recall the definition of and : for any and for all ,
| (205) | ||||
| (206) |
where . Iterations of satisfy the following lemma.
Lemma A.2.
Let be a sub-Riemannian manifold, equipped with a smooth measure , and let be an open relatively compact subset whose boundary is smooth and has no characteristic points. Then, as operators on , we have:
-
(i)
;
-
(ii)
;
-
(iii)
For any ,
(207)
Here we mean that, for any , the operator acts on the functions , . Analogously the right-hand side when evaluated in is a function of .
Proof.
We want to apply Proposition A.1 to for , in order to obtain higher-order asymptotics. However, Lemma A.2 shows that , for , involves time derivatives of which are not well-defined at as . Therefore, we consider the following approximation of and , respectively: fix and define, for any ,
| (212) | ||||
| (213) |
where . We recall that, for any , . Notice that, by the dominated convergence theorem, we have
| (214) |
and, in addition, Lemma A.2 holds unchanged also for and .
Lemma A.3.
Let be a compact sub-Riemannian manifold, equipped with a smooth measure , and let be an open subset whose boundary is smooth and has no characteristic points. Let , and define
| (215) |
Then, for any , the following identities hold:
-
(i)
-
(ii)
-
(iii)
-
(iv)
where, we recall, and .
Remark A.4.
The only difference with respect to [RR21, Lem. A.4] is item , which now holds only as and not for all positive times.
Proof of Lemma A.3.
We claim that, for any ,
| (216) |
Let us prove it by induction: for , applying the divergence theorem, we have
| (217) |
Let us discuss the first term in (217): by divergence theorem applied with respect to the set , we have
| (218) |
then, using [JSC86, Thm. 3] and noticing that , for any and , we conclude that in the limit as , (218) converges to 0. This proves (216), for . For , proceeding by induction, we conclude. Finally, using the co-area formula (94), we complete the proof of the statement as in the usual argument of [Sav98, Lem. 5.6]. ∎
Remark A.5.
In the non-compact case, under the assumption of Theorem 7.3, the above lemma holds. In particular, on the one hand, the divergence theorem holds since has compact support. On the other hand, notice that the time derivative estimates (185) are enough to ensure that (218) converges to as , regardless of the compactness of the set of integration. The same is true for , where higher-order time derivatives appear.
The next step is to apply the iterated Duhamel’s principle (200) to , which now satisfies its assumptions, then, pass to the limit as . The computations are long but straightforward: we report here the result at order .
Lemma A.6.
The expression (129) is a direct consequence of A.6. Moreover, we can apply it, when the base set is chosen to be . Then, evaluating the difference between and we obtain the asymptotic equality (144), which is proved in the next lemma. We use the shorthands , for and respectively.
Lemma A.7.
Under the same assumptions of Lemma A.3, let . Then, as , we have:
| (224) | ||||
| (225) | ||||
| (226) |
where is the operator given by
| (227) |
with the signed distance function from .
Proof.
Firstly, we apply Lemma A.6 to : we obtain exactly the expression (219), with the operator . Secondly, for the outside contribution, recall that we have the following equality of smooth functions:
| (228) |
Therefore, when we apply Lemma A.6 to , we replace with the function . Moreover, the operator defined in (204), for the set , is equal to , since the inward-pointing normal to is . Therefore, writing the difference of the two contributions, and noticing that and its complement share the boundary, we have:
| (229) | ||||
| (230) | ||||
| (231) | ||||
| (232) | ||||
| (233) |
To conclude, it is enough to notice that the functional can be explicitly computed:
| (234) |
Thus, the terms in (231) and (233) are remainder of order . ∎
Applying Lemma A.6 to the sum of and instead, we obtain (162). The proof of this result is not provided here, as it is similar to the proof of Lemma A.7.
Lemma A.8.
Under the same assumptions of Lemma A.3, let . Then, as , we have:
| (235) | ||||
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