Abstract
We consider a family of semiclassically scaled second-order elliptic differential operators on high tensor powers of a Hermitian line bundle (possibly, twisted by an auxiliary Hermitian vector bundle of arbitrary rank) on a Riemannian manifold of bounded geometry. We establish an off-diagonal Gaussian upper bound for the associated heat kernel. The proof is based on some tools from the theory of operator semigroups in a Hilbert space, results on Sobolev spaces adapted to the current setting, and weighted estimates with appropriate exponential weights.
Keywords:
heat kernel estimates; manifold of bounded geometry; Bochner Laplacian; elliptic differential operator; Hermitian line bundle; semiclassical asymptotics MSC:
58J35
1. Introduction
In this paper, we consider a family of second-order elliptic differential operators , , acting on sections of tensor powers of a Hermitian line bundle L of bounded geometry on a complete Riemannian manifold of bounded geometry. Additionally, one can take an auxiliary Hermitian vector bundle E and consider operators acting on sections of . It is well known in geometric quantization (see, for instance, [1]) that the parameter plays the role of semiclassical parameter and the limit can be treated as the semiclassical limit. We put an appropriate semiclassical scaling in the operator and impose some conditions on its ellipticity constant and coefficients (see Section 2 below, in particular, Equation (5) and Assumption 1 for more details). Similar operators have been studied in connection with the Demailly holomorphic Morse inequalities for the Dolbeault cohomology associated with high tensor powers of a holomorphic Hermitian line bundle over a compact complex manifold [2] (see also [3,4,5] and references therein). They may also be viewed as elliptic models for the geometric Fokker–Planck operators (see, for instance, [6,7]).
The main result of the paper is an off-diagonal Gaussian upper bound for the heat kernel associated with the operator in the semiclassical limit . Such Gaussian estimates are, of course, well known for a fixed p (see, for instance, [8] for results on manifolds of bounded geometry). The main point is to obtain estimates, explicitly depending on p. Similar exponential-type estimates were obtained for the Bergman kernels associated with the Bochner Laplacian on Hermitian line bundles on symplectic manifolds of bounded geometry in [9] (see also the references therein for a previous work). The proof in [9] strongly uses the spectral gap property of the Bochner Laplacian. It combines methods developed in [5,10,11] and is inspired by the analytic localization technique of Bismut–Lebeau [12], with exponential weighted estimates as in [8].
The results of the paper are a kind of generalization of the results of [9]. The heat kernel estimates are more general and a bit different. On the one hand, they do not require the spectral gap property and, therefore, there is no need to assume the manifold to be symplectic as in [9]. On the other hand, for the heat kernel estimates, we need to study the resolvents of the operator in unbounded domains of the complex plane (see, for instance, ref. [5] for the related study in the current setting). Actually, for our purposes, we would need to study the resolvent in the right half-plane. In this paper, we use a slightly different strategy. We work with operator semigroups instead of the resolvents and use some tools from the theory of operator semigroups in a Hilbert space. As in [9], we combine these methods with the results on Sobolev spaces adapted to the current setting and weighted estimates with appropriate exponential weights.
The paper is organized as follows. Section 2 contains a more precise presentation of the problem and a statement of the main result. In Section 3, we give some necessary information on differential operators and Sobolev spaces in the current setting. Section 4 is devoted to the study of the Bochner Laplacian. In Section 5, we establish some properties of the operator in Sobolev spaces. In Section 6, we introduce appropriate weight functions and prove some weighted Sobolev estimates for the operator . Finally, in Section 7, we complete the proof of the main theorem, Theorem 1.
2. Preliminaries and the Main Result
Let be a complete Riemannian manifold of dimension d, a Hermitian line bundle on X with a Hermitian connection and a Hermitian vector bundle of rank r on X with a Hermitian connection . We suppose that is a manifold of bounded geometry and L and E have bounded geometry. This means that the curvatures , and of the Levi–Civita connection , connections and , respectively, and their covariant derivatives of any order are uniformly bounded on X in the norm induced by g, and , and the injectivity radius of is positive.
For any , let be the pth tensor power of L and let
be the Hermitian connection on induced by and . Denote by the induced Bochner Laplacian acting on by
where stands for the formal adjoint of the operator .
Given a Hermitian vector bundle on X equipped with a Hermitian connection , the Levi–Civita connection and the connection define a metric connection on each vector bundle for , that allows us to introduce the operator
for every . If has bounded geometry, we denote by , , the space of all such that
where is the norm in defined by g and .
For any , we consider a second order elliptic differential operator acting on of the form
where and . Here we use the canonical identification , which holds for any line bundle L on X. The endomorphism is given by the contraction. If is a local frame in defined in some open set and we write with some , then
If E is the trivial line bundle, then is a vector field on X and
Assumption 1.
We assume that , and, for any , the -norms of and defined by (4) are uniformly bounded on p.
Note that we do not assume that is self-adjoint. Instead, we will use the fact that it is a second order uniformly elliptic differential operator with positive principal symbol and, therefore, generates a holomorphic semigroup in [8]. The heat operator associated to the operator is well defined for any . Let and be the projections of on the first and second factor, respectively. The Schwartz kernel of with respect to the Riemannian volume form is a smooth section .
The main result of the paper is the following upper bound for the heat kernel.
Theorem 1.
For any and , there exists such that for any , there exists such that for any , and with , we have
where
Here we denote by the distance function of and by the pointwise -seminorm of the section at a point , which is the sum of the norms induced by and g of the derivatives up to order k of with respect to the connection and the Levi–Civita connection evaluated at . Finally, stands for the integer part of .
The proof of the main theorem is based on some tools from the theory of operator semigroups in a Hilbert space, a particular choice of Sobolev norms, and a refined form of the Sobolev embedding theorem adapted to a particular sequence of vector bundles as in [13] and weighted estimates with appropriate exponential weights as in [8].
3. Preliminaries on Sobolev Spaces
This section contains the necessary information on Sobolev spaces in the current setting. In particular, we will need a specific choice of the Sobolev norm as well as a slightly refined form of the Sobolev embedding theorem as in [13]. We refer the reader to [8,9,13] for more information on differential operators and Sobolev spaces on manifolds of bounded geometry. We will keep the setting described in Section 2.
Denote by the Riemannian volume form of . The -norm on is given by
For any integer , we introduce the norm on by the formula
The completion of with respect to is the Sobolev space of order m. Denote by the corresponding inner product on . For any integer , we define the norm in the Sobolev space by duality. For any bounded linear operator , , we will denote its operator norm by .
Given a Hermitian vector bundle on X, any differential operator A of order q acting in can be written as
where and the endomorphism · is given by the contraction (see (6) for the case ). If has bounded geometry, we denote by the space of differential operators A of order q in with coefficients in .
We will say that a family is bounded in p, if
and, for any , the family is bounded in the Frechet space . In particular, by Assumption 1, the operator defined by (5) belongs to , and the family is a bounded family.
Proposition 1.
Any operator defines a bounded operator
for any . Moreover, if a family is bounded in p, then for any , there exists such that, for all ,
The following proposition is a refined form of the Sobolev embedding theorem adapted to the sequence ([13], Lemma 2). As observed in [13], its proof does not use the positivity condition for the line bundle L.
Proposition 2.
For any with , we have an embedding
Moreover, there exists such that, for any and ,
For any and , we define the delta-section as a linear functional on given by
where stands for the Hermitian inner product in the fibers of .
As an immediate consequence of Proposition 2, we have [9].
Proposition 3.
For any and , with the following norm estimate
Proof.
By the definition of the Sobolev norm , for any , and , we have
Using (14) and Proposition 2, we get
this completes the proof. □
Proposition 4.
For any , there exist and such that, for any , we have
Proof.
By (10), we have
Using Cauchy inequality and Proposition 1, we get for any
and taking into account (18),
Since for and , we complete the proof. □
4. The Bochner Laplacian
Recall that the Bochner Laplacian is self-adjoint as an unbounded linear operator in the Hilbert space with domain (see, for instance, [8,9]). For the quadratic form of this operator, we have
We establish an elliptic estimate for the operator , uniform in p. For fixed p, this result is proved in [8]. To show a uniform estimate, we use the arguments of [14], where such an estimate is proved for a slightly larger class of manifolds.
Theorem 2.
For any , there exists such that for all we have
Proof.
As above, we write
From the proof of [14] (Lemma 2.1), we can see that the operator
is a first order differential operator of class , uniformly bounded in p. The crucial fact is that the endomorphism
is uniformly bounded in p. We infer that for all
Here and below all constants C are independent of p and u.
We now move to the right and use the above property of repeatedly. After several steps, we arrive at the following inequality:
where
Now we apply the same procedure as above to the second factor in the inner product in the right-hand side of (21). More precisely, put
Using (19), for any , we can write
It follows that
and, as a consequence,
Using (20), the last estimate implies that for all and ,
Applying this estimate to the second term several times, we get (18). □
5. The Operator
Now we consider an operator family given by (5) and establish some properties. As already mentioned, the operator is non-self-adjoint. From the general theory of uniformly elliptic differential operators with bounded coefficients on manifolds of bounded geometry [8], we know that the operator is closed as an unbounded operator in with domain and generates a holomorphic semigroup of angle . It is easy to see that the adjoint of satisfy the same assumption as , that is, it is a second-order differential operator acting on of the form
where and with -norms, uniformly bounded on p.
By (5), Proposition 1 and Proposition 4, for any , , and , we have
where the constants are independent of p, and u.
This estimate allows us to state a uniform elliptic estimate for .
Proposition 5.
For any , there exists such that for we have
Proof.
By Theorem 2, for any , there exists such that for all and we have
Choosing so that , we immediately prove (25). □
Finally, we state the following Garding type inequality for .
Proposition 6.
There exist and such that for we have
Proof.
By Proposition 1, there exists such that, for any and ,
and
It follows that, for any , there exists such that for all ,
Choosing , this completes the proof. □
By Proposition 6, the operator is semi-bounded from below uniformly on p and the constant defined by (8) is finite. Thus, for any , the operator is an accretive operator in . Since this is also true for the operator , the operator is a maximal accretive operator in . By [15] (Theorem 1.24 of Chapter 9), it generates a holomorphic semigroup of angle and
Moreover, for any , there exists such that for every and the element belongs to the domain of and satisfies the following estimates
By elliptic regularity, the domain of coincides with the Sobolev space . This fact allows us to claim that the Schwartz kernel of the operator with respect to the Riemannian volume form is a smooth section . This follows immediately from the formula
where is the delta-section defined by (14).
6. Weighted Estimates
In this section, we first introduce a class of exponential weight functions as in [8] and then study the operator in the corresponding weighted Sobolev spaces.
Recall that d denotes the distance function on X. As shown in [8] (Proposition 4.1) (see also [9] (Section 3.1)), for any and , there exists a function (for simplicity of notation, we skip the index ) , satisfying the following conditions:
(1) We have
(2) For any , there exists (depending on ) such that, for any multi-index with ,
For any , and , we introduce a weight function by
where is a smooth function on X given by
We do not introduce explicitly the weighted Sobolev spaces associated with . Instead, given an operator family , we will consider the operator families
First, we observe that
where is given by
This immediately implies that, if , then, for any and , the operator is in .
Next, we have [9]
where and are given by
Let us estimate the difference of and . For all , , , , and , we have
where C is independent of p, , y, and u.
By Proposition 4 with with an arbitrary , for any , there exist and such that
and, plugging this estimate in (35),
By (25), it follows that for any , , , , , and , we have
As above, we use these estimates to prove a uniform elliptic estimate for .
Proposition 7.
For any , there exists such that for , and , we have
Proof.
Choosing such that , we infer that
Proposition 8.
For , there exists such that, for all , and , we have
7. Proof of the Main Theorem
Now we complete the proof of Theorem 1.
As above, by (38), for any , , and , the operator is a maximal accretive operator in . By [15] (Theorem 1.24 of Chapter 9), the operator generates a holomorphic semigroup of angle and
Moreover, for any , there exists such that for any , , , , and , the section belongs to and satisfies
It is important to note that the constant C is independent of p, , y and t. This follows from the fact that is a contractive semigroup (cf. (39)), Hille–Yosida theorem and Cauchy integral formula.
By (37), it follows that
Similar estimate holds for . Therefore, by duality we have
and
Using the fact that, for any , the function is bounded on , we conclude that
Thus, for any , , , , , , and such that , we have
where depends only on m and .
By (27), (13), (42), Proposition 3 and (28), for any , we have
with . (Here we use the facts that and .)
It follows that there exists such that for any , , and , we have
Using the fact that the Schwartz kernel of the operator is equal to and applying the same argument to this operator, one can prove a similar estimate for any derivative of the heat kernel. Namely, for any , there exists such that for any , , and , we have
Recall that this estimate holds for any , with , and . Put
For any , with and , we get
This completes the proof of Theorem 1.
8. Discussion
Gaussian estimates, similar to the estimates (7), are proved in [13] (Theorem 4) in the case when the operator is the square of the spin Dirac operator. In that case, the operator is self-adjoint and the authors use spectral theorem and the arguments based on the finite propagation speed of solutions of symmetric linear hyperbolic equations. In our case, the operator, generally, is not self-adjoint and we use semigroup theory and weighted estimates instead.
Our estimates are, clearly, non-optimal near the diagonal. In particular, the power singularity is not exact. In [16], a similar technique based on weighted estimates along with logarithmic Sobolev inequalities were used to get optimal bounds for the Laplace–Beltrami operator and some other second-order elliptic differential operators. It should be noted that, in [16], the author still requires some additional information about the behavior of the heat kernel near diagonal.
There are several approaches to study asymptotic behavior of the heat kernel of the Bochner Laplacian on a compact manifold near the diagonal. The leading term of the scaling asymptotics has been studied in connection with the Demailly holomorphic Morse inequalities [2]. We refer to [3,4,5,17] and references therein for the heat kernel proofs of the Demailly inequalities and related topics. Asymptotic expansions for the heat kernel near the diagonal are studied in [5,10]. An asymptotic expansion for the semiclassically scaled heat kernel along the diagonal was proved in [18]. In all these cases, the operators are self-adjoint, but it looks likely that the results can be extended to the case under consideration.
9. Conclusions
The technique developed in this paper is quite simple and effective. It allows us to establish rather optimal estimates for the heat kernel, which hold for small and large times and large distances. It can be easily extended to the case of a general second-order elliptic differential operator with an arbitrary positive second-order part, not necessarily given by the Bochner Laplacian. This assumption has been made in the paper just to simplify the exposition. We also believe that our methods can be extended to higher-order elliptic operators as well as to some other cases when the classical parametrix method does not work, for instance, to the case of a second-order hypoelliptic operator like the geometric Fokker–Planck operator.
Funding
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Acknowledgments
We would also like to thank the anonymous referees for their careful reading and useful suggestions to improve the quality of the manuscript.
Conflicts of Interest
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
Abbreviations
a Hermitian vector bundle of rank r on X, p. 2 | |
a Hermitian line bundle on X, p. 2 | |
, | the coefficient in the formula for , p. 10 |
the space of differential operators of order q in with coefficients in , p. 4 | |
the operator, p. 2 | |
the operator, p. 10 | |
the Sobolev space of order m, p. 4 | |
the curvatures of the connection , p. 2 | |
the curvature of the connection , p. 2 | |
the curvature of the Levi–Civita connection , p.2 | |
the Bochner Laplacian acting on , p. 2 | |
the operator, p. 10 | |
the delta-section, p. 4 | |
the inner product on , p. 4 | |
a Hermitian connection on , p. 2 | |
a Hermitian connection on , p. 2 | |
the Levi–Civita connection of , p. 2 | |
the smoothed distance function, p. 9 | |
p. 10 | |
, | the coefficients of , p. 2 |
the coefficient of , p. 10 | |
d | the dimension of X, p. 2 |
the distance function of , p. 3 | |
the Riemannian volume form of , p. 3 | |
the weight function, p. 9 | |
the space of all -sections of , p. 2 | |
a complete Riemannian manifold, p. 2 | |
the Hermitian connection on , p. 2 | |
, p. 2 | |
the pth tensor power of L, p. 2 |
References
- Berezin, F.A. Quantization. Izv. Akad. Nauk SSSR Ser. Mat. 1974, 38, 1116–1175. [Google Scholar] [CrossRef]
- Demailly, J.-P. Champs magnétiques et inégalités de Morse pour la d′′-cohomologie. Ann. Inst. Fourier 1985, 35, 189–229. [Google Scholar] [CrossRef]
- Demailly, J.-P. Holomorphic Morse inequalities. In Several Complex Variables and Complex Geometry, Part 2 (Santa Cruz, CA, 1989); Proceedings of Symposia in Pure Mathematics, 52, Part 2; American Mathematical Society: Providence, RI, USA, 1991; pp. 93–114. [Google Scholar]
- Bismut, J.-M. Demailly’s asymptotic Morse inequalities: A heat equation proof. J. Funct. Anal. 1987, 72, 263–278. [Google Scholar] [CrossRef]
- Ma, X.; Marinescu, G. Holomorphic Morse Inequalities and Bergman Kernels; Birkhäuser: Basel, Switzerland, 2007. [Google Scholar]
- Helffer, B.; Nier, F. Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians; Lecture Notes in Mathematics, 1862; Springer: Berlin, Germany, 2005. [Google Scholar]
- Lebeau, G. Geometric Fokker-Planck equations. Port. Math. (N.S.) 2005, 62, 469–530. [Google Scholar]
- Kordyukov, Y.A. Lp-theory of elliptic differential operators on manifolds of bounded geometry. Acta Appl. Math. 1991, 23, 223–260. [Google Scholar]
- Kordyukov, Y.A.; Ma, X.; Marinescu, G. Generalized Bergman kernels on symplectic manifolds of bounded geometry. Comm. Partial Differ. Equ. 2019, 44, 1037–1071. [Google Scholar] [CrossRef]
- Dai, X.; Liu, K.; Ma, X. On the asymptotic expansion of Bergman kernel. J. Differ. Geom. 2006, 72, 1–41. [Google Scholar] [CrossRef]
- Ma, X.; Marinescu, G. Generalized Bergman kernels on symplectic manifolds. Adv. Math. 2008, 217, 1756–1815. [Google Scholar] [CrossRef]
- Bismut, J.-M.; Lebeau, G. Complex immersions and Quillen metrics. Inst. Hautes Études Sci. Publ. Math. 1991, 74, 1–291. [Google Scholar] [CrossRef]
- Ma, X.; Marinescu, G. Exponential estimate for the asymptotics of Bergman kernels. Math. Ann. 2015, 362, 1327–1347. [Google Scholar] [CrossRef]
- Salomonsen, G. Equivalence of Sobolev spaces. Results Math. 2001, 2001 39, 115–130. [Google Scholar] [CrossRef]
- Kato, T. Perturbation Theory for Linear Operators; Springer: New York, NY, USA, 1966. [Google Scholar]
- Davies, E.B. Explicit constants for Gaussian upper bounds on heat kernels. Am. J. Math. 1987, 109, 319–333. [Google Scholar] [CrossRef]
- Bouche, T. Convergence de la métrique de Fubini-Study d’un fibré linéaire positif. Ann. Inst. Fourier 1990, 40, 117–130. [Google Scholar] [CrossRef][Green Version]
- Ma, X.; Marinescu, G.; Zelditch, S. Scaling asymptotics of heat kernels of line bundles. In Analysis, Complex Geometry, and Mathematical Physics: In Honor of Duong H. Phong; Contemporary Mathematics, 644; American Mathematical Society: Providence, RI, USA, 2015; pp. 175–202. [Google Scholar]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).