Heat Kernels Estimates for Hermitian Line Bundles on Manifolds of Bounded Geometry

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.

1. Introduction

In this paper, we consider a family of second-order elliptic differential operators $D_p$, $p\in \mathbb{N}$, acting on sections of tensor powers $L^p$ 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 $L^p\otimes E$. It is well known in geometric quantization (see, for instance, [1]) that the parameter $h=\frac{1}{p}$ plays the role of semiclassical parameter and the limit $p\to \infty$ can be treated as the semiclassical limit. We put an appropriate semiclassical scaling in the operator $D_p$ 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 $D_p$ in the semiclassical limit $p\to \infty$. 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 $D_p$ in Sobolev spaces. In Section 6, we introduce appropriate weight functions and prove some weighted Sobolev estimates for the operator $D_p$. Finally, in Section 7, we complete the proof of the main theorem, Theorem 1.

2. Preliminaries and the Main Result

Let $(X,g)$ be a complete Riemannian manifold of dimension d, $(L,h^L)$ a Hermitian line bundle on X with a Hermitian connection ${\nabla }^L$ and $(E,h^E)$ a Hermitian vector bundle of rank r on X with a Hermitian connection ${\nabla }^E$. We suppose that $(X,g)$ is a manifold of bounded geometry and L and E have bounded geometry. This means that the curvatures $R^{TX}$, $R^L$ and $R^E$ of the Levi–Civita connection ${\nabla }^{TX}$, connections ${\nabla }^L$ and ${\nabla }^E$, respectively, and their covariant derivatives of any order are uniformly bounded on X in the norm induced by g, $h^L$ and $h^E$, and the injectivity radius of $(X,g)$ is positive.

For any $p\in \mathbb{N}$, let $L^p:=L^{\otimes p}$ be the pth tensor power of L and let

$${\nabla }^{E_p}:C^{\infty }(X,E_p)\to C^{\infty }(X,T^{\ast }X\otimes E_p)$$

(1)

be the Hermitian connection on $E_p:=L^p\otimes E$ induced by ${\nabla }^L$ and ${\nabla }^E$. Denote by ${\Delta }^{E_p}$ the induced Bochner Laplacian acting on $C^{\infty }(X,E_p)$ by

$${\Delta }^{E_p}={\left({\nabla }^{E_p}\right)}^{\ast }{\nabla }^{E_p},$$

(2)

where ${\left({\nabla }^{E_p}\right)}^{\ast }:C^{\infty }(X,T^{\ast }X\otimes E_p)\to C^{\infty }(X,E_p)$ stands for the formal adjoint of the operator ${\nabla }^{E_p}$.

Given a Hermitian vector bundle $\mathcal{F}$ on X equipped with a Hermitian connection ${\nabla }^{\mathcal{F}}$, the Levi–Civita connection ${\nabla }^{TX}$ and the connection ${\nabla }^{\mathcal{F}}$ define a metric connection ${\nabla }^{\mathcal{F}}:C^{\infty }(X,{\left(T^{\ast }X\right)}^{\otimes j}\otimes \mathcal{F})\to C^{\infty }(X,{\left(T^{\ast }X\right)}^{\otimes (j+1)}\otimes \mathcal{F})$ on each vector bundle ${\left(T^{\ast }X\right)}^{\otimes j}\otimes \mathcal{F}$ for $j\in \mathbb{N}$, that allows us to introduce the operator

$${\left({\nabla }^{\mathcal{F}}\right)}^{\ell }:C^{\infty }(X,\mathcal{F})\to C^{\infty }(X,{\left(T^{\ast }X\right)}^{\otimes \ell }\otimes \mathcal{F})$$

(3)

for every $\ell \in \mathbb{N}$. If $\mathcal{F}$ has bounded geometry, we denote by $C_b^k(X,\mathcal{F})$, $k\in {\mathbb{Z}}_{+}$, the space of all $u\in C^k(X,\mathcal{F})$ such that

$${\parallel u\parallel }_{C_b^k}=\underset{x\in X,\ell \le k}{sup}{\left|{\left({\nabla }^{\mathcal{F}}\right)}^{\ell }u\left(x\right)\right|}_x<\infty ,$$

(4)

where ${|·|}_x$ is the norm in ${\left(T_x^{\ast }X\right)}^{\otimes \ell }\otimes {\mathcal{F}}_x$ defined by g and $h^{\mathcal{F}}$.

For any $p\in \mathbb{N}$, we consider a second order elliptic differential operator $D_p$ acting on $C^{\infty }(X,E_p)$ of the form

$$D_p=\frac{1}{p}{\Delta }^{E_p}+\frac{1}{\sqrt{p}}{a}_{p,1}·{\nabla }^{E_p}+a_{p,0},$$

(5)

where $a_{p,0}\in C^{\infty }(X,End\left(E\right))$ and $a_{p,1}\in C^{\infty }(X,TX\otimes End\left(E\right))$. Here we use the canonical identification $End\left(L^p\right)=\mathbb{C}$, which holds for any line bundle L on X. The endomorphism $·:(TX\otimes End\left(E\right))\otimes (T^{\ast }X\otimes E)\to E$ is given by the contraction. If $\{e_j:j=1,\dots ,d\}$ is a local frame in $TX$ defined in some open set $U\subset X$ and we write $a_{p,1}={\sum }_{j=1}^d{a}_{p,1}^j{e}_j$ with some $a_{p,1}^j\in C^{\infty }(U,End\left(E\left|{}_U\right.\right))$, then

$$a_{p,1}·{\nabla }^{E_p}=\sum _{j=1}^d{a}_{p,1}^j{\nabla }_{e_j}^{E_p}.$$

(6)

If E is the trivial line bundle, then $a_{p,1}$ is a vector field on X and

$$a_{p,1}·{\nabla }^{E_p}={\nabla }_{a_{p,1}}^{L^p}.$$

Assumption 1.

We assume that $a_{p,0}\in C_b^{\infty }(X,End\left(E\right))$, $a_{p,1}\in C_b^{\infty }(X,TX\otimes End\left(E\right))$ and, for any $k\in {\mathbb{Z}}_{+}$, the $C_b^k$-norms of $a_{p,0}$ and $a_{p,1}$ defined by (4) are uniformly bounded on p.

Note that we do not assume that $D_p$ 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 $L^2(X,E_p)$ [8]. The heat operator $exp(-t{D}_p)$ associated to the operator $D_p$ is well defined for any $t>0$. Let ${\pi }_1$ and ${\pi }_2$ be the projections of $X\times X$ on the first and second factor, respectively. The Schwartz kernel of $exp(-t{D}_p)$ with respect to the Riemannian volume form $d{v}_X$ is a smooth section $exp\left(-t{D}_p\right)(·,·)\in C^{\infty }(X\times X,{\pi }_1^{\ast }{E}_p\otimes {\pi }_2^{\ast }{E}_p^{\ast })$.

The main result of the paper is the following upper bound for the heat kernel.

Theorem 1.

For any ${\epsilon }_0>0$ and $\epsilon >0$, there exists $A>0$ such that for any $k\in {\mathbb{Z}}_{+}$, there exists $C_k>0$ such that for any $p\in \mathbb{N}$, $t>0$ and $x,y\in X$ with $d(x,y)>\frac{{\epsilon }_0}{\sqrt{p}}$, we have

$${\left|exp\left(-t{D}_p\right)(x,y)\right|}_{C^k}\le \frac{C_k}{t^{[d/2]+1+k/2}}exp\left((-{\lambda }_0+\epsilon )t-\frac{pd{(x,y)}^2}{At}\right),$$

(7)

where

$${\lambda }_0:=\underset{p\in \mathbb{N}}{inf}\underset{u\in L^2(X,L^p\otimes E),u\ne 0}{inf}\frac{\Re {〈D_{p}u,u〉}_{p,0}}{{\parallel u\parallel }_{p,0}}>-\infty .$$

(8)

Here we denote by $d(x,y)$ the distance function of $(X,g)$ and by ${\left|exp\left(-t{D}_p\right)(x,y)\right|}_{C^k}$ the pointwise $C^k$-seminorm of the section $exp\left(-t{D}_p\right)(·,·)$ at a point $(x,y)\in X\times X$, which is the sum of the norms induced by $h^L,h^E$ and g of the derivatives up to order k of $exp\left(-t{D}_p\right)(·,·)$ with respect to the connection ${\nabla }^{E_p}$ and the Levi–Civita connection ${\nabla }^{TX}$ evaluated at $(x,y)$. Finally, $[d/2]$ stands for the integer part of $d/2$.

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 $L^p\otimes E,p\in \mathbb{N},$ 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 $d{v}_X$ the Riemannian volume form of $(X,g)$. The $L^2$-norm on $L^2(X,E_p)$ is given by

$${\parallel u\parallel }_{p,0}^2={\int }_X{\left|u\left(x\right)\right|}^{2}d{v}_X\left(x\right),u\in L^2(X,E_p).$$

(9)

For any integer $m>0$, we introduce the norm ${\parallel ·\parallel }_{p,m}$ on $C_c^{\infty }(X,E_p)$ by the formula

$${\parallel u\parallel }_{p,m}^2=\sum _{\ell =0}^m{\int }_X{\left|{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{\ell }u\left(x\right)\right|}^{2}d{v}_X\left(x\right),u\in H^m(X,E_p).$$

(10)

The completion of $C_c^{\infty }(X,E_p)$ with respect to ${\parallel ·\parallel }_{p,m}$ is the Sobolev space $H^m(X,E_p)$ of order m. Denote by ${\langle ·,·\rangle }_{p,m}$ the corresponding inner product on $H^m(X,E_p)$. For any integer $m<0$, we define the norm in the Sobolev space $H^m(X,E_p)$ by duality. For any bounded linear operator $A:H^m(X,E_p)\to H^{m^{\prime }}(X,E_p)$, $m,m^{\prime }\in \mathbb{Z}$, we will denote its operator norm by ${\parallel A\parallel }_p^{m,m^{\prime }}$.

Given a Hermitian vector bundle $\mathcal{F}$ on X, any differential operator A of order q acting in $C^{\infty }(X,\mathcal{F})$ can be written as

$$A=\sum _{\ell =0}^q{a}_{\ell }·{\left({\nabla }^{\mathcal{F}}\right)}^{\ell },$$

(11)

where $a_{\ell }\in C^{\infty }(X,{\left(TX\right)}^{\otimes \ell }\otimes End\left(\mathcal{F}\right))$ and the endomorphism · is given by the contraction (see (6) for the case $\ell =1$). If $\mathcal{F}$ has bounded geometry, we denote by $B{D}^q(X,\mathcal{F})$ the space of differential operators A of order q in $C_c^{\infty }(X,\mathcal{F})$ with coefficients $a_{\ell }$ in $C_b^{\infty }(X,{\left(TX\right)}^{\otimes \ell }\otimes End\left(\mathcal{F}\right))$.

We will say that a family $\{A_p\in B{D}^q(X,E_p),p\in \mathbb{N}\}$ is bounded in p, if

$$A_p=\sum _{\ell =0}^q{a}_{p,\ell }·{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{\ell },a_{p,\ell }\in C_b^{\infty }(X,{\left(TX\right)}^{\otimes \ell }\otimes End\left(E\right)),$$

(12)

and, for any $\ell =0,1,\dots ,q$, the family $\{a_{p,\ell },p\in \mathbb{N}\}$ is bounded in the Frechet space $C_b^{\infty }(X,{\left(TX\right)}^{\otimes \ell }\otimes End\left(E\right))$. In particular, by Assumption 1, the operator $D_p$ defined by (5) belongs to $B{D}^2(X,E_p)$, and the family $\{D_p,p\in \mathbb{N}\}$ is a bounded family.

Proposition 1.

Any operator $A\in B{D}^q(X,E_p)$ defines a bounded operator

$$A:H^{m+q}(X,E_p)⟶H^m(X,E_p)$$

for any $m\in \mathbb{Z}$. Moreover, if a family $\{A_p\in B{D}^q(X,E_p),p\in \mathbb{N}\}$ is bounded in p, then for any $m\in \mathbb{Z}$, there exists $C_m>0$ such that, for all $p\in \mathbb{N}$,

$$\parallel A_p{u\parallel }_{p,m}\le C_m{\parallel u\parallel }_{p,m+q},u\in H^{m+q}(X,E_p).$$

The following proposition is a refined form of the Sobolev embedding theorem adapted to the sequence $E_p,p\in \mathbb{N}$ ([13], Lemma 2). As observed in [13], its proof does not use the positivity condition for the line bundle L.

Proposition 2.

For any $k,m\in {\mathbb{Z}}_{+}$ with $m>k+d/2$, we have an embedding

$$H^m(X,E_p)\subset C_b^k(X,E_p).$$

Moreover, there exists $C_{m,k}>0$ such that, for any $p\in \mathbb{N}$ and $u\in H^m(X,E_p)$,

$${\parallel u\parallel }_{C_b^k}\le C_{m,k}{p}^{d/4+k/2}{\parallel u\parallel }_{p,m}.$$

(13)

For any $x\in X$ and $v\in E_{p,x}$, we define the delta-section ${\delta }_v\in C^{-\infty }(X,E_p)$ as a linear functional on $C_c^{\infty }(X,E_p)$ given by

$$\langle {\delta }_v,\phi \rangle ={\langle v,\phi \left(x\right)\rangle }_{h^{E_p}},\phi \in C_c^{\infty }(X,E_p),$$

(14)

where ${\langle ·,·\rangle }_{h^{E_p}}$ stands for the Hermitian inner product in the fibers of $E_p$.

As an immediate consequence of Proposition 2, we have [9].

Proposition 3.

For any $m>d/2$ and $v\in E_p$, ${\delta }_v\in H^{-m}(X,E_p)$ with the following norm estimate

$$\underset{\left|v\right|=1}{sup}{p}^{-d/4}{\parallel {\delta }_v\parallel }_{p,-m}<\infty .$$

(15)

Proof. 

By the definition of the Sobolev norm ${\parallel ·\parallel }_{p,-m}$, for any $m>d/2$, $x\in X$ and $v\in E_{p,x}$, we have

$$\parallel {\delta }_v{\parallel }_{p,-m}=\underset{\phi \in H^m(X,L^p\otimes E),\phi \ne 0}{sup}\frac{|\langle {\delta }_v,\phi \rangle |}{{\parallel \phi \parallel }_{p,m}}.$$

Using (14) and Proposition 2, we get

$$|\langle {\delta }_v,\phi \rangle |=|{\langle v,\phi \left(x\right)\rangle }_{h^{E_p}}{|\le \parallel \phi \parallel }_{C_b^0}\left|v\right|\le C{p}^{d/4}{\parallel \phi \parallel }_{p,m}\left|v\right|,$$

this completes the proof. □

Proposition 4.

For any $m\in {\mathbb{Z}}_{+}$, there exist $c_1>0$ and $c_2>0$ such that, for any $\epsilon >0$, we have

$${\parallel u\parallel }_{p,m+1}\le \epsilon {∥u∥}_{p,m+2}+\left(\frac{c_1}{\epsilon }+c_2\right){\parallel u\parallel }_{p,m},u\in H^m(X,E_p).$$

Proof. 

By (10), we have

$${\parallel u\parallel }_{p,m+1}^2={∥u∥}_{p,m}^2+{∥\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes m}\otimes E_p}{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{m}u∥}_{p,0}^2.$$

(16)

Using Cauchy inequality and Proposition 1, we get for any $\epsilon >0$

$$\begin{array}{c}{∥\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes m}\otimes E_p}{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{m}u∥}_{p,0}^2\hfill \\ ={〈{\left(\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes m}\otimes E_p}\right)}^{\ast }\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes m}\otimes E_p}{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{m}u,{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{m}u〉}_{p,0}\\ \le {∥{\left(\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes m}\otimes E_p}\right)}^{\ast }\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes m}\otimes E_p}{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{m}u∥}_{p,0}\\ \hfill \times {∥{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{m}u∥}_{p,0}\le C_m{∥u∥}_{p.m+2}{\parallel u\parallel }_{p,m}\le {\epsilon }^2{∥u∥}_{p.m+2}^2+\frac{1}{4C_m^2{\epsilon }^2}{\parallel u\parallel }_{p,m}^2\end{array}$$

and taking into account (18),

$${\parallel u\parallel }_{p,m+1}^2\le {\epsilon }^2{∥u∥}_{p.m+2}^2+\left(\frac{1}{4C_m^2{\epsilon }^2}+1\right){\parallel u\parallel }_{p,m}^2.$$

Since $\sqrt{a^2+b^2}\le a+b$ for $a\ge 0$ and $b\ge 0$, we complete the proof. □

4. The Bochner Laplacian

Recall that the Bochner Laplacian ${\Delta }^{E_p}$ is self-adjoint as an unbounded linear operator in the Hilbert space $L^2(X,E_p)$ with domain $H^2(X,E_p)$ (see, for instance, [8,9]). For the quadratic form of this operator, we have

$${〈\frac{1}{p}{\Delta }^{E_p}u,u〉}_{p,0}={∥\frac{1}{\sqrt{p}}{\nabla }^{E_p}u∥}_{p,0}^2,u\in H^2(X,E_p).$$

(17)

We establish an elliptic estimate for the operator $\frac{1}{p}{\Delta }^{E_p}$, 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 $m\in {\mathbb{Z}}_{+}$, there exists $C_m>0$ such that for all $p\in \mathbb{N}$ we have

$${\parallel u\parallel }_{p,m+2}\le C_m\left({∥\frac{1}{p}{\Delta }^{E_p}u∥}_{p,m}+{\parallel u\parallel }_{p,0}\right),u\in H^{m+2}(X,E_p).$$

(18)

Proof. 

As above, we write

$$\begin{array}{c}{\parallel u\parallel }_{p,m+2}^2={∥u∥}_{p,m+1}^2+\langle {\left(\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes (m+1)}\otimes E_p}\right)}^{\ast }\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes (m+1)}\otimes E_p}\times \hfill \\ \hfill \times {\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{m+1}u,{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{m+1}u{\rangle }_{p,0}^2.\end{array}$$

From the proof of [14] (Lemma 2.1), we can see that the operator

$$\begin{array}{c}{R}_m={\left(\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes (m+1)}\otimes E_p}\right)}^{\ast }\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes (m+1)}\otimes E_p}\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes m}\otimes E_p}\hfill \\ \hfill -\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes m}\otimes E_p}{\left(\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes m}\otimes E_p}\right)}^{\ast }\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes m}\otimes E_p}\end{array}$$

(19)

is a first order differential operator of class $B{D}^1$, uniformly bounded in p. The crucial fact is that the endomorphism

$$\frac{1}{p}{R}^{E_p}=R^L+\frac{1}{p}{R}^E$$

is uniformly bounded in p. We infer that for all $p\in \mathbb{N}$

$$\begin{array}{c}{\parallel u\parallel }_{p,m+2}^2={\parallel u\parallel }_{p,m+1}^2+\langle \frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes m}\otimes E_p}{\left(\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes m}\otimes E_p}\right)}^{\ast }\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes m}\otimes E_p}\times \hfill \\ \times {\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{m}u,{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{m+1}u{\rangle }_{p,0}+{〈R_m{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{m}u,{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{m+1}u〉}_{p,0}\\ \le \langle \frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes m}\otimes E_p}{\left(\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes m}\otimes E_p}\right)}^{\ast }\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes m}\otimes E_p}\times \\ \hfill \times {\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{m}u,{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{m+1}u{\rangle }_{p,0}+C{\parallel u\parallel }_{p,m+1}^2.\end{array}$$

Here and below all constants C are independent of p and u.

We now move ${\nabla }^{\ast }$ to the right and use the above property of $R_m$ repeatedly. After several steps, we arrive at the following inequality:

$${\parallel u\parallel }_{p,m+2}^2\le |I_0{|+C\parallel u\parallel }_{p,m+1}^2,p\in \mathbb{N},$$

(20)

where

$$\begin{array}{c}{I}_0={〈{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{m+1}{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{\ast }\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)u,{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{m+1}u〉}_{p,0}\hfill \\ \hfill ={〈{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{m+1}\left(\frac{1}{p}{\Delta }^{E_p}\right)u,{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{m+1}u〉}_{p,0}.\end{array}$$

(21)

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

$$\begin{array}{c}{I}_k=\langle {\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^m\left(\frac{1}{p}{\Delta }^{E_p}\right)u,{\left(\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes (m-k)}\otimes E_p}\right)}^k{\left(\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes (m-k+1)}\otimes E_p}\right)}^{\ast }\times \hfill \\ \hfill \times {\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{m-k+1}u{\rangle }_{p,0},k=0,1,\dots ,m.\end{array}$$

(22)

It is clear that, for $k=0$, (22) agrees with (21), and

$$\begin{array}{c}{I}_m={〈{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^m\left(\frac{1}{p}{\Delta }^{E_p}\right)u,{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^m{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{\ast }\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)u〉}_{p,0}\hfill \\ \hfill ={〈{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^m\left(\frac{1}{p}{\Delta }^{E_p}\right)u,{\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^m\left(\frac{1}{p}{\Delta }^{E_p}\right)u〉}_{p,0}={∥\frac{1}{p}{\Delta }^{E_p}u∥}_{p,m}^2.\end{array}$$

Using (19), for any $k=0,\dots ,m-1$, we can write

$$\begin{array}{c}{I}_k=\langle {\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^m\left(\frac{1}{p}{\Delta }^{E_p}\right)u,{\left(\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes (m-k+1)}\otimes E_p}\right)}^k\times \hfill \\ \times {\left(\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes (m-k+1)}\otimes E_p}\right)}^{\ast }\left(\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes (m-k+1)}\otimes E_p}\right)\left(\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes (m-k)}\otimes E_p}\right)\times \\ \times {\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{m-k-1}u{\rangle }_{p,0}\\ =I_{k+1}+\langle {\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^m\left(\frac{1}{p}{\Delta }^{E_p}\right)u,{\left(\frac{1}{\sqrt{p}}{\nabla }^{{\left(T^{\ast }X\right)}^{\otimes (m-k+1)}\otimes E_p}\right)}^k{R}_{m-k}\times \\ \hfill \times {\left(\frac{1}{\sqrt{p}}{\nabla }^{E_p}\right)}^{m-k-1}u{\rangle }_{p,0}.\end{array}$$

It follows that

$$|I_k|\le |I_{k+1}|+C_k{∥\left(\frac{1}{p}{\Delta }^{E_p}\right)u∥}_{p,m}{\parallel u\parallel }_{p,m},k=0,\dots ,m-1,p\in \mathbb{N},$$

and, as a consequence,

$$|I_0|\le |I_m|+C{∥\left(\frac{1}{p}{\Delta }^{E_p}\right)u∥}_{p,m}{\parallel u\parallel }_{p,m}\le C\left({∥\frac{1}{p}{\Delta }^{E_p}u∥}_{p,m}^2+{\parallel u\parallel }_{p,m}^2\right),p\in \mathbb{N}.$$

Using (20), the last estimate implies that for all $p\in \mathbb{N}$ and $u\in H^{m+2}(X,E_p)$,

$${\parallel u\parallel }_{p,m+2}\le C\left({∥\frac{1}{p}{\Delta }^{E_p}u∥}_{p,m}+{\parallel u\parallel }_{p,m}\right).$$

Applying this estimate to the second term ${\parallel u\parallel }_{p,m}$ several times, we get (18). □

5. The Operator ${\mathit{D}}_{\mathit{p}}$

Now we consider an operator family $\{D_p,p\in \mathbb{N}\}$ given by (5) and establish some properties. As already mentioned, the operator $D_p$ 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 $D_p$ is closed as an unbounded operator in $L^2(X,E_p)$ with domain $H^2(X,E_p)$ and generates a holomorphic semigroup of angle $\pi /2$. It is easy to see that the adjoint $D_p^{\ast }$ of $D_p$ satisfy the same assumption as $D_p$, that is, it is a second-order differential operator acting on $C^{\infty }(X,E_p)$ of the form

$$D_p^{\ast }=\frac{1}{p}{\Delta }^{E_p}+\frac{1}{\sqrt{p}}{a}_{p,1}^{\prime }·{\nabla }^{E_p}+a_{p,0}^{\prime },$$

where $a_{p,1}^{\prime }\in C_b^{\infty }(X,TX\otimes End\left(E\right))$ and $a_{p,0}^{\prime }\in C_b^{\infty }(X,End\left(E\right))$ with $C_b^k$-norms, uniformly bounded on p.

By (5), Proposition 1 and Proposition 4, for any $m\in {\mathbb{Z}}_{+}$, $p\in \mathbb{N}$, $\epsilon >0$ and $u\in H^m(X,E_p)$, we have

$$\begin{array}{cc}{∥\left(D_p-\frac{1}{p}{\Delta }^{E_p}\right)u∥}_{p,m}\hfill & \le C{∥u∥}_{p,m+1}\hfill \end{array}$$

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& \hfill \le C_1\epsilon {∥u∥}_{p,m+2}+\left(\frac{C_2}{\epsilon }+C_3\right){\parallel u\parallel }_{p,m}\end{array}$$

(23)

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& \hfill \le C_4\epsilon {∥\left(1+\frac{1}{p}{\Delta }^{E_p}\right)u∥}_{p,m}+\left(\frac{C_5}{\epsilon }+C_6\right){\parallel u\parallel }_{p,m}.,\end{array}$$

(24)

where the constants are independent of p, $\epsilon$ and u.

This estimate allows us to state a uniform elliptic estimate for $D_p$.

Proposition 5.

For any $m\in {\mathbb{Z}}_{+}$, there exists $C_m>0$ such that for $p\in \mathbb{N}$ we have

$${\parallel u\parallel }_{p,m+2}\le C_m\left({∥D_{p}u∥}_{p,m}+{\parallel u\parallel }_{p,0}\right),u\in H^{m+2}(X,E_p).$$

(25)

Proof. 

By Theorem 2, for any $m\in {\mathbb{Z}}_{+}$, there exists $C>0$ such that for all $p\in \mathbb{N}$ and $u\in H^{m+2}(X,E_p)$ we have

$$\begin{array}{cc}\hfill {\parallel u\parallel }_{p,m+2}\le & C\left({∥\frac{1}{p}{\Delta }^{E_p}u∥}_{p,m}+{\parallel u\parallel }_{p,0}\right)\hfill \\ \hfill \le & C\left({∥D_{p}u∥}_{p,m}+{∥\left(D_p-\frac{1}{p}{\Delta }^{E_p}\right)u∥}_{p,m}+{\parallel u\parallel }_{p,0}\right).\hfill \end{array}$$

By (23), there exist $C_1>0$, $C_2>0$ and $C_3>0$, depending only on m, such that, for any $\epsilon >0$, we get

$$\begin{array}{cc}\hfill {\parallel u\parallel }_{p,m+2}\le & C\left({∥D_{p}u∥}_{p,m}+C_1\epsilon {∥u∥}_{p,m+2}+\left(\frac{C_2}{\epsilon }+C_3\right){\parallel u\parallel }_{p,m}+{\parallel u\parallel }_{p,0}\right).\hfill \end{array}$$

Choosing $\epsilon >0$ so that $C{C}_1\epsilon <1$, we immediately prove (25). □

Finally, we state the following Garding type inequality for $D_p$.

Proposition 6.

There exist $C_1>0$ and $C_2$ such that for $p\in \mathbb{N}$ we have

$$\Re {〈D_{p}u,u〉}_{p,0}\ge C_1{\parallel u\parallel }_{p,1}^2+C_2{\parallel u\parallel }_{p,0}^2,u\in H^2(X,E_p).$$

(26)

Proof. 

By (5) and (17), for all $p\in \mathbb{N}$ and $u\in H^2(X,E_p)$, we have

$$\Re {〈D_{p}u,u〉}_{p,0}={∥\frac{1}{\sqrt{p}}{\nabla }^{E_p}u∥}_{p,0}^2+\Re {〈\left(\frac{1}{\sqrt{p}}{a}_{p,1}·{\nabla }^{E_p}\right)u,u〉}_{p,0}+\Re {〈a_{p,0}u,u〉}_{p,0}.$$

By Proposition 1, there exists $c>0$ such that, for any $\epsilon >0$ and $p\in \mathbb{N}$,

$$\Re {〈\left(\frac{1}{\sqrt{p}}{a}_{p,1}·{\nabla }^{E_p}\right)u,u〉}_{p,0}\ge -{c\parallel u\parallel }_{p,1}{\parallel u\parallel }_{p,0}\ge -{\epsilon \parallel u\parallel }_{p,1}^2-\frac{c^2}{4\epsilon }{\parallel u\parallel }_{p,0}^2$$

and

$$\Re {〈a_{p,0}u,u〉}_{p,0}\ge -c{\parallel u\parallel }_{p,0}^2$$

It follows that, for any $\epsilon >0$, there exists $C_{\epsilon }\in \mathbb{R}$ such that for all $p\in \mathbb{N}$,

$$\Re {〈D_{p}u,u〉}_{p,0}\ge {(1-\epsilon )\parallel u\parallel }_{p,1}^2-C_{\epsilon }{\parallel u\parallel }_{p,0}^2,u\in H^2(X,E_p).$$

Choosing $\epsilon <1$, this completes the proof. □

By Proposition 6, the operator $\Re D_p:=\frac{1}{2}(D_p+D_p^{\ast })$ is semi-bounded from below uniformly on p and the constant ${\lambda }_0$ defined by (8) is finite. Thus, for any $p\in \mathbb{N}$, the operator $D_p-{\lambda }_0$ is an accretive operator in $L^2(X,E_p)$. Since this is also true for the operator $D_p^{\ast }-{\lambda }_0$, the operator $D_p$ is a maximal accretive operator in $L^2(X,E_p)$. By [15] (Theorem 1.24 of Chapter 9), it generates a holomorphic semigroup $exp(-z{D}_p)$ of angle $\pi /2$ and

$$\parallel exp(-t{D}_p){\parallel }_p^{0,0}\le C_1{e}^{-{\lambda }_{0}t},t>0.$$

Moreover, for any $m\in {\mathbb{Z}}_{+}$, there exists $C_m>0$ such that for every $t>0$ and $u\in L^2(X,E_p)$ the element $exp(-t{D}_p)u$ belongs to the domain of $D_p^m$ and satisfies the following estimates

$$\parallel D_p^{m}exp(-t{D}_p){u\parallel }_{p,0}\le \frac{C_m}{t^m}{e}^{-{\lambda }_{0}t}{\parallel u\parallel }_{p,0},t>0.$$

By elliptic regularity, the domain of $D_p^m$ coincides with the Sobolev space $H^{2m}(X,E_p)$. This fact allows us to claim that the Schwartz kernel of the operator $exp(-t{D}_p)$ with respect to the Riemannian volume form $d{v}_X$ is a smooth section $exp\left(-t{D}_p\right)(·,·)\in C^{\infty }(X\times X,{\pi }_1^{\ast }{E}_p\otimes {\pi }_2^{\ast }{E}_p^{\ast })$. This follows immediately from the formula

$$exp(-t{D}_p)(x,y)v=[exp(-t{D}_p){\delta }_v]\left(x\right),v\in E_{p,y},$$

(27)

where ${\delta }_v\in C^{-\infty }(X,E_p)$ 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 $D_p$ 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 $p\in \mathbb{N}$ and $\gamma >0$, there exists a function ${\tilde{d}}_{p,\gamma }={\tilde{d}}_p$ (for simplicity of notation, we skip the index $\gamma$) , satisfying the following conditions:

(1) We have

$$|{\tilde{d}}_p(x,y)-d(x,y)|<\frac{\gamma }{\sqrt{p}},x,y\in X;$$

(28)

(2) For any $k>0$, there exists $c_k>0$ (depending on $\gamma$) such that, for any multi-index $\beta$ with $\left|\beta \right|=k$,

$$\left|{\left(\frac{1}{\sqrt{p}}\right)}^k{\partial }_x^{\beta }{\tilde{d}}_p(x,y)\right|<\frac{c_k}{\sqrt{p}},x,y\in X.$$

(29)

For any $\alpha \in \mathbb{R}$, $p\in \mathbb{N}$ and $y\in X$, we introduce a weight function $f_{\alpha ,p,y}\in C^{\infty }\left(X\right)$ by

$$f_{\alpha ,p,y}\left(x\right)=e^{\alpha {\tilde{d}}_{p,y}\left(x\right)},x\in X,$$

(30)

where ${\tilde{d}}_{p,y}$ is a smooth function on X given by

$${\tilde{d}}_{p,y}\left(x\right)={\tilde{d}}_p(x,y),x\in X.$$

(31)

We do not introduce explicitly the weighted Sobolev spaces associated with $f_{\alpha ,p,y}$. Instead, given an operator family $\{A_p:C_c^{\infty }(X,E_p)\to C^{-\infty }(X,E_p),p\in \mathbb{N}\}$, we will consider the operator families

$$A_{p;\alpha ,y}=f_{\alpha ,p,y}{A}_p{f}_{\alpha ,p,y}^{-1},p\in \mathbb{N},\alpha \in \mathbb{R},y\in X.$$

(32)

First, we observe that

$${\nabla }_{\alpha ,y}^{E_p}:=f_{\alpha ,p,y}{\nabla }^{E_p}{f}_{\alpha ,p,y}^{-1}={\nabla }^{E_p}+\alpha c_{p,y},$$

(33)

where $c_{p,y}\in C_b^{\infty }(X,T^{\ast }X)$ is given by

$$c_{p,y}\left(v\right)=-v\left({\tilde{d}}_{p,y}\right),v\in TX.$$

This immediately implies that, if $Q\in B{D}^q(X,E_p)$, then, for any $\alpha \in \mathbb{R}$ and $y\in X$, the operator $f_{\alpha ,p,y}Q{f}_{\alpha ,p,y}^{-1}$ is in $B{D}^q(X,E_p)$.

Next, we have [9]

$${\Delta }_{p;\alpha ,y}:=f_{\alpha ,p,y}{\Delta }^{E_p}{f}_{\alpha ,p,y}^{-1}={\Delta }^{E_p}+\alpha A_{p;y}+{\alpha }^2{B}_{p;y},$$

(34)

where $A_{p;y}\in B{D}^1(X,E_p)$ and $B_{p;y}\in B{D}^0(X,E_p)$ are given by

$$A_{p;y}=-2\nabla {\tilde{d}}_{p,y}·{\nabla }^{E_p}+\Delta {\tilde{d}}_{p,y},B_{p;y}=-{|\nabla {\tilde{d}}_{p,y}|}^2.$$

Note that the families $\{\frac{1}{\sqrt{p}}{A}_{p;y}:p\in \mathbb{N},y\in X\}$ and $\{B_{p;y}:p\in \mathbb{N},y\in X\}$ are uniformly bounded in p. By (33) and (34), we infer that

$$D_{p;\alpha ,y}:=f_{\alpha ,p,y}{D}_p{f}_{\alpha ,p,y}^{-1}=D_p+\frac{\alpha }{\sqrt{p}}\left(\frac{1}{\sqrt{p}}{A}_{p;y}+a_{p,1}·c_{p,y}\right)+\frac{{\alpha }^2}{p}{B}_{p;y}.$$

Let us estimate the difference of $D_{p;\alpha ,y}$ and $D_p$. For all $p\in \mathbb{N}$, $m\in {\mathbb{Z}}_{+}$, $\alpha \in \mathbb{R}$, $y\in X$, and $u\in H^{m+2}(X,E_p)$, we have

$$\begin{array}{c}{∥\left(D_{p;\alpha ,y}-D_p\right)u∥}_{p,m}={∥\left(\frac{\alpha }{\sqrt{p}}\left(\frac{1}{\sqrt{p}}{A}_{p;y}+a_{p,1}·c_{p,y}\right)+\frac{{\alpha }^2}{p}{B}_{p;y}\right)u∥}_{p,m}\hfill \\ \hfill \le C\left(\frac{\alpha }{\sqrt{p}}{\parallel u\parallel }_{p,m+1}+\frac{{\alpha }^2}{p}{\parallel u\parallel }_{p,m}\right).\end{array}$$

(35)

where C is independent of p, $\alpha$, y, and u.

By Proposition 4 with $\epsilon =\kappa \frac{\sqrt{p}}{\alpha }$ with an arbitrary $\kappa >0$, for any $m\in {\mathbb{Z}}_{+}$, there exist $c_1>0$ and $c_2>0$ such that

$${\parallel u\parallel }_{p,m+1}\le \kappa \frac{\sqrt{p}}{\alpha }{∥u∥}_{p,m+2}+\left(c_1\frac{\alpha }{\sqrt{p}\kappa }+c_2\right){\parallel u\parallel }_{p,m},u\in H^m(X,E_p).$$

and, plugging this estimate in (35),

$${∥\left(D_{p;\alpha ,y}-D_p\right)u∥}_{p,m}\le C\left({\kappa \parallel u\parallel }_{p,m+2}+\left(c_1\frac{{\alpha }^2}{p\kappa }+c_2\frac{\alpha }{\sqrt{p}}\right){\parallel u\parallel }_{p,m}\right).$$

By (25), it follows that for any $\kappa >0$, $p\in \mathbb{N}$, $m\in {\mathbb{Z}}_{+}$, $\alpha \in \mathbb{R}$, $y\in X$, and $u\in H^{m+2}(X,E_p)$, we have

$${∥\left(D_{p;\alpha ,y}-D_p\right)u∥}_{p,m}\le C\left(\kappa \parallel D_p{u\parallel }_{p,m}+\left(c_2\frac{{\alpha }^2}{p\kappa }+c_3\frac{\alpha }{\sqrt{p}}\right){\parallel u\parallel }_{p,m}\right).$$

(36)

As above, we use these estimates to prove a uniform elliptic estimate for $D_{p;\alpha ,y}$.

Proposition 7.

For any $m\in {\mathbb{Z}}_{+}$, there exists $C_1,C_2,C_3>0$ such that for $p\in \mathbb{N}$, $\alpha \in \mathbb{R}$ and $y\in X$, we have

$${\parallel u\parallel }_{p,m+2}\le C_1\parallel D_{p;\alpha ,y}{u\parallel }_{p,m}+\left(C_2\frac{{\alpha }^2}{p}+C_3\frac{\alpha }{\sqrt{p}}\right){\parallel u\parallel }_{p,m},u\in H^{m+2}(X,E_p).$$

(37)

Proof. 

By (36), for any $\kappa >0$, $p\in \mathbb{N}$, $m\in {\mathbb{Z}}_{+}$, $\alpha \in \mathbb{R}$, $y\in X$, and $u\in H^{m+2}(X,E_p)$, we have

$$\begin{array}{cc}\hfill {∥D_{p}u∥}_{p,m}\le & \parallel D_{p;\alpha ,y}{u\parallel }_{p,m}+{\parallel (D_{p;\alpha ,y}-D_p)u\parallel }_{p,m}\hfill \\ \hfill \le & \parallel D_{p;\alpha ,y}{u\parallel }_{p,m}+C\left(\kappa \parallel D_p{u\parallel }_{p,m}+\left(c_2\frac{{\alpha }^2}{p\kappa }+c_3\frac{\alpha }{\sqrt{p}}\right){\parallel u\parallel }_{p,m}\right),\hfill \end{array}$$

where C is independent of $\kappa$, p, $\alpha$, y, and u. It follows that

$$(1-C\kappa ){∥D_{p}u∥}_{p,m}\le \parallel D_{p;\alpha ,y}{u\parallel }_{p,m}+C\left(c_2\frac{{\alpha }^2}{p\kappa }+c_3\frac{\alpha }{\sqrt{p}}\right){\parallel u\parallel }_{p,m}.$$

Choosing $\kappa >0$ such that $C\kappa <1$, we infer that

$${∥D_{p}u∥}_{p,m}\le \frac{1}{1-C\kappa }\parallel D_{p;\alpha ,y}{u\parallel }_{p,m}+\frac{C}{1-C\kappa }\left(c_2\frac{{\alpha }^2}{p\kappa }+c_3\frac{\alpha }{\sqrt{p}}\right){\parallel u\parallel }_{p,m}.$$

Plugging this estimate in (25), we get (37). □

Proposition 8.

For $\epsilon >0$, there exists $A_{\epsilon }>0$ such that, for all $p\in \mathbb{N}$, $\alpha \in \mathbb{R}$ and $y\in X$, we have

$$\Re {〈D_{p;\alpha ,y}u,u〉}_{p,0}\ge \left({\lambda }_0-\epsilon -A_{\epsilon }\frac{{\alpha }^2}{p}\right){\parallel u\parallel }_{p,0}^2,u\in H^2(X,E_p).$$

(38)

Proof. 

We have (see, for instance, [9] (3.18))

$$\Re {\Delta }_{p;\alpha ,y}:=\frac{1}{2}({\Delta }_{p;\alpha ,y}+{\Delta }_{p;\alpha ,y}^{\ast })={\Delta }^{E_p}-{\alpha }^2{|\nabla {\tilde{d}}_{p,y}|}^2.$$

Consider the first order term $R_p=\frac{1}{\sqrt{p}}{a}_{p,1}·{\nabla }^{E_p}$ in $D_p$. Using (33), we compute

$$R_{p;\alpha ,y}=a_{p,1}·{\nabla }_{\alpha ,y}^{E_p}=R_p-\frac{\alpha }{\sqrt{p}}{a}_{p,1}\left({\tilde{d}}_{p,y}\right).$$

Thus, we get

$$\Re D_{p;\alpha ,y}=\Re D_p-\frac{{\alpha }^2}{p}{|\nabla {\tilde{d}}_{p,y}|}^2-\frac{\alpha }{\sqrt{p}}\Re a_{p,1}\left({\tilde{d}}_{p,y}\right).$$

Along with (8), this immediately implies that, for all $p\in \mathbb{N}$, $\alpha \in \mathbb{R}$ and $y\in X$,

$$\Re {〈D_{p;\alpha ,y}u,u〉}_{p,0}\ge \left({\lambda }_0-{\lambda }_1\frac{\alpha }{\sqrt{p}}-{\lambda }_2\frac{{\alpha }^2}{p}\right){\parallel u\parallel }_{p,0}^2,u\in H^2(X,E_p),$$

where

$${\lambda }_1=\underset{p\in \mathbb{N},x,y\in X}{sup}|\Re a_{p,1}\left({\tilde{d}}_{p,y}\right)\left(x\right)|,{\lambda }_2=\underset{p\in \mathbb{N},x,y\in X}{sup}{|\nabla {\tilde{d}}_{p,y}\left(x\right)|}^2.$$

It remains to use the inequality $\left|{\lambda }_1\frac{\alpha }{\sqrt{p}}\right|\le \epsilon +{\lambda }_1\frac{{\alpha }^2}{4\epsilon p}$ to complete the proof of (38) with $A_{\epsilon }={\lambda }_2+\frac{{\lambda }_1}{4\epsilon }$. □

7. Proof of the Main Theorem

Now we complete the proof of Theorem 1.

As above, by (38), for any $p\in \mathbb{N}$, $\alpha \in \mathbb{R}$, $y\in X$ and $\epsilon >0$, the operator $D_{p;\alpha ,y}-{\lambda }_0+\epsilon +A_{\epsilon }\frac{{\alpha }^2}{p}$ is a maximal accretive operator in $L^2(X,E_p)$. By [15] (Theorem 1.24 of Chapter 9), the operator $D_{p;\alpha ,y}$ generates a holomorphic semigroup $exp(-z{D}_{p;\alpha ,y})$ of angle $\pi /2$ and

$$\parallel exp(-z{D}_{p;\alpha ,y}){\parallel }_p^{0,0}\le exp\left(\left(-{\lambda }_0+\epsilon +A_{\epsilon }\frac{{\alpha }^2}{p}\right)\Re z\right),\Re z>0.$$

(39)

Moreover, for any $m\in {\mathbb{Z}}_{+}$, there exists $C>0$ such that for any $t>0$, $p\in \mathbb{N}$, $\alpha \in \mathbb{R}$, $y\in X$, $\epsilon >0$ and $u\in L^2(X,E_p)$, the section $exp(-t{D}_{p;\alpha ,y})u$ belongs to $H^{2m}(X,E_p)$ and satisfies

$$\parallel D_{p;\alpha ,y}^{m}exp(-t{D}_{p;\alpha ,y}){u\parallel }_{p,0}\le \frac{C}{t^m}exp\left(\left(-{\lambda }_0+\epsilon +A_{\epsilon }\frac{{\alpha }^2}{p}\right)t\right){\parallel u\parallel }_{p,0}.$$

It is important to note that the constant C is independent of p, $\alpha$, y and t. This follows from the fact that $exp(-z{D}_{p;\alpha ,y})$ is a contractive semigroup (cf. (39)), Hille–Yosida theorem and Cauchy integral formula.

By (37), it follows that

$${∥exp\left(-t{D}_{p;\alpha ,y}\right)∥}_p^{0,2m}\le C_1{\left(\frac{1}{t}+\frac{{\alpha }^2}{p}\right)}^{m}exp\left(\left(-{\lambda }_0+\epsilon +A\frac{{\alpha }^2}{p}\right)t\right).$$

(40)

Similar estimate holds for $D_{p;\alpha ,y}^{\ast }$. Therefore, by duality we have

$${∥exp\left(-t{D}_{p;\alpha ,y}\right)∥}_p^{-2m^{\prime }}\le C_2{\left(\frac{1}{t}+\frac{{\alpha }^2}{p}\right)}^{m^{\prime }}exp\left(\left(-{\lambda }_0+\epsilon +A\frac{{\alpha }^2}{p}\right)t\right).$$

(41)

and

$${∥exp\left(-t{D}_{p;\alpha ,y}\right)∥}_p^{-2m^{\prime },2m}\le C_3{\left(\frac{1}{t}+\frac{{\alpha }^2}{p}\right)}^{m+m^{\prime }}exp\left(\left(-{\lambda }_0+\epsilon +A\frac{{\alpha }^2}{p}\right)t\right).$$

Using the fact that, for any $\lambda >0$, the function $f\left(z\right)={(1+z)}^m{e}^{-\lambda z}$ is bounded on $(0,\infty )$, we conclude that

$${∥exp\left(-t{D}_{p;\alpha ,y}\right)∥}_p^{-2m^{\prime },2m}\le \frac{C_4}{t^{m+m^{\prime }}}exp\left(\left(-{\lambda }_0+2\epsilon +A\frac{{\alpha }^2}{p}\right)t\right).$$

Thus, for any $m\in {\mathbb{Z}}_{+}$, $m^{\prime }\in {\mathbb{Z}}_{+}$, $t>0$, $p\in \mathbb{N}$, $\alpha \in \mathbb{R}$, $y\in X$, $\epsilon >0$ and $f\in C^{-\infty }(X,E_p)$ such that $e^{\alpha {\tilde{d}}_{p,y}}f\in H^{-2m^{\prime }}(X,E_p)$, we have

$$\begin{array}{c}\parallel e^{\alpha {\tilde{d}}_{p,y}}exp(-t{D}_p){f\parallel }_{p,2m}\hfill \\ \hfill \le \frac{C}{t^{m+m^{\prime }}}exp\left(\left(-{\lambda }_0+2\epsilon +A\frac{{\alpha }^2}{p}\right)t\right){\parallel e^{\alpha {\tilde{d}}_{p,y}}f\parallel }_{p,-2m^{\prime }},t>0,\end{array}$$

(42)

where $C>0$ depends only on m and $m^{\prime }$.

By (27), (13), (42), Proposition 3 and (28), for any $x,y\in X$, we have

$$\begin{array}{cc}|e^{\alpha {\tilde{d}}_{p,y}\left(x\right)}exp(-t{D}_p)(x,y)|\hfill & \\ & \le C_1{p}^{d/4}\underset{v\in {\left(E_p\right)}_y,\parallel v\parallel =1}{sup}{\parallel e^{\alpha {\tilde{d}}_{p,y}}exp(-t{D}_p){\delta }_v\parallel }_{p,m}\hfill \\ & \le \frac{C_2}{t^{(m+m^{\prime })/2}}{p}^{d/4}exp\left(\left(-{\lambda }_0+2\epsilon +A\frac{{\alpha }^2}{p}\right)t\right)\underset{v\in {\left(E_p\right)}_y,\parallel v\parallel =1}{sup}{\parallel e^{\alpha {\tilde{d}}_{p,y}}{\delta }_v\parallel }_{p,-2m^{\prime }}\hfill \\ & \le \frac{C_3}{t^{(m+m^{\prime })/2}}{p}^{d/2}exp\left(\left(-{\lambda }_0+2\epsilon +A\frac{{\alpha }^2}{p}\right)t+\alpha \frac{\gamma }{\sqrt{p}}\right)\hfill \end{array}$$

with $m=m^{\prime }=[d/2]+1$. (Here we use the facts that $e^{\alpha {\tilde{d}}_{p,y}}{\delta }_v=e^{\alpha {\tilde{d}}_{p,y}\left(y\right)}{\delta }_v$ and $|{\tilde{d}}_{p,y}\left(y\right)|\le \frac{\gamma }{\sqrt{p}}$.)

It follows that there exists $C>0$ such that for any $p\in \mathbb{N}$, $x,y\in X$, $t>0$ and $\alpha >0$, we have

$$|exp(-t{D}_p)(x,y)|\le \frac{C}{t^{[d/2]+1}}{p}^{d/2}exp\left(\left(-{\lambda }_0+2\epsilon +A\frac{{\alpha }^2}{p}\right)t+\alpha \frac{\gamma }{\sqrt{p}}-\alpha {\tilde{d}}_{p,y}\left(x\right)\right).$$

Using the fact that the Schwartz kernel of the operator $D_p^{N}exp(-t{D}_p)D_p^M$ is equal to ${\left(D_p^N\right)}_x{\left({\left(D_p^{\ast }\right)}^M\right)}_{y}exp(-t{D}_p)(x,y)$ and applying the same argument to this operator, one can prove a similar estimate for any derivative of the heat kernel. Namely, for any $k\in {\mathbb{Z}}_{+}$, there exists $C_k>0$ such that for any $p\in \mathbb{N}$, $x,y\in X$, $t>0$ and $\alpha >0$, we have

$$\begin{array}{c}|exp(-t{D}_p)(x,y){|}_{C^k}\hfill \\ \hfill \le \frac{C_k}{t^{[d/2]+1+k/2}}{p}^{(d+k)/2}exp\left(\left(-{\lambda }_0+2\epsilon +A\frac{{\alpha }^2}{p}\right)t+\alpha \frac{\gamma }{\sqrt{p}}-\alpha {\tilde{d}}_{p,y}\left(x\right)\right).\end{array}$$

(43)

Now assume that $d(x,y)>\frac{{\epsilon }_0}{\sqrt{p}}$ with some ${\epsilon }_0>0$. Let us take the functions ${\tilde{d}}_{p,y}$ with $\gamma =\frac{{\epsilon }_0}{4}$. Since

$${\tilde{d}}_{p,y}\left(x\right)-\frac{\gamma }{\sqrt{p}}\ge d(x,y)-|{\tilde{d}}_{p,y}\left(x\right)-d(x,y)|-\frac{{\epsilon }_0}{4\sqrt{p}}\ge d(x,y)-\frac{{\epsilon }_0}{2\sqrt{p}}>\frac{1}{2}d(x,y),$$

it follows from (43) that

$$|exp(-t{D}_p)(x,y){|}_{C^k}\le \frac{C_k}{t^{[d/2]+1+k/2}}{p}^{(d+k)/2}exp\left(\left(-{\lambda }_0+2\epsilon +A\frac{{\alpha }^2}{p}\right)t-\frac{1}{2}\alpha d(x,y)\right).$$

Recall that this estimate holds for any $p\in \mathbb{N}$, $x,y\in X$ with $d(x,y)>\frac{{\epsilon }_0}{\sqrt{p}}$, $t>0$ and $\alpha >0$. Put

$$\alpha =\frac{pd(x,y)}{4At}.$$

For any $p\in \mathbb{N}$, $x,y\in X$ with $d(x,y)>\frac{{\epsilon }_0}{\sqrt{p}}$ and $t>0$, we get

$${\left|exp\left(-t{D}_p\right)(x,y)\right|}_{C^k}\le \frac{C_k}{t^{[d/2]+1+k/2}}{p}^{(d+k)/2}exp\left((-{\lambda }_0+2\epsilon )t-\frac{pd{(x,y)}^2}{8At}\right).$$

(44)

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 $D_p$ is the square of the spin${}^c$ 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 $1/t^{[d/2]+1+k/2}$ 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.