Donaldson's $Q$-operators for symplectic manifolds

TL;DR

This paper establishes an estimate for Donaldson's $Q$-operator on compact symplectic manifolds, contributing to a symplectic analogue of scalar curvature bounds relevant in geometric analysis.

Contribution

It provides a key estimate for Donaldson's $Q$-operator, advancing the understanding of scalar curvature bounds in symplectic geometry.

Findings

Derived a new estimate for Donaldson's $Q$-operator

Supported a symplectic generalization of scalar curvature bounds

Contributed to the proof of a lower bound for Hermitian scalar curvature

Abstract

We prove an estimate for Donaldson's $Q$-operator on a prequantized compact symplectic manifold. This estimate is an ingredient in the recent result of Keller and Lejmi about a symplectic generalization of Donaldson's lower bound for the $L^2$-norm of the Hermitian scalar curvature.

Equations

$$\frac{\sqrt{-1}}{2\pi }{R}^L=\omega .$$

$$\big{\langle}s_{1},s_{2}\big{\rangle}=\int_{X}\big{\langle}s_{1}(x),s_{2}(x)\big{\rangle}_{\!h^{L^{p}\otimes E}}\,dv_{X}(x).$$

$$\Delta^{L^{p}\otimes E}=-\sum_{k}\Big{[}\big{(}\nabla_{e_{k}}^{L^{p}\otimes E}\big{)}^{2}-\nabla_{\nabla_{e_{k}}^{TX}e_{k}}^{L^{p}\otimes E}\Big{]}.$$

$${\Delta }_{p,\Phi }={\Delta }^{L^p\otimes E}-2\pi np+\Phi .$$

$$\text{Spec}({\Delta }_{p,\Phi })\subset [-C_L,C_L]\cup [4\pi p-C_L,+\infty ),$$

$${\mathcal{H}}_p:=\underset{\lambda \in [-C_L,C_L]}{⨁}\mathrm{Ker}({\Delta }_{p,\Phi }-\lambda )\subset {\mathcal{C}}^{\infty }(X,L^p\otimes E)$$

$$\text{dim}{\mathcal{H}}_p={\int }_X\text{Td}(T^{(1,0)}X)\text{ch}(L^p\otimes E),$$

$$\displaystyle K_{p}(x,x^{\prime})=\big{|}P_{\mathcal{H}_{p}}(x,x^{\prime})\big{|}^{2},\ \ R_{p}:=(\dim\mathcal{H}_{p})/\textup{Vol}(X,dv_{X}).$$

$$(K_{p}f)(x)={\int }_X{K}_p(x,y)f(y)d{v}_X(y),Q_{K_p}=\frac{1}{R_p}{K}_{p}f.$$

$$\displaystyle\big{\|}Q_{K_{p}}(f)-f\big{\|}_{\mathscr{C}^{m}(X)}\leqslant\frac{C}{p}\big{\|}f\big{\|}_{\mathscr{C}^{m+2}(X)}.$$

$$R^L(W,\overline{Y})=⟨{\dot{R}}^{L}W,\overline{Y}⟩.$$

$$\tau =\mathrm{Tr}{|}_{T^{(1,0)}X}{\dot{R}}^L,{\mu }_0=\underset{u\in T_x^{(1,0)}X,x\in X}{\inf }{R}_x^L(u,\overline{u})/|u{|}_{g^{TX}}^2>0.$$

$${\Delta }_{p,\Phi }={\Delta }^{L^p\otimes E}-\tau p+\Phi .$$

$$\text{Spec}({\Delta }_{p,\Phi })\subset [-C_L,C_L]\cup [2{\mu }_{0}p-C_L,+\infty ),$$

$$\displaystyle\Big{|}P_{\mathcal{H}_{p}}(x,x^{\prime})\Big{|}_{\mathscr{C}^{m}(X\times X)}\leqslant Cp^{-l},\ \ \textup{if}\ d(x,x^{\prime})>\varepsilon p^{-\frac{\theta}{2}}.$$

$$(Z_1,\dots ,Z_{2n})\in {\mathbb{R}}^{2n}⟼\sum _i{Z}_i{e}_i\in T_{x_0}X,$$

$$\rho (v)=1\text{if}|v|<2;\rho (v)=0\text{if}|v|>4.$$

$${\nabla }^{L_0}{|}_Z={\phi }_{\epsilon }^{\ast }{\nabla }^L+\frac{1}{2}(1-{\rho }^2(|Z|/\epsilon ))R_{x_0}^L(\mathcal{R},\cdot ).$$

$${\tau }_0=\frac{\sqrt{-1}}{2}\sum _i{R}^{L_0}(e_i,J_0{e}_i).$$

$$\text{Spec}({\Delta }_{p,{\Phi }_0}^{X_0})\subset [-C_{L_0},C_{L_0}]\cup [{\mu }_{0}p-C_{L_0},+\infty ).$$

$$\displaystyle\Big{|}\big{(}P_{0,\mathcal{H}_{p}}-P_{\mathcal{H}_{p}}\big{)}(x,x^{\prime})\Big{|}_{\mathscr{C}^{m}(X\times X)}\leqslant C_{l,m}p^{-l},$$

$$d{v}_{X_0}(Z)=\kappa (Z)d{v}_{TX}(Z),$$

$$\begin{array}{rl}(S_{t}s)(Z)=& s(Z/t),{\nabla }_t=t{S}_t^{-1}{\kappa }^{\frac{1}{2}}{\nabla }^{L_0}{\kappa }^{-\frac{1}{2}}{S}_t,\\ {\mathcal{L}}_t=& S_t^{-1}{\kappa }^{\frac{1}{2}}{t}^2{\Delta }_{p,{\Phi }_0}^{X_0}{\kappa }^{-\frac{1}{2}}{S}_t.\end{array}$$

$$\displaystyle\textup{Spec}(\mathscr{L}_{t})\subset\big{[}\!-C_{L_{0}}t^{2},C_{L_{0}}t^{2}\,\big{]}\cup[\frac{1}{2}\mu_{0},+\infty).$$

$$\Vert s{\Vert }_{t,0}^2=$$

$$\Vert s{\Vert }_{t,m}^2=$$

$$\displaystyle\sup_{|\alpha|+|\alpha^{\prime}|\leq m^{\prime}}\bigg{|}\frac{\partial^{|\alpha|+|\alpha^{\prime}|}}{\partial Z^{\alpha}\partial Z^{\prime\alpha^{\prime}}}\frac{\partial^{r}}{\partial t^{r}}\mathcal{P}_{0,t}(Z,Z^{\prime})\bigg{|}_{\mathscr{C}^{m}(X)}\leqslant C(1+|Z|+|Z^{\prime}|)^{M_{r,m^{\prime},m}}$$

$$M_{r,m^{\prime },m}=2n+2+2r+m^{\prime }+2m.$$

$${\mathcal{P}}_{0,t}=\frac{1}{2\pi \sqrt{-1}}{\int }_{\delta }{\lambda }^{k-1}(\lambda -{\mathcal{L}}_t{)}^{-k}d\lambda .$$

$$\displaystyle\big{\|}Q\mathcal{P}_{0,t}Q^{\prime}\big{\|}^{0,0}_{t}\leqslant C_{m},\ \ \textup{for}\ \ Q,Q^{\prime}\in\mathcal{Q}^{m}.$$

$$\frac{1}{C}(1+q{)}^{-m}\Vert s{\Vert }_{t,m}⩽\Vert s{\Vert }_m⩽C(1+q{)}^m\Vert s{\Vert }_{t,m}.$$

Full text

W. L. partially supported by NNSFC 11401232

X. M. partially supported by NNSFC 11528103 and funded through the Institutional Strategy of the University of Cologne within the German Excellence Initiative

G. M. partially supported by DFG funded project SFB TRR 191

1. Introduction

The $Q$-operator is an integral operator whose kernel is the square norm of the Bergman kernel of a positive line bundle (see (1.8), (1.9)). It was introduced by Donaldson [5] in order to find explicit numerical approximations of Kähler-Einstein metrics on projective manifolds, and have attracted much attention recently, see e.g., [1, 6, 8, 9, 10, 16].

Using the full asymptotic expansion of the Bergman kernel [2], Liu and Ma [10, Theorem 0.1] verified a statement of Donaldson [5, Section 4.2] about the relation of the asymptotics of $Q_{K_{p}}$ to the heat kernel. Such statement was needed for the convergence of the approximation procedure in [5]. In [6], Liu and Ma improved the statement to a $\mathscr{C}^{m}$-estimate for $Q_{K_{p}}$ on Kähler manifolds, as a crucial step towards the result of [6] about the convergence of the balancing flow to the Calabi flow. This is a parabolic analogue of Donaldson’s theorem relating balanced embeddings to metrics with constant scalar curvature [3]. Besides, such results also turn out to be important in Cao and Keller’s work [1] on Calabi’s problem.

The purpose of the note is to extend the $\mathscr{C}^{m}$-estimates of the operators $Q_{K_{p}}$ to the case of symplectic manifolds. This result, together with [11], plays an important role in the recent work of Keller and Lejmi [8] about a lower bound for the $L^{2}$-norm of the Hermitian scalar curvature. Such a lower bound was obtained in the Kähler case by Donaldson [4]. Our proof is based on the asymptotic expansion of the (generalized) Bergman kernel, which in our case is the kernel of the spectral projection on lower lying eigenstates of the normalized Bochner Laplacian. We refer the readers to the monograph [14] (see also [15], [12]) for more information on the Bergman kernel on symplectic manifolds.

Let us describe our result in detail. Let $(X,\omega)$ be a compact symplectic manifold of real dimension $2n$. Let $(L,h^{L})$ be a Hermitian line bundle on $X$, and let $\nabla^{L}$ be a Hermitian connection on $(L,h^{L})$ with curvature $R^{L}=(\nabla^{L})^{2}$. Let $(E,h^{E})$ be an auxiliary Hermitian vector bundle with Hermitian connection $\nabla^{E}$. We will assume throughout the paper that $(L,h^{L})$ satisfies the pre-quantization condition

[TABLE]

We choose an almost complex structure $J$ on $TX$ (i.e., $J\in\textup{End}(TX)$ and $J^{2}=-1$) such that $\omega$ is $J$-invariant and $\omega(\cdot,J\cdot)>0$. The almost complex structure $J$ induces a splitting $TX\otimes_{\mathbb{R}}\mathbb{C}=T^{(1,0)}X\oplus T^{(0,1)}X$, where $T^{(1,0)}X$ and $T^{(0,1)}X$ are the eigenbundles of $J$ corresponding to the eigenvalues $\sqrt{-1}$ and $-\sqrt{-1}$, respectively.

Let $g^{TX}(\cdot,\cdot):=\omega(\cdot,J\cdot)$ be the Riemannian metric on $TX$ induced by $\omega$ and $J$. The Riemannian volume form $dv_{X}$ of $(X,g^{TX})$ has the form $dv_{X}=\omega^{n}/n!$. We denote by $L^{p}:=L^{\otimes p}$ the tensor powers of $L$ for $p\in\mathbb{N}$ and by $h^{L^{p}}:=(h^{L})^{\otimes p}$, $h^{L^{p}\otimes E}=h^{L^{p}}\otimes h^{E}$, the induced Hermitian metrics on $L^{p}$ and $L^{p}\otimes E$, respectively. The $L^{2}$-Hermitian product on the space $\mathscr{C}^{\infty}(X,L^{p}\otimes E)$ of smooth sections of $L^{p}\otimes E$ on $X$ is given by

[TABLE]

Let $\nabla^{TX}$ be the Levi-Civita connection on $(X,g^{TX})$, and let $\nabla^{L^{p}\otimes E}$ be the connection on $L^{p}\otimes E$ induced by $\nabla^{L}$ and $\nabla^{E}$. Let $\{e_{k}\}$ be a local orthonormal frame of $(TX,g^{TX})$. The Bochner Laplacian acting on $\mathscr{C}^{\infty}(X,L^{p}\otimes E)$ is given by

[TABLE]

Let $\Phi\in\mathscr{C}^{\infty}(X,\textup{End}(E))$ be Hermitian (i.e., self-adjoint with respect to $h^{E}$). The renormalized Bochner Laplacian is defined by

[TABLE]

By [7], [13, Corollary 1.2] there exists $C_{L}>0$ independent of $p$ such that

[TABLE]

where $\textup{Spec}(A)$ denotes the spectrum of the operator $A$. Since $\Delta_{p,\Phi}$ is an elliptic operator on a compact manifold, it has discrete spectrum and its eigensections are smooth. Let

[TABLE]

be the direct sum of eigenspaces of $\Delta_{p,\Phi}$ corresponding to the eigenvalues lying in $[-C_{L},C_{L}]$. In mathematical physics terms, the operator $\Delta_{p,\Phi}$ is a semiclassical Schrödinger operator and the space $\mathcal{H}_{p}$ is the space of its bound states as $p\to\infty$. By [14, Theorem 8.3.1],

[TABLE]

where $\textup{Td}(\cdot),\textup{ch}(\cdot)$ denote the Todd class and the Chern character of the corresponding complex vector bundle. The formula (1.7) agrees with the Riemann-Roch-Hirzebruch theorem and Kodaira vanishing theorem in the Kähler case. The space $\mathcal{H}_{p}$ proves to be an appropriate replacement for the space of holomorphic sections $H^{0}(X,L^{p}\otimes E)$ from the Kähler case.

Let $P_{\mathcal{H}_{p}}$ be the orthogonal projection from $\mathscr{C}^{\infty}(X,L^{p}\otimes E)$ onto $\mathcal{H}_{p}$. The kernel $P_{\mathcal{H}_{p}}(x,x^{\prime})$ of $P_{\mathcal{H}_{p}}$ with respect to $dv_{X}(x^{\prime})$ is called a generalized Bergman kernel [15]. Note that $P_{\mathcal{H}_{p}}(x,x^{\prime})\in(L^{p}\otimes E)_{x}\otimes(L^{p}\otimes E)_{x^{\prime}}^{\ast}$. Set $\textup{Vol}(X,dv_{X})=\int_{X}dv_{X}$. Following Donaldson [5, § 4], we set

[TABLE]

Let $K_{p},Q_{K_{p}}$ be the integral operators associated to $K_{p}$ which is defined by for $f\in\mathscr{C}^{\infty}(X)$,

[TABLE]

The operator $Q_{K_{p}}$ has been studied by Donaldson [5], Liu-Ma [6, Appendix], [10], and Ma-Marinescu [16, § 6] in the case of Kähler manifolds.

The main result of the note is as follows. For Kähler manifolds it was obtained by Liu-Ma [6, Appendix], [10].

Theorem 1.1**.**

For any integer $m\geqslant 0$, there exists a constant $C>0$ such that for any $f\in\mathscr{C}^{\infty}(X)$,

[TABLE]

Moreover, (1.10) is uniform in the following sense. Consider $Q_{K_{p}}$ as a function of the parameters $g^{TX},h^{L},\nabla^{L},h^{E},\nabla^{E}$ and $\Phi$, that is, $Q_{K_{p}}=Q_{K_{p}}(g^{TX},h^{L},\nabla^{L},h^{E},\nabla^{E},\Phi)$. Let $\mathcal{M}$ be a subset of the infinite dimensional manifold $\mathscr{D}$ of all compatible tuples $g^{TX},h^{L},\nabla^{L},h^{E},\nabla^{E}$ and $\Phi$. Assume that :

• (i)

*the covariant derivatives in the direction $X$ of order $\ell\leqslant 2n+m+6$ of elements of $\mathcal{M}$ form a set of tensors on $X\times\mathcal{M}$ which is bounded in the $\mathscr{C}^{0}$- norm calculated in the direction of $\mathcal{M}$ ; *

• (ii)

the projection of $\mathcal{M}$ on the space of Riemannian metrics is bounded below in the $\mathscr{C}^{0}$- norm.

Then there exists $C=C_{m}(\mathcal{M})$ such that (1.10) holds for all tuples of parameters from $\mathcal{M}$. Moreover, the $\mathscr{C}^{m}$-norm in (1.10) can be taken on $X\times\mathcal{M}$.

The organization of this paper is as follows. In Section 2, we establish the asymptotic expansion of the generalized Bergman kernel which extends [14, §8.3]. In Section 3, we prove Theorem 1.1.

2. Asymptotic expansion of the generalized Bergman kernel

In this section, we assume that $g^{TX}$ is an arbitrary $J$-invariant Riemannian metric on $X$. Let $\Delta^{L^{p}\otimes E}$ be the Bochner Laplacian acting on $\mathscr{C}^{\infty}(X,L^{p}\otimes E)$ associated with $g^{TX}$ and $\nabla^{L^{p}\otimes E}$. Let $\Phi\in\mathscr{C}^{\infty}(X,\textup{End}(E)$ be Hermitian.

Let $dv_{X}$ be the Riemannian volume form on $(X,g^{TX})$. Now the Hermitian product on $\mathscr{C}^{\infty}(X,L^{p}\otimes E)$ is induced by $h^{L},h^{E}$ and $dv_{X}$.

We identify the two form $R^{L}$ with the Hermitian matrix $\dot{R}^{L}\in\operatorname{End}(T^{(1,0)}X)$ such that for $W,Y\in T^{(1,0)}X$,

[TABLE]

Set

[TABLE]

Note that if $g^{TX}=\omega(\cdot,J\cdot)$ then $\tau=2\pi n$ and $\mu_{0}=2\pi$.

Then the renormalized Bochner Laplacian is defined as

[TABLE]

By the same references as in Introduction, there exists $C_{L}>0$ independent of $p$ such that

[TABLE]

Thus $\mathcal{H}_{p}$ in (1.6) is still well-defined and (1.7) holds.

Let $P_{\mathcal{H}_{p}}(x,x^{\prime})$ be the smooth kernel of the orthogonal projection $P_{\mathcal{H}_{p}}$ from $\mathscr{C}^{\infty}(X,L^{p}\otimes E)$ onto $\mathcal{H}_{p}$ with respect to $dv_{X}(x^{\prime})$. In this section, we study the asymptotics of $P_{\mathcal{H}_{p}}(x,x^{\prime})$ as $p\rightarrow\infty$.

Let $a^{X}$ be the injectivity radius of $(X,g^{TX})$. We fix $\varepsilon\in(0,a^{X}/4)$. Let $d(x,y)$ denote the Riemannian distance from $x$ to $y$ on $(X,g^{TX})$. By [14, Prop. 8.3.5] and the argument after [14, Prop. 8.3.5], we get for any $l,m\in\mathbb{N}$ and $0<\theta<1$, there exists $C>0$ such that

[TABLE]

Now we still need to understand the asymptotics of $P_{\mathcal{H}_{p}}(x,x^{\prime})$ for $d(x,x^{\prime})>\varepsilon p^{-\frac{\theta}{2}}$.

We recall first the procedure of [15, § 1.2] and [14, § 8.3].

Denote by $B^{X}(x,\varepsilon)$ and $B^{T_{x}X}(0,\varepsilon)$ the open balls in $X$ and $T_{x}X$ with center $x$ and radius $\varepsilon$, respectively. We identify $B^{T_{x}X}(0,\varepsilon)$ with $B^{X}(x,\varepsilon)$ by using the exponential map of $(X,g^{TX})$.

We fix $x_{0}\in X$. For $Z\in B^{T_{x_{0}}X}(0,\varepsilon)$, we identify $L_{Z},E_{Z}$ and $(L^{p}\otimes E)_{Z}$ to $L_{x_{0}},E_{x_{0}}$ and $(L^{p}\otimes E)_{x_{0}}$ by parallel transport with respect to the connections $\nabla^{L},\nabla^{E}$ and $\nabla^{L^{p}\otimes E}$ along the curve $\gamma_{Z}:[0,1]\ni u\rightarrow\textup{exp}^{X}_{x_{0}}(uZ)$. Then under our identification, $P_{\mathcal{H}_{p}}(Z,Z^{\prime})$ is a function on $Z,Z^{\prime}\in T_{x_{0}}X$, $|Z|,|Z^{\prime}|<\varepsilon$. We denote it by $P_{\mathcal{H}_{p},x_{0}}(Z,Z^{\prime})$. Let $\pi:TX\times_{X}TX\rightarrow X$ be the natural projection from the fiberwise product of $TX$ on $X$. Then we can view $P_{{\mathcal{H}_{p}},x_{0}}(Z,Z^{\prime})$ as a smooth function over $TX\times_{X}TX$ by identifying a section $s\in\mathscr{C}^{\infty}(TX\times_{X}TX,\pi^{\ast}(\textup{End}(E)))$ with the family $(s_{x})_{x\in X}$, where $s_{x}=s|_{\pi^{-1}(x)}$.

Let $\{e_{i}\}_{i}$ be an oriented orthonormal basis of $T_{x_{0}}X$, and let $\{e^{i}\}_{i}$ be its dual basis. For $\varepsilon>0$ small enough, we will extend the geometric objects from $B^{T_{x_{0}}X}(0,\varepsilon)$ to $\mathbb{R}^{2n}\simeq T_{x_{0}}X$ where the identification is given by

[TABLE]

such that $\Delta_{p,\Phi}$ is the restriction of a renormalized Bochner-Laplacian on $\mathbb{R}^{2n}$ associated with a Hermitian line bundle with positive curvature. In this way, we replace $X$ by $\mathbb{R}^{2n}$.

At first, we denote by $L_{0},E_{0}$ the trivial bundles with fiber $L_{x_{0}},E_{x_{0}}$ on $X_{0}=\mathbb{R}^{2n}$. We still denote by $\nabla^{L},\nabla^{E},h^{L}$ etc the connections and metrics on $L_{0},E_{0}$ on $B^{T_{x_{0}}X}(0,4\varepsilon)$ induced by the above identification. Then $h^{L},h^{E}$ is identified to the constant metrics $h^{L_{0}}=h^{L_{x_{0}}},h^{E_{0}}=h^{E_{x_{0}}}$.

Let $\rho:\mathbb{R}\rightarrow[0,1]$ be a smooth even function such that

[TABLE]

Let $\varphi_{\varepsilon}:\mathbb{R}^{2n}\rightarrow\mathbb{R}^{2n}$ is the map defined by $\varphi_{\varepsilon}(Z)=\rho(|Z|/\varepsilon)Z$. Then $\Phi_{0}=\Phi\circ\varphi_{\varepsilon}$ is a smooth function on $X_{0}$. Let $g^{TX_{0}}(Z)=g^{TX}(\varphi_{\varepsilon}(Z))$ be the metric on $X_{0}$. Set $\nabla^{E_{0}}=\varphi_{\varepsilon}^{\ast}\nabla^{E}$. Then $\nabla^{E_{0}}$ is the extension of $\nabla^{E}$ on $B^{T_{x_{0}}X}(0,\varepsilon)$. Denote by $\mathcal{R}=\sum_{i}Z_{i}e_{i}=Z$ the radial vector field on $\mathbb{R}^{2n}$. We define the Hermitian connection $\nabla^{L_{0}}$ on $(L^{0},h^{L_{0}})$ by

[TABLE]

Let $R^{L_{0}}$ denote the curvature of $\nabla^{L_{0}}$ and $\{e_{i}\}_{i}$ be an orthonormal frame of $(TX_{0},g^{TX_{0}})$. Let $J_{0}$ be an almost complex structure on $X_{0}$ compatible with $g^{TX_{0}}$ and $\frac{\sqrt{-1}}{2\pi}R^{L_{0}}$ such that $J_{0}=J$ on $B^{T_{x_{0}}X}(0,2\varepsilon)$ and $J_{0}=J_{x_{0}}$ outside $B^{T_{x_{0}}X}(0,4\varepsilon)$. Set (cf. (2.2))

[TABLE]

Let $\Delta^{X_{0}}_{p,\Phi_{0}}=\Delta^{L_{0}^{p}\otimes E_{0}}-p\tau_{0}+\Phi_{0}$ be the renormalized Bochner-Laplacian on $X_{0}$ associated to the above data as in $(\ref{0.4a})$. By [15, (1.23)] there exists $C_{L_{0}}>0$ such that

[TABLE]

Let $S_{L}$ be an unit vector of $L_{0}$. Using $S_{L}$ and the above discussion, we get an isometry $L^{p}_{0}\simeq\mathbb{C}$. Let $P_{0,\mathcal{H}_{p}}$ be the spectral projection of $\Delta^{X_{0}}_{p,\Phi_{0}}$ from $\mathscr{C}^{\infty}(X_{0},L^{p}_{0}\otimes E_{0})\simeq\mathscr{C}^{\infty}(X_{0},E_{0})$ corresponding to the interval $[-C_{L_{0}},C_{L_{0}}]$, and let $P_{0,\mathcal{H}_{p}}(x,x^{\prime})$ be the smooth kernel of $P_{0,\mathcal{H}_{p}}$ with respect to the volume form $dv_{X_{0}}(x^{\prime})$. By [15, Proposition 1.3] (for $q=0$ therein), for any $l,m\in\mathbb{N}$, there exists $C_{l,m}>0$ such that for $x,x^{\prime}\in B^{T_{x_{0}}X}(0,\varepsilon)$, we have

[TABLE]

here the $\mathscr{C}^{m}$-norm is induced by $\nabla^{TX},\nabla^{L},\nabla^{E},h^{L},h^{E}$ and $g^{TX}$.

Let $dv_{TX}$ be the Riemannian volume form on $(T_{x_{0}}X,g^{T_{x_{0}}X})$. Let $\kappa(Z)$ be the smooth positive function defined by the equation

[TABLE]

with $\kappa(0)=1$. Denote by $\nabla_{U}$ the ordinary differentiation operation on $T_{x_{0}}X$ in the direction $U$. Denote by $t=\frac{1}{\sqrt{p}}$. For $s\in\mathscr{C}^{\infty}(\mathbb{R}^{2n},E_{0})$ and $Z\in\mathbb{R}^{2n}$, set

[TABLE]

It follows from (2.10) and (2.13) that for $t$ small enough (cf. [15, (1.43)]),

[TABLE]

Let $\delta$ be the counterclockwise oriented circle in $\mathbb{C}$ of center [math] radius $\frac{1}{4}\mu_{0}$. By (2.14), there exists $t_{0}>0$ such that the resolvent $(\lambda-\mathscr{L}_{t})^{-1}$ exists for $\lambda\in\delta$ and $t\in(0,t_{0}]$.

We denote by $\langle\cdot,\cdot\rangle_{0,L^{2}}$ and $\|\cdot\|_{0,L^{2}}$ the scalar product and the $L^{2}$-norm on $\mathscr{C}^{\infty}(X_{0},E_{0})$ induced by $g^{TX_{0}}$ as in (1.2). For $s\in C^{\infty}(X_{0},E_{0})$, set

[TABLE]

We denote by $\langle\cdot,\cdot\rangle$ the inner product on $C^{\infty}(X_{0},E_{0})$ corresponding to $\|\cdot\|_{t,0}$. Let $H^{m}_{t}$ be the Sobolev space of order $m$ with norm $\|\cdot\|_{t,m}$. Let $H^{-1}_{t}$ be the Sobolev space of order $-1$ and let $\|\cdot\|_{t,-1}$ be the norm on $H^{-1}_{t}$ defined by $\|s\|_{t,-1}=\sup_{0\neq s^{\prime}\in H^{1}_{t}}|\langle s,s^{\prime}\rangle_{t,0}|/\|s^{\prime}\|_{t,1}$. If $A\in\mathscr{L}(H^{m},H^{m^{\prime}})$, then we denote by $\|A\|^{m,m^{\prime}}_{t}$ the norm of $A$ with respect to the norms $\|\cdot\|_{t,m}$ and $\|\cdot\|_{t,m^{\prime}}$.

Let $\mathcal{P}_{0,t}$ the orthogonal projection from $(\mathscr{C}^{\infty}(X_{0},E_{0}),\|\cdot\|_{0})$ onto the space of the direct sum of eigenspaces of $\mathscr{L}_{t}$ corresponding to the eigenvalues lying in $[-C_{L_{0}}t^{2},C_{L_{0}}t^{2}]$. Let $\mathcal{P}_{0,t}(Z,Z^{\prime})=\mathcal{P}_{0,t,x_{0}}(Z,Z^{\prime})$ (with $Z,Z^{\prime}\in X_{0}$) be the smooth kernel of $\mathcal{P}_{0,t}$ with respect to $dv_{TX}(Z^{\prime})$. We denote by $\mathscr{C}^{m}(X)$ the $\mathscr{C}^{m}$-norm for the parameter $x_{0}\in X$. By [14, (4.2.9)], we have the following extension of [15, Theorem 1.10] (for $q=0$).

Claim. For any $r,m^{\prime},m\in\mathbb{N}$, there exists $C>0$ such that for $t\in(0,t_{0}]$ and $Z,Z^{\prime}\in T_{x_{0}}X$,

[TABLE]

with

[TABLE]

We will sketch the proof of the claim. The readers to referred to [2], [14, Chapter 4] and [15, § 1] for more details. In fact, by (2.14), for any $k\in\mathbb{N}^{\ast}$ (cf. [15, (1.55)]),

[TABLE]

For $m\in\mathbb{N}$, let $\mathcal{Q}^{m}$ be the set of operators $\{\nabla_{t,e_{i_{1}}}\cdots\nabla_{t,e_{i_{j}}}\}_{j\leqslant m}$. By [15, (1.58)],

[TABLE]

Let $\|\cdot\|_{m}$ be the usual Sobolev norm on $C^{\infty}(\mathbb{R}^{n},E_{0})$ induced by $h^{E_{0}}$ and the volume form $dv_{TX}(Z)$. By [14, (4.29)], there exists $C>0$ such that for $s\in C^{\infty}(X_{0},E_{0})$ with $\textup{supp}(s)\subset B^{T_{x_{0}}X}(0,q)$, $m\geq 0$,

[TABLE]

Now (2.19) and (2.20) together with Sobolev inequalities imply that for $Q,Q^{\prime}\in\mathcal{Q}^{m}$,

[TABLE]

Combining [15, (1.35)] and (2.21) yields (2.16) for $r=m^{\prime}=0$. To obtain (2.16) for $r\geq 1$ and $m^{\prime}=0$, note that by (2.18),

[TABLE]

For $k,r\in\mathbb{N}^{\ast}$, let

[TABLE]

Then there exist $a^{\bf k}_{\bf r}\in\mathbb{R}$ such that

[TABLE]

We can now carry on nearly word by word the corresponding part of the proof of [15, Theorem 1.10] to finish the proof of (2.16). We finish the proof of the claim.

Set (cf. [14, (4.1.65)])

[TABLE]

Let $\mathscr{F}_{r}(Z,Z^{\prime})$ ($Z,Z^{\prime}\in T_{x_{0}}X$) be the smooth kernel of $\mathscr{F}$ with respect to $dv_{TX}(Z^{\prime})$. Then $\mathscr{F}_{r}(Z,Z^{\prime})\in\mathscr{C}^{\infty}(TX\times_{X}TX,\pi^{\ast}\textup{End}(E))$. By the proof of the estimate (2.16), we observe that $\mathscr{F}_{r}$ verifies the similar inequalities as (2.16), i.e., to replace the factor $\frac{\partial^{r}}{\partial t^{r}}\mathcal{P}_{0,r}$ in (2.16) by $\mathscr{F}_{r}$. Using this observation, (2.16) and (2.25), we obtain the extension of [15, Theorem 1.12]: there exists $C>0$ such that for $t\in(0,t_{0}]$ and $Z,Z^{\prime}\in T_{x_{0}}X$,

[TABLE]

By (2.25) and (2.26), we have (cf. [15, (1.78)])

[TABLE]

By (2.16), (2.27) and the Taylor expansion

[TABLE]

we obtain the extension of [15, Theorem 1.13]: for any $k,m,m^{\prime}\in\mathbb{N}$, there exists $C>0$ such that for $t\in(0,t_{0}]$, $Z,Z^{\prime}\in T_{x_{0}}X$ and for $|\alpha|+|\alpha^{\prime}|\leqslant m^{\prime}$,

[TABLE]

By (2.12) and (2.13), for $Z,Z^{\prime}\in\mathbb{R}^{2n}$ (cf. [15, (1.112)]),

[TABLE]

Combining (2.11), (2.29) and (2.30), we obtain

[TABLE]

Now we fix $k_{0},m^{\prime},m$. Take

[TABLE]

Then for $|\alpha|+|\alpha^{\prime}|\leqslant m^{\prime}$ and $|Z|,|Z^{\prime}|<p^{-\frac{1}{2}+\theta}$, we have

[TABLE]

To sum up, we have finished the proof of the following result:

Theorem 2.1**.**

For any $k_{0},m^{\prime},m\in\mathbb{N}$, there exists $C>0$ such that for $|\alpha|+|\alpha^{\prime}|\leqslant m^{\prime}$ and $|Z|,|Z^{\prime}|<p^{-\frac{1}{2}+\theta}$ with

[TABLE]

we have

[TABLE]

where $k=k_{0}+m^{\prime}+2$.

We choose $\{w_{j}\}_{j=1}^{n}$ an orthonormal basis of $T_{x_{0}}^{(1,0)}X$ such that

[TABLE]

Then $e_{2j-1}=\tfrac{1}{\sqrt{2}}(w_{j}+\overline{w}_{j})$ and $e_{2j}=\tfrac{\sqrt{-1}}{\sqrt{2}}(w_{j}-\overline{w}_{j})\,,j=1,\dotsc,n$, form an orthonormal basis of $T_{x_{0}}X$. We use the coordinates on $T_{x_{0}}X\simeq\mathbb{R}^{2n}$ induced by $\{e_{i}\}$ as in (2.6) and in what follows we also introduce the complex coordinates $z=(z_{1},\cdots,z_{n})$ on $\mathbb{C}^{n}\simeq\mathbb{R}^{2n}$. Set

[TABLE]

By [15, Theorem 1.18], there exist $J_{r}(Z,Z^{\prime})$ polynomials in $Z,Z^{\prime}$ with the same parity as $r$ and degree $\leq 3r$ such that

[TABLE]

3. Proof of Theorem 1.1

Now $g^{TX}(\cdot,\cdot):=\omega(\cdot,J\cdot)$, thus $a_{j}=2\pi$ in (2.37).

Recall that the classical heat kernel on $\mathbb{C}^{n}$ is $e^{-u\Delta}(Z,Z^{\prime})=(4\pi u)^{-n}e^{-\frac{1}{4u}|Z-Z^{\prime}|^{2}}$. Then

[TABLE]

Note that \big{|}P_{\mathcal{H}_{p},x_{0}}(Z,Z^{\prime})\big{|}^{2}=P_{\mathcal{H}_{p},x_{0}}(Z,Z^{\prime})\overline{P_{\mathcal{H}_{p},x_{0}}(Z,Z^{\prime})}. By (1.8), (LABEL:0.23), (2.38) and (3.1), there exist polynomials $J^{\prime}_{r}(Z,Z^{\prime})$ in $Z,Z^{\prime}$ such that for $|Z|,|Z^{\prime}|<p^{-\frac{1}{2}+\theta}$ with $\theta$ in (2.34),

[TABLE]

with

[TABLE]

For a function $f\in\mathscr{C}^{\infty}(X)$, we denote by $f_{x_{0}}(Z)$ the function $f$ in normal coordinates $Z$ around a point $x_{0}\in X$. We have thus a family $(f_{x_{0}})$ of functions indexed by the parameter $x_{0}\in X$. Combining (1.8), (2.5) with $\theta$ in (2.34), and (3.2), we obtain

[TABLE]

By using Taylor expansion of $f_{x_{0}}(Z^{\prime})$ at [math], we obtain

[TABLE]

Finally, by [15, Theorem 1.18] and [15, (1.97), (1.98), (1.111)], we obtain

[TABLE]

Combining Taylor expansion of $f_{x_{0}}(Z^{\prime})$ at [math], and (3.6) yields

[TABLE]

Combining (3.4) for $k_{0}=0$, (LABEL:0.29) and (3.7) yields

[TABLE]

Then the desired $\mathscr{C}^{m}$-estimate (1.10) follows from (1.9) and (3.8). The proof of the uniformity assertion from Theorem 1.1 is modeled on [14, § 4.1.7], [15, § 1.5]. First we notice that in the proof of the estimate (2.16), we only use the derivatives of the coefficients of $\mathscr{L}_{t}$ with order $\leqslant 2n+m+m^{\prime}+r+2$. Thus, by (2.28), the constants in (2.16), (2.26) (resp. (2.29), (2.31)) are bounded, if with respect to a fixed metric $g_{0}^{TX}$, the $\mathscr{C}^{2n+m+m^{\prime}+r+3}$ (resp. $\mathscr{C}^{2n+m+m^{\prime}+k+4}$)-norms on $X$ of the data $g^{TX},h^{L},\nabla^{L},h^{E},\nabla^{E}$ and $\Phi$ are bounded and $g^{TX}$ is bounded below. Note $k=k_{0}+m^{\prime}+2$ in (LABEL:0.23). Then the constants in (LABEL:0.23) (resp. (3.2), (3.4), (3.8)) are bounded if with respect to a fixed metric $g_{0}^{TX}$, the $\mathscr{C}^{2n+m+2m^{\prime}+k_{0}+6}$ (resp. $\mathscr{C}^{2n+m+k_{0}+6}$, $\mathscr{C}^{2n+m+k_{0}+6}$, $\mathscr{C}^{2n+m+6}$)-norm on $X$ of the data $g^{TX},h^{L},\nabla^{L},h^{E},\nabla^{E}$ and $\Phi$ are bounded and $g^{TX}$ is bounded below. Moreover, taking derivatives with respect to the parameters we obtain a similar equation to (2.22) (cf. [15, (1.65)]). Thus the $\mathscr{C}^{m}$-norm in (3.8) can also include the parameters of the $\mathscr{C}^{m}$-norm if the $\mathscr{C}^{m}$-norms (with respect to the parameter $x_{0}\in X$) of derivatives of the above data with order $\leqslant 2n+6$ are bounded. Thus we can take $C$ in (1.10) independent of $g^{TX}$. The proof of Theorem 1.1 is completed.