Gradient Estimates For The CR Heat Equation On Complete Noncompact Pseudohermitian Manifolds¹¹1 This work is supported by NSFC Grants No.12171091.

The method of gradient estimates is an important tool in geometric analysis, which is originated first in Yau [30] and Cheng and Yau [13] for investigating harmonic functions and further developed in Li and Yau [25] for studying solutions of the heat equation on complete Riemannian manifolds. In [25], Li and Yau established their celebrated parabolic inequality, which asserts that, if $M$ is an $n$-dimensional complete Riemannian manifold with Ricci curvature bounded from below by $-K$, for some constant $K\geq 0$ and $u$ is any positive solution of the heat equation

$$(\bigtriangleup-\frac{\partial}{\partial t})u=0,$$

then

$$\frac{|\nabla u|^{2}}{u^{2}}-\alpha\frac{u_{t}}{u}\leq\frac{n\alpha^{2}}{2t}+\frac{n\alpha^{2}}{2(\alpha-1)}K$$

for all $t>0$, $\alpha>1$. In particular, if $K=0$, then the following more precise inequality holds

$$\frac{|\nabla u|^{2}}{u^{2}}-\frac{u_{t}}{u}\leq\frac{n}{2t}.$$

Since then many improvements or generalizations of Li-Yau’s parabolic inequality have been developed on Riemannian manifolds or more general metric measure spaces, see e.g. [17], [22], [4], [7], [3], [27], [24], [2], [32], [12], [33], [31] and the references therein.

The Li-Yau’s inequality has also been generalized for non elliptic operators that include subelliptic operators on sub-Riemannian manifolds, see e.g. [8], [1], [5], [6], [19], [11], [10], etc. Some of these works concern Li-Yau type inequalities on pseudo-Hermitian manifolds. In [11], Chang et al. derived a CR Li-Yau type estimate in terms of the lower bound of pseudo-Hermitian Ricci curvature essentially for closed Sasakian 3-manifolds. In [5], Baudoin and Garofalo proved, among other results, a CR Li-Yau type inequality on complete Sasakian manifolds under a curvature dimension inequality. In [9], the authors announced a CR Li-Yau gradient estimate by using a generalized curvature-dimension inequality and the maximum principle in a closed pseudo-Hermitian manifold possibly with nonvanishing torsion. Besides, they also established a Li-Yau type inequality for the sum of squares of vector fields up to higher step on a closed manifold, generalizing Cao-Yau’s result ([8]) for operators expressed as the sum of squares of vector fields of step 2. However, we don’t understand their proof for the CR part. Anyhow Cao-Yau’s inequality in [8] almost gave us in particular a Li-Yau type inequality for closed pseudo-Hermitian manifolds (See Remark 3.1).

Let’s recall briefly Cao-Yau’s work in [8] as follows. Suppose $X_{1},....,$ $X_{n}$ are smooth vector fields on a closed manifold $M$ and

$$L=\sum_{i=1}^{n}X_{i}^{2}-X_{0}$$

(1.1)

with $X_{0}=\sum_{i=1}^{n}c_{i}X_{i}$, where $c_{i}$ are smooth functions on $M$. Suppose $X_{1},...,X_{n}$ satisfy the following conditions: for $1\leq i,j,k\leq n$, $[X_{i},[X_{j},X_{k}]]$ can be expressed as linear combinations of $X_{1},...,X_{n}$ and their brackets $[X_{1},X_{2}],....,$
$[X_{n-1},X_{n}]$. Cao and Yau considered a positive solution $u(x,t)$ of

$$\left(L-\frac{\partial}{\partial t}\right)u=0$$

(1.2)

on $M\times(0,\infty)$ and showed that there exists a constant $\delta_{0}>0$ such that for any $\delta>\delta_{0}$, $u$ satisfies

$$\frac{1}{u^{2}}\sum_{i=1}^{n}|X_{i}u|^{2}-\delta\frac{X_{0}u}{u}-\delta\frac{u_{t}}{u}\leq\frac{C_{1}}{t}+C_{2},$$

(1.3)

where $C_{1}$ and $C_{2}$ are positive constants depending on $n$, $\delta_{0}$, $\delta$ and $\{X_{i}\}$.

This paper is devoted to establish a Li-Yau type inequality on a complete pseudo-Hermitian manifold possibly with nonvanishing pseudo-Hermitian torsion. The pseudo-Hermitian manifolds considered here are CR manifolds of hypersurface type which admit positive definite pseudo-Hermitian structures (see §2 for the detailed definition). Let $(M^{2m+1},HM,J,\theta)$ denote a pseudo-Hermitian manifold of dimension $2m+1$. Here $(HM,J)$ is a CR structure of type $(m,1)$, and $\theta$ is a pseudo-Hermitian structure on $M$. We find that a pseudo-Hermitian manifold carries a rich geometric structure, including an almost complex structure $J$ on $HM$, the positive definite Levi form $L_{\theta}$ on $HM$ induced from $\theta$ and $J$, the Webster metric (a Riemannian metric on $M$ extending $L_{\theta}$), the Reeb vector field $\xi$ on $M$, the sub-Laplacian $\bigtriangleup_{b}$ (a subelliptic differential operator) and the horizontal gradient operator $\nabla_{b}$ acting on functions. Note also that the pair $(HM,L_{\theta})$ is a $2$-step sub-Riemannian structure, which induces a Carnot–Carathédory distance $d_{cc}$ on $M$. These geometric data provide us a basis to investigate Li-Yau type inequality on a pseudo-Hermitian manifold. We will consider a positive solution of the following CR heat equation

$$\frac{\partial u}{\partial t}=\bigtriangleup_{b}u$$

(1.4)

on a complete pseudo-Hermitian manifold, and establish a Li-Yau type inequality for $u$. The main ingredients of Li-Yau’s method [25] or Cao-Yau’s method ([8]) involve the Bochner type formula, a parabolic differential inequality for a suitable auxiliary function and the maximum principle. For any smooth function $f$ on the pseudo-Hermitian manifold, one has two CR Bochner formulas for $|\nabla_{b}f|^{2}$ and $f_{0}^{2}$ respectively, where $f_{0}=\xi(f)$ (see (2.8) and (2.9) in §2). Now set $f=\ln u$. Following Cao-Yau’s idea, we will consider the auxiliary functions

$$\mathcal{F}=t\left(|\nabla_{b}f|^{2}+t^{2\lambda-1}\left(1+f_{0}^{2}\right)^{\lambda}-\delta f_{t}\right)$$

(1.5)

or

$$\mathcal{G}=t\left(|\nabla_{b}f|^{2}+\left(1+f_{0}^{2}\right)^{\lambda}-\delta f_{t}\right)$$

(1.6)

according to the ranges of $t$. Some parabolic differential inequalities for $\mathcal{F}$ and $\mathcal{G}$ can be derived from the CR Bochner formulas. Following the technique in [25], we may multiply $\mathcal{F}$ and $\mathcal{G}$ by a suitable cut-off function $\phi$ to localize the problem. By applying the maximum principle to $\phi\mathcal{F}$ and $\phi\mathcal{G}$, and using the CR sub-Laplacian comparison theorem in [15], we are able to establish the following local Li-Yau gradient estimate.

$\mathbf{Theorem\ 1.1}$ Let $(M^{2m+1},HM,J,\theta)$ be a complete noncompact pseudo-Hermitian manifold with

$\displaystyle Ric_{b}+2(m-2)Tor_{b}\geq-k,\ and\ |A|,|\nabla_{b}A|\leq k_{1},$

and $u$ be a positive solution of the CR heat equation

$\displaystyle\frac{\partial u}{\partial t}=\Delta_{b}u$

on $B_{p}(2R)\times(0,\infty)\ with\ R\geq 1$, where $B_{p}(r)$ denotes the Riemannian ball of radius $r$ with respect to the Webster metric $g_{\theta}$. Then for any constant $\frac{1}{2}<\lambda<\frac{2}{3}$ and any constant $\delta>1+\frac{4}{m\lambda(2\lambda-1)}$, there exists a constant $C$ depending on $m,k,k_{1},\lambda,\delta$, such that

$\displaystyle\frac{|\nabla_{b}u|^{2}}{u^{2}}-\delta\frac{u_{t}}{u}\leq C(1+\frac{1}{t}+\frac{1}{R^{\lambda}}+\frac{1}{tR^{\lambda}})$

(1.7)

on $B_{p}(R)\times(0,\infty)$.

Letting $R\rightarrow\infty$ in Theorem 1.1, we get immediately the global Li-Yau type gradient estimate.

$\mathbf{Theorem\ 1.2}$ Let $(M^{2m+1},HM,J,\theta)$ be a complete noncompact pseudo-Hermitian manifold with

$\displaystyle Ric_{b}+2(m-2)Tor_{b}\geq-k,\ and\ |A|,|\nabla_{b}A|\leq k_{1},$

and $u$ be a positive solution of the heat equation

$\displaystyle\frac{\partial u}{\partial t}=\Delta_{b}u$

on $M\times(0,\infty)$. Then for any constant $\frac{1}{2}<\lambda<\frac{2}{3}$ and any constant $\delta>1+\frac{4}{m\lambda(2\lambda-1)}$, there exists a constant $C$ depending on $m,k,k_{1},\lambda,\delta$, such that

$\displaystyle\frac{|\nabla_{b}u|^{2}}{u^{2}}-\delta\frac{u_{t}}{u}\leq C+\frac{C}{t}$

(1.8)

on $M\times(0,\infty)$.

As applications of the above gradient estimates, we give a Harnack inequality for the positive solutions of the CR heat equation, and then obtain an upper bound estimate for the heat kernel of the CR heat equation.

$\mathbf{Theorem\ 1.3}$ Let $(M^{2m+1},HM,J,\theta)$ be a complete noncompact pseudo-Hermitian manifold with

$\displaystyle Ric_{b}+2(m-2)Tor_{b}\geq-k,\ and\ |A|,|\nabla_{b}A|\leq k_{1},$

and $u$ be a positive solution of the heat equation

$\displaystyle\frac{\partial u}{\partial t}=\Delta_{b}u$

on $M\times(0,\infty)$. Then for any constant $\frac{1}{2}<\lambda<\frac{2}{3}$ and any constant $\delta>1+\frac{4}{m\lambda(2\lambda-1)}$, there exists a constant $C$ which is given by Theorem 1.2 such that for any $0<t_{1}<t_{2}$ and $x,y\in M$, we have

$\displaystyle u(x,t_{1})\leq u(y,t_{2})(\frac{t_{2}}{t_{1}})^{\frac{C}{\delta}}exp(\frac{C}{\delta}(t_{2}-t_{1})+\frac{\delta d_{cc}^{2}(x,y)}{4(t_{2}-t_{1})}).$

(1.9)

$\mathbf{Theorem\ 1.4}$ Let $(M^{2m+1},HM,J,\theta)$ be a complete noncompact pseudo-Hermitian manifold with

$\displaystyle Ric_{b}+2(m-2)Tor_{b}\geq-k,\ and\ |A|,|\nabla_{b}A|\leq k_{1},$

and $H(x,y,t)$ be the heat kernel of (1.4). Then for any constants $\frac{1}{2}<\lambda<\frac{2}{3}$, $\delta>1+\frac{4}{m\lambda(2\lambda-1)}$ and $0<\epsilon<1$, there exists constants $C^{{}^{\prime}}\ and\ C^{{}^{\prime\prime}}$ depending on $m,k,k_{1},\lambda,\delta,\epsilon$, such that $H(x,y,t)$ satisfies

$\displaystyle H(x,y,t)\leq C^{{}^{\prime}}[Vol(B_{cc}(x,\sqrt{t}))]^{-\frac{1}{2}}[Vol(B_{cc}(y,\sqrt{t}))]^{-\frac{1}{2}}exp(C^{{}^{\prime\prime}}\epsilon t-\frac{d^{2}_{cc}(x,y)}{(4+\epsilon)t}),$

(1.10)

where the $B_{cc}(x,r)$ is the ball with respect to Carnot-Carath$\acute{e}$odory distance. The constant $C^{{}^{\prime}}\rightarrow\infty$ as $\epsilon\rightarrow 0$.

In this section we introduce some basic notations in pseudo-Hermitian geometry (cf. [18, 29, 28] for details), and then give the CR Bochner formulas for functions on a pseudo-Hermitian manifold. Next, we will derive parabolic differential inequalities for the auxiliary functions $\mathcal{F}$ and $\mathcal{G}$.

Let $M^{2m+1}$ be a real $2m+1$ dimensional orientable $C^{\infty}$ manifold. A CR structure on $M$ is a complex subbundle $H^{1,0}M$ of $TM\otimes\mathbb{C}$ satisfying

$$H^{1,0}M\cap H^{0,1}M=\{0\},\ \ [\Gamma(H^{1,0}M),\Gamma(H^{1,0}M)]\subseteq\Gamma(H^{1,0}M)$$

(2.1)

where $H^{0,1}M=\overline{H^{1,0}M}$. Equivalently, the CR structure may also be described by the real bundle $HM=Re\{H^{1,0}M\oplus H^{0,1}M\}$ and an almost complex structure $J$ on $HM$, where $J(X+\overline{X})=\sqrt{-1}(X-\overline{X})$ for any $X\in H^{1,0}M$. Then $(M,HM,J)$ is said to be a CR manifold.

We denote by $E$ the conormal bundle of $HM$ in $T^{*}M$, whose fiber at each point $x\in M$ is given by

$\displaystyle E_{x}=\{\omega\in T_{x}^{*}M|\omega(H_{x}M)=0\}.$

(2.2)

It turns out that $E$ is a trivial line bundle. Therefore there exist globally defined nowhere vanishing sections $\theta\in\Gamma(E)$. A section $\theta\in\Gamma(E\backslash\{0\})$ is called a pseudo-Hermitian structure on $M$. The Levi form $L_{\theta}$ of a pseudo-Hermitian structure $\theta$ is defined by

$\displaystyle L_{\theta}(X,Y)=d\theta(X,JY)$

for any $X,Y\in HM$. The integrability condition in (2.1) implies that $L_{\theta}$ is $J$-invariant, and thus symmetric. When $L_{\theta}$ is positive definite on $HM$ for some $\theta$, then $(M,HM,J)$ is said to be strictly pseudoconvex. From now on, we will always assume that $(M,HM,J)$ is a strictly pseudoconvex CR manifold endowed with $\theta$, such that $L_{\theta}$ is positive definite. Then the quadruple $(M,HM,J,\theta)$ is referred to as a pseudo-Hermitian manifold.

For a pseudo-Hermitian manifold $(M,HM,J,\theta)$, due to the positivity of $L_{\theta}$, we have a sub-Riemannian structure $(HM,L_{\theta})$ of step-2 on $M$. We say that a Lipschitz curve $\gamma:[0,l]\rightarrow M$ is horizontal if $\gamma^{{}^{\prime}}\in H_{\gamma(t)}M$ a.e. in $[0,l]$. For any two points $p,q\in M$, by the well-known theorem of Chow-Rashevsky([16, 26] ), there always exist such horizontal curves joining $p$ and $q$. Therefore we may define the Carnot-Carath$\acute{e}$odory distance as follows:

$\displaystyle d_{cc}(p,q)=inf\{\int_{0}^{l}\sqrt{L_{\theta}(\gamma^{{}^{\prime}},\gamma^{{}^{\prime}})}dt\ |\ \gamma\in\Gamma(p,q)\},$

where $\Gamma(p,q)$ denotes the set of all horizontal curves joining $p$ and $q$. Clearly $d_{cc}$ induces to a metric space structure on $M$, in which its metric ball centered at $x$ with radius $r$ is given by

$\displaystyle B_{cc}(x,r)=\{y\in M\ |\ d_{cc}(y,x)<r\}.$

For a pseudo-Hermitian manifold $(M,HM,J,\theta)$, it is clear that $\theta$ is a contact form on $M$. Consequently there exists a unique vector field $\xi$ such that

$\displaystyle\theta(\xi)=1,\ d\theta(\xi,\cdot)=0.$

(2.3)

This vector field $\xi$ is called the Reeb vector field. From (2.2) and (2.3), it is easy to see that $TM$ admits the following direct sum decomposition

$\displaystyle TM=HM\oplus R\xi,$

(2.4)

which induces a natural projection $\pi_{b}:TM\rightarrow HM$. In terms of $\theta$ and the decomposition (2.4), the Levi form $L_{\theta}$ can be extended to a Riemannian metric

$\displaystyle g_{\theta}=L_{\theta}+\theta\otimes\theta,$

which is called the Webster metric. We will denote by $r$ the corresponding Riemannian distance and by $B_{p}(R)$ the Riemannian ball of radius $R$ centered at $p$. One may extend the complex structure $J$ on $HM$ to an endomorphism of $TM$, still denoted by $J$, by requiring

$\displaystyle J\xi=0.$

It is known that there exists a canonical connection $\nabla$ on a pseudo-Hermitian manifold, called the Tanaka-Webster connection (cf. [18, 28, 29]), such that

		$\displaystyle 1.$	$\displaystyle\ \nabla_{X}\Gamma(HM)\subseteq\Gamma(HM),\ for\ any\ X\in\Gamma(TM);$
		$\displaystyle 2.$	$\displaystyle\ \nabla g_{\theta}=0\ and\ \nabla J=0;$
		$\displaystyle 3.$	$\displaystyle\ T_{\nabla}(X,Y)=2d\theta(X,Y)\xi\ and\ T_{\nabla}(\xi,JX)+JT_{\nabla}(\xi,X)=0,$
			$\displaystyle for\ any\ X,Y\in HM,\ where\ T_{\nabla}\ denotes\ the\ torsion\ of$
			$\displaystyle the\ connection\ \nabla.$

The pseudo-Hermitian torsion of $\nabla$ is an important pseudo-Hermitian invariant, which is an $HM$-valued 1-form defined by

$\displaystyle\tau(X)=T_{\nabla}(\xi,X)$

for any $X\in TM$. Note that is $\tau$ trace-free and self-adjoint with respect to the Webster metric $g_{\theta}$ (cf. [18]). Set $A(X,Y)=g_{\theta}(T_{\nabla}(\xi,X),Y)$ for any $X,Y\in TM$, then we have

$\displaystyle A(X,Y)=A(Y,X).$

(2.5)

We say that $M$ is Sasakian if $\tau=0$ (or equivalently, $A=0$).

Let $(M,HM,J,\theta)$ be a complete pseudo-Hermitian manifold of dimension $2m+1$. We choose a local orthonormal frame field $\{e_{A}\}_{A=0}^{2m}=\{\xi,e_{1},\cdots,e_{m},\\ e_{m+1},\cdots,e_{2m}\}$ with respect to the Webster metric $g_{\theta}$ such that

$\displaystyle\{e_{m+1},\cdots,e_{2m}\}=\{Je_{1},\cdots,Je_{m}\}.$

Set

$\displaystyle\eta_{\alpha}=\frac{1}{\sqrt{2}}(e_{\alpha}-\sqrt{-1}Je_{\alpha}),\quad\eta_{\bar{\alpha}}=\frac{1}{\sqrt{2}}(e_{\alpha}+\sqrt{-1}Je_{\alpha}),\ (\alpha=1,\cdots,m).$

Then $\{\eta_{\alpha}\}_{\alpha=1}^{m}$ is a unitary frame field of $H^{1,0}M$ with respect to $g_{\theta}$. Let $\{\theta^{1},\cdots,\theta^{m}\}$ be the dual frame field of $\{\eta_{\alpha}\}_{\alpha=1}^{m}$. According to the property 3 of the Tanaka-Webster connection, one may write

	$\displaystyle\tau$	$\displaystyle=$	$\displaystyle\tau^{\alpha}\eta_{\alpha}+\tau^{\bar{\alpha}}\eta_{\bar{\alpha}}$
		$\displaystyle=$	$\displaystyle A_{\bar{\beta}}^{\alpha}\theta^{\bar{\beta}}\otimes\eta_{\alpha}+A_{\beta}^{\bar{\alpha}}\theta^{\beta}\otimes\eta_{\bar{\alpha}}.$

We will also write $A_{\alpha\beta}=A_{\beta}^{\bar{\alpha}}$ and $A_{\bar{\alpha}\bar{\beta}}=A_{\bar{\beta}}^{\alpha}$. Then (2.5) means that $A_{\alpha\beta}=A_{\beta\alpha}$ and $A_{\bar{\alpha}\bar{\beta}}=A_{\bar{\beta}\bar{\alpha}}$. From [29], we have the following structure equations of the Tanaka-Webster connection $\nabla$:

	$\displaystyle d\theta$	$\displaystyle=$	$\displaystyle 2\sqrt{-1}\theta^{\alpha}\wedge\theta^{\bar{\alpha}},$
	$\displaystyle d\theta^{\alpha}$	$\displaystyle=$	$\displaystyle\theta^{\beta}\wedge\theta^{\alpha}_{\beta}+A_{\bar{\alpha}\bar{\beta}}\theta\wedge\theta^{\beta},$		(2.6)
	$\displaystyle d\theta^{\alpha}_{\beta}$	$\displaystyle=$	$\displaystyle\theta^{\gamma}_{\beta}\wedge\theta^{\alpha}_{\gamma}+\Pi^{\alpha}_{\beta}$

with

$\displaystyle\Pi^{\alpha}_{\beta}=2\sqrt{-1}(\theta^{\alpha}\wedge\tau^{\bar{\beta}}-\tau^{\alpha}\wedge\theta^{\bar{\beta}})+R^{\alpha}_{\beta\lambda\bar{\mu}}\theta^{\lambda}\wedge\theta^{\bar{\mu}}+W^{\alpha}_{\beta\bar{\gamma}}\theta\wedge\theta^{\bar{\gamma}}-W^{\alpha}_{\beta\gamma}\theta\wedge\theta^{{\gamma}},$

where $W^{\alpha}_{\beta\bar{\gamma}}=A^{\alpha}_{\bar{\gamma},\beta},\ W^{\alpha}_{\beta\gamma}=A^{\bar{\gamma}}_{\beta,\bar{\alpha}}$ are the are the covariant derivatives of $A$, and $R_{\beta\lambda\bar{\mu}}^{\alpha}$ are the components of curvature tensor of the Tanaka-Webster connection. Set

$\displaystyle R_{\alpha\bar{\beta}}=R_{\gamma\alpha\bar{\beta}}^{\gamma},$

then $R_{\alpha\bar{\beta}}=R_{\bar{\beta}\alpha}$ (cf. [18]). For any $X=a^{\alpha}\eta_{\alpha}+b^{\bar{\alpha}}\eta_{\bar{\alpha}}$ and $Y=c^{\beta}\eta_{\beta}+d^{\bar{\beta}}\eta_{\bar{\beta}}\in HM\otimes\mathbb{C}$, we define

$\displaystyle Ric_{b}(X,Y)=R_{\alpha\bar{\beta}}a^{\alpha}d^{\bar{\beta}}+R_{\bar{\alpha}\beta}b^{\bar{\alpha}}c^{\beta},$

whose components are given by

	$\displaystyle Ric_{b}(\eta_{\alpha},\eta_{\bar{\beta}})$	$\displaystyle=$	$\displaystyle R_{\alpha\bar{\beta}},\ Ric_{b}(\eta_{\bar{\alpha}},\eta_{\beta})=R_{\bar{\alpha}\beta},$
	$\displaystyle Ric_{b}(\eta_{\alpha},\eta_{{\beta}})$	$\displaystyle=$	$\displaystyle Ric_{b}(\eta_{\bar{\alpha}},\eta_{\bar{\beta}})=0.$

The 2-tensor $Ric_{b}$ will be referred to as the pseudo-Hermitian Ricci tensor. For any $X=X^{\alpha}\eta_{\alpha}+X^{\bar{\alpha}}\eta_{\bar{\alpha}}$ and $Y=Y^{\beta}\eta_{\beta}+Y^{\bar{\beta}}\eta_{\bar{\beta}}\in HM\otimes\mathbb{C}$, we introduce

	$\displaystyle Tor_{b}(X,Y)$	$\displaystyle=$	$\displaystyle A(X,JY)$
		$\displaystyle=$	$\displaystyle\sqrt{-1}A(X^{\alpha}\eta_{\alpha}+X^{\bar{\alpha}}\eta_{\bar{\alpha}},Y^{\beta}\eta_{\beta}-Y^{\bar{\beta}}\eta_{\bar{\beta}})$
		$\displaystyle=$	$\displaystyle\sqrt{-1}(A_{\alpha\beta}X^{\alpha}Y^{\beta}-A_{\bar{\alpha}\bar{\beta}}X^{\bar{\alpha}}Y^{\bar{\beta}}).$

Clearly both $Ric_{b}$ and $Tor_{b}$ are real symmetric, fiberwise 2-tensors on $HM$.

For a $C^{2}$ function $f:M\rightarrow R$, its differential $df$ and gradient $\nabla f$ can be expressed as

$\displaystyle df=f_{0}\theta+f_{\alpha}\theta^{\alpha}+f_{\bar{\alpha}}\theta^{\bar{\alpha}}$

and

$\displaystyle\nabla f=f_{0}\xi+f_{\bar{\alpha}}\eta_{\alpha}+f_{\alpha}\eta_{\bar{\alpha}},$

where $f_{0}=\xi(f),f_{\alpha}=\eta_{\alpha}(f),f_{\bar{\alpha}}=\eta_{\bar{\alpha}}(f)$. Then the horizontal gradient of $f$ is given by

$\displaystyle\nabla_{b}f=f_{\bar{\alpha}}\eta_{\alpha}+f_{\alpha}\eta_{\bar{\alpha}}.$

Let $\nabla df$ be the covariant derivative of the differential $df\in\Gamma(T^{*}M)$ with respect to the Tanaka-Webster connection. Then $\nabla df$ may be expressed as

	$\displaystyle\nabla df$	$\displaystyle=$	$\displaystyle f_{\alpha\beta}\theta^{\alpha}\otimes\theta^{\beta}+f_{\alpha\bar{\beta}}\theta^{\alpha}\otimes\theta^{\bar{\beta}}+f_{\bar{\alpha}\beta}\theta^{\bar{\alpha}}\otimes\theta^{\beta}+f_{\bar{\alpha}\bar{\beta}}\theta^{\bar{\alpha}}\otimes\theta^{\bar{\beta}}$
			$\displaystyle+f_{0\alpha}\theta\otimes\theta^{\alpha}+f_{0\bar{\alpha}}\theta\otimes\theta^{\bar{\alpha}}+f_{\alpha 0}\theta^{\alpha}\otimes\theta+f_{\bar{\alpha}0}\theta^{\bar{\alpha}}\otimes\theta.$

The following communication relations are known (see, e.g., Chapter 9 in [18], or §3 in [14]):

$\displaystyle f_{\alpha\beta}=f_{\beta\alpha},\quad f_{\alpha\bar{\beta}}-f_{\bar{\beta}\alpha}=2\sqrt{-1}f_{0}\delta_{\alpha}^{\beta},\quad f_{0\alpha}-f_{\alpha 0}=f_{\bar{\beta}}A_{\alpha}^{\bar{\beta}}.$

(2.7)

The horizontal Hessian of $f$ is defined by

	$\displaystyle Hess_{b}(f)$	$\displaystyle=$	$\displaystyle(\nabla df)(\pi_{b},\pi_{b})$
		$\displaystyle=$	$\displaystyle f_{\alpha\beta}\theta^{\alpha}\otimes\theta^{\beta}+f_{\alpha\bar{\beta}}\theta^{\alpha}\otimes\theta^{\bar{\beta}}+f_{\bar{\alpha}\beta}\theta^{\bar{\alpha}}\otimes\theta^{\beta}+f_{\bar{\alpha}\bar{\beta}}\theta^{\bar{\alpha}}\otimes\theta^{\bar{\beta}}.$

Consequently

$\displaystyle|\nabla_{b}f|^{2}=2f_{\alpha}f_{\bar{\alpha}},\quad|Hess_{b}(f)|^{2}=2(f_{\alpha\beta}f_{\bar{\alpha}\bar{\beta}}+f_{\alpha\bar{\beta}}f_{\bar{\alpha}\beta}).$

The sub-Laplacian of $f$ is defined by

	$\displaystyle\Delta_{b}f$	$\displaystyle=$	$\displaystyle tr\{Hess_{b}(f)\}$
		$\displaystyle=$	$\displaystyle f_{\alpha\bar{\alpha}}+f_{\bar{\alpha}\alpha}.$

From [20], [21] (see also [18] and [14]), we have the following Bochner formulas

	$\displaystyle\frac{1}{2}\Delta_{b}|\nabla_{b}f|^{2}$	$\displaystyle=$	$\displaystyle 2(f_{\alpha\beta}f_{\bar{\alpha}\bar{\beta}}+f_{\alpha\bar{\beta}}f_{\bar{\alpha}\beta})+f_{\bar{\alpha}}(f_{\beta\bar{\beta}}+f_{\bar{\beta}\beta})_{\alpha}+f_{\alpha}(f_{\beta\bar{\beta}}+f_{\bar{\beta}\beta})_{\bar{\alpha}}$
			$\displaystyle+2R_{\alpha\bar{\beta}}f_{\bar{\alpha}}f_{\beta}+2\sqrt{-1}(m-2)(A_{\alpha\beta}f_{\bar{\alpha}}f_{\bar{\beta}}-A_{\bar{\alpha}\bar{\beta}}f_{\alpha}f_{\beta})$
			$\displaystyle+4\sqrt{-1}(f_{\bar{\alpha}}f_{0\alpha}-f_{\alpha}f_{0\bar{\alpha}})$
		$\displaystyle=$	$\displaystyle|Hess_{b}(f)|^{2}+\langle\nabla_{b}f,\nabla_{b}\Delta_{b}f\rangle+4\langle J\nabla_{b}f,\nabla_{b}f_{0}\rangle$
			$\displaystyle+(Ric_{b}+2(m-2)Tor_{b})(\nabla_{b}f,\nabla_{b}f)$

and

	$\displaystyle\frac{1}{2}\Delta_{b}f_{0}^{2}$	$\displaystyle=$	$\displaystyle|\nabla_{b}f_{0}|^{2}+f_{0}(\Delta_{b}f)_{0}$
			$\displaystyle+2f_{0}(f_{\beta}A_{\bar{\beta}\bar{\alpha},\alpha}+f_{\bar{\beta}}A_{\beta\alpha,\bar{\alpha}}+f_{\beta\alpha}A_{\bar{\beta}\bar{\alpha}}+f_{\bar{\beta}\bar{\alpha}}A_{\beta\alpha})$
		$\displaystyle=$	$\displaystyle|\nabla_{b}f_{0}|^{2}+f_{0}(\Delta_{b}f)_{0}+2f_{0}\mathrm{Im}Qf,$

where $Q$ is the purely holomorphic second-order operator defined by ([20])

$\displaystyle Qf=2\sqrt{-1}(A_{\bar{\beta}\bar{\alpha}}f_{\beta})_{\alpha}.$

Note that the coefficient before the ’mixed term’ $\langle J\nabla_{b}f,\nabla_{b}f_{0}\rangle$ in (2.8) is slight different from that in [21].

$\mathbf{Lemma\ 2.1}$ Let $(M^{2m+1},HM,J,\theta)$ be a pseudo-Hermitian manifold and $u$ be a positive solution of the CR heat equation (1.4). Set $f=\mathrm{ln}\ u$. Then for any $0<\lambda\leq 1$, we have

	$\displaystyle(\Delta_{b}-\partial_{t})|\nabla_{b}f|^{2}$	$\displaystyle\geq$	$\displaystyle\frac{1}{m}(\Delta_{b}f)^{2}+4mf^{2}_{0}+4f_{\alpha\beta}f_{\bar{\alpha}\bar{\beta}}$		(2.10)
			$\displaystyle-2\langle\nabla_{b}|\nabla_{b}f|^{2},\nabla_{b}f\rangle+8\langle\nabla_{b}f_{0},J\nabla_{b}f\rangle$
			$\displaystyle+2(Ric_{b}+2(m-2)Tor_{b})(\nabla_{b}f,\nabla_{b}f)$

and

	$\displaystyle(\Delta_{b}-\partial_{t})(1+f^{2}_{0})^{\lambda}$	$\displaystyle\geq$	$\displaystyle 2\lambda(2\lambda-1)(1+f^{2}_{0})^{\lambda-1}|\nabla_{b}f_{0}|^{2}$		(2.11)
			$\displaystyle-2\lambda(1+f^{2}_{0})^{\lambda-1}(\langle\nabla_{b}f^{2}_{0},\nabla_{b}f\rangle-2f_{0}A(\nabla_{b}f,\nabla_{b}f))$
			$\displaystyle+4\lambda(1+f^{2}_{0})^{\lambda-1}f_{0}\mathrm{Im}Qf.$

$\mathbf{Proof}$ A direct computation gives

$\displaystyle(\Delta_{b}-\partial_{t})f=-|\nabla_{b}f|^{2}.$

(2.12)

Using (2.7), we may estimate the term in $|Hess_{b}(f)|^{2}$:

	$\displaystyle f_{\alpha\bar{\beta}}f_{\bar{\alpha}\beta}$	$\displaystyle\geq$	$\displaystyle\sum\limits_{\alpha=1}^{m}f_{\alpha\bar{\alpha}}f_{\bar{\alpha}\alpha}=\frac{1}{4}\sum\limits_{\alpha=1}^{m}(|f_{\alpha\bar{\alpha}}-f_{\bar{\alpha}\alpha}|^{2}+|f_{\alpha\bar{\alpha}}+f_{\bar{\alpha}\alpha}|^{2})$		(2.13)
		$\displaystyle\geq$	$\displaystyle\frac{1}{4m}|\sum\limits_{\alpha=1}^{m}(f_{\alpha\bar{\alpha}}+f_{\bar{\alpha}\alpha})|^{2}+\frac{1}{4}\sum\limits_{\alpha=1}^{m}|f_{\alpha\bar{\alpha}}-f_{\bar{\alpha}\alpha}|^{2}$
		$\displaystyle=$	$\displaystyle\frac{1}{4m}(\Delta_{b}f)^{2}+mf_{0}^{2}.$

Then (2.10) follows immediately from (2.8), (2.12) and (2.13). From (2.9), we get

$\displaystyle(\Delta_{b}-\partial_{t})f_{0}^{2}=2|\nabla_{b}f_{0}|^{2}+2f_{0}(\Delta_{b}f-\partial_{t}f)_{0}+4f_{0}\mathrm{Im}Qf.$

(2.14)

Using (2.14), we derive that

			$\displaystyle(\Delta_{b}-\partial_{t})(1+f^{2}_{0})^{\lambda}$
		$\displaystyle=$	$\displaystyle 2\lambda(\lambda-1)(1+f^{2}_{0})^{\lambda-2}(f^{2}_{0})_{\alpha}(f^{2}_{0})_{\bar{\alpha}}+\lambda(1+f^{2}_{0})^{\lambda-1}(f^{2}_{0})_{\alpha\bar{\alpha}+\bar{\alpha}\alpha}$
			$\displaystyle-\partial_{t}(1+f^{2}_{0})^{\lambda}$
		$\displaystyle=$	$\displaystyle 4\lambda(\lambda-1)(1+f^{2}_{0})^{\lambda-2}f^{2}_{0}|\nabla_{b}f_{0}|^{2}+\lambda(1+f^{2}_{0})^{\lambda-1}(\Delta_{b}-\partial_{t})(f^{2}_{0})$
		$\displaystyle=$	$\displaystyle 4\lambda(\lambda-1)(1+f^{2}_{0})^{\lambda-2}f^{2}_{0}|\nabla_{b}f_{0}|^{2}+2\lambda(1+f^{2}_{0})^{\lambda-1}|\nabla_{b}f_{0}|^{2}$
			$\displaystyle+2\lambda(1+f^{2}_{0})^{\lambda-1}f_{0}\nabla_{\xi}(\Delta_{b}f-f_{t})$
			$\displaystyle+4\lambda(1+f^{2}_{0})^{\lambda-1}f_{0}\mathrm{Im}Qf$
		$\displaystyle=$	$\displaystyle 2\lambda(1+f^{2}_{0})^{\lambda-2}|\nabla_{b}f_{0}|^{2}((2\lambda-1)f^{2}_{0}+1)$
			$\displaystyle-2\lambda(1+f^{2}_{0})^{\lambda-1}(\langle\nabla_{b}f^{2}_{0},\nabla_{b}f\rangle-2f_{0}A(\nabla_{b}f,\nabla_{b}f))$
			$\displaystyle+4\lambda(1+f^{2}_{0})^{\lambda-1}f_{0}\mathrm{Im}Qf$
		$\displaystyle\geq$	$\displaystyle 2\lambda(2\lambda-1)(1+f^{2}_{0})^{\lambda-1}|\nabla_{b}f_{0}|^{2}$
			$\displaystyle-2\lambda(1+f^{2}_{0})^{\lambda-1}(\langle\nabla_{b}f^{2}_{0},\nabla_{b}f\rangle-2f_{0}A(\nabla_{b}f,\nabla_{b}f))$
			$\displaystyle+4\lambda(1+f^{2}_{0})^{\lambda-1}f_{0}\mathrm{Im}Qf.$

This completes the proof of Lemma 2.1. ∎