Global Hypoellipticity and Compactness of Resolvent for Fokker-Planck Operator

TL;DR

This paper investigates conditions on the potential in the Fokker-Planck operator to ensure global hypoellipticity and resolvent compactness, improving previous results with weaker assumptions.

Contribution

It provides new global hypoelliptic estimates for the Fokker-Planck operator under less restrictive conditions on the potential.

Findings

Established conditions for hypoellipticity of the Fokker-Planck operator.

Proved compactness of the resolvent under certain potential behaviors.

Improved upon previous results by weakening assumptions on the potential.

Abstract

In this paper we study the Fokker-Planck operator with potential V(x), and analyze some kind of conditions imposed on the potential to ensure the validity of global hypoelliptic estimates. As a consequence, we obtain the compactness of resolvent of the Fokker-Planck operator if either the Witten Laplacian on 0-forms has a compact resolvent or some additional assumption on the behavior of the potential at infinity is fulfilled. This work improves the previous results of H\'erau-Nier and Helffer-Nier, by obtaining a better global hypoelliptic estimate under weaker assumptions on the potential.

Equations

$$P=y\cdot {\partial }_x-{\partial }_{x}V(x)\cdot {\partial }_y-{△}_y+\frac{{\left|y\right|}^2}{4}-\frac{n}{2},(x,y)\in {\mathbb{R}}^{2n}$$

$$\forall \left|\alpha \right|=2,\exists C_{\alpha }>0,\left|{\partial }_x^{\alpha }V(x)\right|\le C_{\alpha }{\left(1+{\left|{\partial }_{x}V(x)\right|}^2\right)}^{\frac{s}{2}}\mathrm{with}\mathrm{s}<\frac{4}{3}.$$

$$\displaystyle\big{\|}\left|\partial_{x}V(x)\right|^{2\over 3}u\big{\|}_{L^{2}}\leq C\left\{\,\,\big{\|}Pu\big{\|}_{L^{2}}+\big{\|}u\big{\|}_{L^{2}}\,\,\right\},$$

$$\displaystyle\big{\|}\left(1-\triangle_{x}\right)^{\frac{\delta}{2}}u\big{\|}_{L^{2}}+\big{\|}\left(1-\triangle_{y}+\left|y\right|^{2}\right)^{1\over 2}u\big{\|}_{L^{2}}\leq C\Big{\{}\,\big{\|}Pu\big{\|}_{L^{2}}+\big{\|}u\big{\|}_{L^{2}}\Big{\}},$$

$$\forall~{}u\in C_{0}^{\infty}\big{(}\mathbb{R}^{2n}\big{)},\quad\big{\|}\left|\partial_{x}V(x)\right|^{2\over 3}u\big{\|}_{L^{2}}+\big{\|}\left(1-\triangle_{x}\right)^{\frac{1}{3}}u\big{\|}_{L^{2}}\leq C\Big{\{}\big{\|}Pu\big{\|}_{L^{2}}+\big{\|}u\big{\|}_{L^{2}}\Big{\}}.$$

$$\forall~{}u\in C_{0}^{\infty}\big{(}\mathbb{R}^{2n}\big{)},\quad\big{\|}\left(1-\triangle_{y}+\left|y\right|^{2}\right)u\big{\|}_{L^{2}}\leq C\Big{\{}\,\big{\|}Pu\big{\|}_{L^{2}}+\big{\|}u\big{\|}_{L^{2}}\Big{\}}.$$

$$\frac{1}{C}{⟨x⟩}^{2M-1}\le {\left(1+{\left|{\partial }_{x}V(x)\right|}^2\right)}^{\frac{1}{2}}\mathrm{and}\forall \left|\gamma \right|\ge 0,\left|{\partial }_x^{\gamma }V(x)\right|\le C_{\gamma }{⟨x⟩}^{2M-|\gamma |},$$

$$\displaystyle\big{\|}\Lambda_{x,y}^{1\over 4}u\big{\|}_{L^{2}}\leq C\left\{\,\big{\|}Pu\big{\|}_{L^{2}}+\big{\|}u\big{\|}_{L^{2}}\,\right\}$$

$$\displaystyle\frac{1}{C}\left<x\right>^{\frac{1}{k}}\leq\Big{(}1+|\partial_{x}V(x)|^{2}\Big{)}^{1\over 2}\leq C\left<x\right>^{k}~{}\textrm{and}~{}\forall\left|\gamma\right|\geq 0,~{}\left|\partial_{x}^{\gamma}V(x)\right|\leq C_{\gamma}\big{(}1+\left|\partial_{x}V(x)\right|^{2}\big{)}^{1\over 2}.$$

$$\forall \left|\alpha \right|=2,\left|{\partial }_x^{\alpha }V(x)\right|\le C_{\alpha }{\left(1+{\left|{\partial }_{x}V(x)\right|}^2\right)}^{\frac{1-\rho }{2}}\mathrm{with}\rho >\frac{1}{3},$$

$$\displaystyle\big{\|}\left|\partial_{x}V(x)\right|^{2\over 3}u\big{\|}_{L^{2}}\leq C\left\{\,\,\big{\|}Pu\big{\|}_{L^{2}}+\big{\|}u\big{\|}_{L^{2}}\,\,\right\}.$$

$${△}_{V/2}^{(0)}=-{△}_x+\frac{1}{4}{\left|{\partial }_{x}V(x)\right|}^2-\frac{1}{2}{△}_{x}V(x).$$

$$\forall \left|\gamma \right|\ge 0,\forall x\in {\mathbb{R}}^{2n},\left|{\partial }_x^{\gamma }V(x)\right|\le C_{\gamma }{\left(1+{\left|{\partial }_{x}V(x)\right|}^2\right)}^{\frac{1}{2}},$$

$$\exists M,C>1,\forall x\in {\mathbb{R}}^{2n},\left|{\partial }_{x}V(x)\right|\le C{⟨x⟩}^M,$$

$$\exists \kappa >0,\forall \left|\alpha \right|=2,\forall x\in {\mathbb{R}}^{2n},\left|{\partial }_x^{\alpha }V(x)\right|\le C_{\alpha }{\left(1+{\left|{\partial }_{x}V(x)\right|}^2\right)}^{\frac{1}{2}}{⟨x⟩}^{-\kappa }.$$

$$D_{x_j}=-i{\partial }_{x_j},D_{y_j}=-i{\partial }_{y_j}\mathrm{and}{D}_x=(D_{x_1},\cdots ,D_{x_n}),D_y=(D_{y_1},\cdots ,D_{y_n}).$$

$${\Lambda }_y={\left(1+\frac{1}{2}{\left|y\right|}^2-{△}_y\right)}^{\frac{1}{2}}.$$

$$f(x)={\left(1+{\left|{\partial }_{x}V(x)\right|}^2\right)}^{\frac{1}{2}}.$$

$$P=iQ+\sum _{j=1}^n{L}_j^{\ast }{L}_j.$$

$$\displaystyle\forall~{}u\in C_{0}^{\infty}\big{(}\mathbb{R}^{2n}\big{)},\quad\sum_{j=1}^{n}\big{\|}L_{j}u\big{\|}_{L^{2}}^{2}\leq{\rm Re}\left<Pu,~{}u\right>_{L^{2}},$$

$$\displaystyle\forall u\in C_{0}^{\infty}\big{(}\mathbb{R}^{2n}\big{)},\quad\big{\|}\Lambda_{y}u\big{\|}_{L^{2}}^{2}\leq C\Big{\{}\left|\left<Pu,~{}u\right>_{L^{2}}\right|+\big{\|}u\big{\|}_{L^{2}}^{2}\Big{\}}.$$

$$\displaystyle\big{\|}\Lambda_{y}^{2}u\big{\|}_{L^{2}}\leq C\Big{\{}\big{\|}\left|\partial_{x}V(x)\right|^{2\over 3}u\big{\|}_{L^{2}}+\big{\|}\left(1-\triangle_{x}\right)^{1\over 3}u\big{\|}_{L^{2}}+\big{\|}Pu\big{\|}_{L^{2}}\Big{\}},$$

$$\displaystyle\sum_{j=1}^{n}\big{\|}L_{j}L_{j}^{*}u\big{\|}_{L^{2}}\leq C\left\{\,\big{\|}\left|\partial_{x}V(x)\right|^{2\over 3}~{}u\big{\|}_{L^{2}}+\big{\|}\left(1-\triangle_{x}\right)^{1\over 3}~{}u\big{\|}_{L^{2}}+\big{\|}Pu\big{\|}_{L^{2}}\,\right\}.$$

$$\displaystyle\big{\|}L_{j}L_{j}^{*}u\big{\|}_{L^{2}}^{2}$$

$$\big{\|}L_{j}L_{j}^{*}u\big{\|}_{L^{2}}^{2}\leq 2\left|\left<[P,~{}L_{j}^{*}]u,~{}L_{j}^{*}u\right>_{L^{2}}\right|+4\big{\|}Pu\big{\|}_{L^{2}}^{2}.$$

$$\displaystyle\left|\left<[P,L_{j}^{*}]u,L_{j}^{*}u\right>_{L^{2}}\right|\leq\varepsilon\big{\|}L_{j}L_{j}^{*}u\big{\|}_{L^{2}}^{2}+C_{\varepsilon}\Big{\{}\big{\|}|\partial_{x}V|^{2\over 3}u\big{\|}_{L^{2}}^{2}+\big{\|}(1-\triangle_{x})^{1\over 3}u\big{\|}_{L^{2}}^{2}+\big{\|}Pu\big{\|}_{L^{2}}^{2}\Big{\}}.$$

$$[iQ,L_j^{\ast }]=-\frac{1}{2}{\partial }_{x_j}V(x)+{\partial }_{x_j},[L_j,L_k]=[L_j^{\ast },L_k^{\ast }]=0,[L_j,L_k^{\ast }]={\delta }_{jk};$$

$$[P,L_j^{\ast }]=-\frac{1}{2}{\partial }_{x_j}V(x)+{\partial }_{x_j}+L_j^{\ast }.$$

$$\left|{⟨[P,L_j^{\ast }]u,L_j^{\ast }u⟩}_{L^2}\right|$$

$$\displaystyle\big{\|}L_{j}\left|\partial_{x}V(x)\right|^{1\over 3}u\big{\|}_{L^{2}}^{2}$$

Full text

1. Introduction and main results

In this work we consider the Fokker-Planck operator

[TABLE]

where $x$ denotes the space variable and $y$ denotes the velocity variable, and $V(x)$ is a potential defined in the whole space $\mathbb{R}_{x}^{n}$. There have been extensive works concerned with the operator $P$, with various techniques from different fields such as partial differential equation, spectral theory and statistical physics. In this paper we will focus on analyzing some kind of conditions imposed on the potential $V(x),$ so that the Fokker-Planck operator $P$ admits a global hypoelliptic estimate and has a compact resolvent. This problem is linked closely with the trend to equilibrium for the Fokker-Planck operator, and has been studied by Desvillettes-Villani, Helffer-Nier, Hérau-Nier and some other authors (see [2, 3, 5] and the references therein). It is believed that the global estimate and the compactness of resolvent are related to the properties of the potential $V(x).$ In the particular case of quadratic potential, the theory is well developed. As far as general potential is concerned, different kind of assumptions on $V(x)$ had been explored firstly by Hérau-Nier [5] and then generalized by Helffer-Nier [3]. This work is motivated by the previous works of Hérau-Nier and Helffer-Nier, and can be seen as an improvement of their results. Our main result is the following.

\theoremname** 1.1**

Let $V(x)\in C^{2}(\mathbb{R}^{n})$ be a real-valued function satisfying that

[TABLE]

Then there is a constant $C,$ such that for any u\in C_{0}^{\infty}\big{(}\mathbb{R}^{2n}\big{)} one has

[TABLE]

and

[TABLE]

where $\delta$ equals to $\frac{2}{3}$ if $s\leq\frac{2}{3}$, $\frac{4}{3}-s$ if ${2\over 3}<s\leq\frac{10}{9},$ and $\frac{2}{3}-{s\over 2}$ if ${{10}\over 9}<s<\frac{4}{3}$. Here and throughout the paper we will use $\|\cdot\|_{L^{2}}$ to denote the norm of the complex Hilbert space L^{2}\big{(}\mathbb{R}^{2n}\big{)}, and denote by $C_{0}^{\infty}\left(\mathbb{R}^{2n}\right)$ the set of smooth compactly supported functions.

\remarkname** 1.2**

In particular, if the assumption (1.2) is fulfilled with $s=\frac{2}{3}$, then we have the following hypoelliptic estimate which seems to be optimal:

[TABLE]

Moreover one can deduce from the above estimate a better regularity in the velocity variable $y,$ that is,

[TABLE]

This can be seen in Proposition 2.1 in the next section.

\corollaryname** 1.3**

The Fokker-Planck operator $P$ has a compact resolvent if the potential $V(x)$ satisfies (1.2) and that $\lim\limits_{\left|x\right|\rightarrow+\infty}\left|\partial_{x}V(x)\right|=+\infty.$

To analyze the compactness of resolvent of the operator $P,$ the hypoellipticity techniques are an efficient tool, one of which is referred to Kohn’s method [7] and another is based on nilpotent Lie groups (see [4, 8]). Kohn’s method had been used by Hérau-Nier [5] to study such a potential $V(x)$ that behaves at infinity as a high-degree homogeneous function. More precisely, if $V(x)$ satisfies that for some $C,M\geq 1,$

[TABLE]

where $\left<x\right>=\left(1+\left|x\right|^{2}\right)^{1\over 2}$, then Hérau-Nier established the following isotropic hypoelliptic estimate, by use of the global pseudo-differential calculus,

[TABLE]

with $\Lambda_{x,y}=\left(1-\triangle_{x}-\triangle_{y}+\frac{1}{2}\left|V(x)\right|^{2}+\frac{1}{2}\left|y\right|^{2}\right)^{1\over 2}.$ By developing the approach of Hérau-Nier, Helffer-Nier [3] obtained the same estimate as above for more general $V(x)$ which satisfies that for some $C,k\geq 1,$

[TABLE]

As for the Kohn’s proof for the hypoellipticity, the exponent ${1\over 4}$ in (1.6) is not optimal. A better exponent, which seems to be ${2\over 3}$ as seen in [8], can be obtained via explicit method in the particular case when $V(x)$ is a non-degenerate quadratic form. Moreover Helffer-Nier [3] studied such a $V(x)$ that satisfies

[TABLE]

and obtained the estimate

[TABLE]

This generalized the quadratic potential case, and their main tool is the nilpotent technique that initiated by [8] and then developed by [4]. Although the estimate (1.9) is better, the condition (1.8) is stronger than (1.7) for the second derivatives, and comparing with (1.6), we see that in (1.9) some information on the Sobolev regularity in $x$ is missing. In (1.2) we get rid of the assumptions on the behavior of $\partial_{x}V(x)$ at infinity. This generalizes the conditions (1.5) and (1.7). Moreover, the exponent in (1.3) is ${2\over 3}$, better than ${1\over 4}$ established in (1.6). Besides, we have relaxed the condition (1.8) by allowing the number $\rho$ there to take values in the interval $]-{1\over 3},~{}+\infty[.$ As seen in the proof presented in Section 3, our approach is direct, which seems simpler for it doesn’t touch neither complicated nilpotent group techniques nor pseudo-differential calculus.

Another direction to get the compact resolvent is to analyze the relationship between $P$ and the Witten Laplace operator $\triangle_{V/2}^{(0)}$ defined by

[TABLE]

In [3], Helffer-Nier stated a conjecture which says that the Fokker-Planck operator $P$ has a compact resolvent if and only if the Witten Laplacian $\triangle_{V/2}^{(0)}$ has a compact resolvent. The necessity part is well-known, and under rather weak assumptions on the potential $V,$ saying V\in C^{\infty}\big{(}\mathbb{R}^{2n}\big{)} for instance, if the Fokker-Planck operator $P$ has a compact resolvent then the Witten Laplacian $\triangle_{V/2}^{(0)}$ has a compact resolvent. The reverse implication still remains open, and some partial answers have been obtained by [3, 5]. For example, suppose V\in C^{\infty}\big{(}\mathbb{R}^{2n}\big{)} such that

[TABLE]

[TABLE]

and

[TABLE]

Then the operator $P$ has a compact resolvent if the Witten Laplace operator $\triangle_{V/2}^{(0)}$ has a compact resolvent (see Corollary 5.10 of [3]). Due to Theorem 1.1, we can generalize the previous results as follows.

\corollaryname** 1.4**

Let $V(x)$ satisfy the condition (1.2). Then the Fokker-Planck operator $P$ has a compact resolvent if the Witten Laplacian $\triangle_{V/2}^{(0)}$ has a compact resolvent.

The paper is organized as follow. In the next section we introduce some notations used throughout the paper, and then present some regularity results on the velocity variable $y$. Since the proof of Theorem 1.1 is quite lengthy, we divide it into two parts and proceed to handle them in Section 3 and Section 4. The proof of Corollary 1.4 will be presented in Section 5.

2. Notation and regularity in velocity variable

We firstly list some notations used throughout the paper in Subsection 2.1, and then establish the regularity in the velocity variable $y$ in Subsection 2.2. This will give the desired estimate on the second term on the left of (1.4).

2.1. Notation

Throughout the paper we denote by $(\xi,\eta)$ the dual variables of $(x,y),$ and denote by $\left<\cdot,~{}\cdot\right>_{L^{2}}$ the inner product of the complex Hilbert space L^{2}\big{(}\mathbb{R}^{2n}\big{)}. Set

[TABLE]

Let $\Lambda_{y}$ be the operator given by

[TABLE]

Observing $\left|\partial_{x}V(x)\right|$ is only continuous, we have to replace it sometimes by the equivalent $C^{1}$ function $f(x)$ given by

[TABLE]

Denoting $Q=y\cdot D_{x}-\partial_{x}V(x)\cdot D_{y}$ and $L_{j}=\partial_{y_{j}}+\frac{y_{j}}{2},j=1,\cdots n,$ we can write the operator $P$ given in (1.1) as

[TABLE]

2.2. Regularity in the velocity variable

In view of the expression (2.1), we see that the required estimate on the term \big{\|}\Lambda_{y}u\big{\|}_{L^{2}} is easy to get, without any assumption on the potential $V(x)$. Indeed, As a result of (2.1), we have

[TABLE]

from which one can deduce that

[TABLE]

This gives the desired estimate on the second term on the left of (1.4).

For constant potential, i.e., $\partial_{x}V(x)=0,$ starting from the regularity in $x$, we can derive a better Sobolev exponent, which is known to be $2,$ for the regularity in $y$ variable (see for instant [1]). When general potential is involved, we have the following estimate.

\propositionname** 2.1**

There exists a constant $C$ such that for any u\in C_{0}^{\infty}\big{(}\mathbb{R}^{2n}\big{)},

[TABLE]

or equivalently,

[TABLE]

Proof. In this proof we show (2.5). Using (2.2) gives

[TABLE]

Hence

[TABLE]

Now assume the following estimate holds, for any $\varepsilon>0$,

[TABLE]

Then combining the above two inequalities and then letting $\varepsilon$ small enough, we get the desired estimate (2.5). In order to show (2.6), we make use of the following commutation relations satisfied by $iQ,L_{j},L_{k}^{*},j,k=1,2,\cdots,n,$

[TABLE]

this gives

[TABLE]

Then

[TABLE]

Moreover, note that

[TABLE]

and hence

[TABLE]

Similarly,

[TABLE]

These inequalities yield (2.6). The proof of Proposition 2.1 is thus completed. ∎

3. Proof of Theorem

1.1: the first part

In this section we only show (1.3) and postpone (1.4) to the next section. Let $V(x)$ satisfy the assumption (1.2). Then using the notation

[TABLE]

we have

[TABLE]

The following is the main result of this section.

\propositionname** 3.1**

Suppose $f$ satisfies the condition (3.1). Then

[TABLE]

Proof. To simplify the notation, we will use the capital letter $C$ to denote different suitable constants. Let R\in C^{1}\big{(}\mathbb{R}^{2n}\big{)} be a real-valued function given by

[TABLE]

We can verify that

[TABLE]

Recall $P=iQ+\sum_{j=1}^{n}L_{j}^{*}L_{j}$ with $Q=y\cdot D_{x}-\partial_{x}V(x)\cdot D_{y}$ and $L_{j}=\partial_{y_{j}}+\frac{y_{j}}{2}.$ Then the above inequalities together with the relation

[TABLE]

yield

[TABLE]

Next we will proceed to treat the terms on both sides of (3.3) by the following three steps.

Step I**.** Firstly we will show that for any $\varepsilon>0$ there exists a constant $C_{\varepsilon}>0$ such that

[TABLE]

To confirm this, we use (2.3) to get

[TABLE]

The upper bound of the term ${\rm Re}\left<Pu,~{}f(x)^{2\over 3}u\right>_{L^{2}}$ can be obtained by Cauchy-Schwarz’s inequality. Then the required estimate (3.4) will follow if the following inequality holds: for any $\varepsilon_{1},\varepsilon_{2}>0,$ there exists a constant $C_{\varepsilon_{1},\varepsilon_{2}}$ such that

[TABLE]

To prove (3.5), we use (3.1); this gives

[TABLE]

and hence

[TABLE]

Since $s-{2\over 3}<{2\over 3}$ for $s<{4\over 3}$ then the following interpolation inequality holds:

[TABLE]

Now combination of the above inequalities yields (3.5).

Step II**.** Next we will show that there exists a constant $C>0$ such that

[TABLE]

Since $Q=y\cdot D_{x}-\partial_{x}V(x)\cdot D_{y}$ and $R=2f(x)^{-{2\over 3}}\partial_{x}V(x)\cdot y$, then it’s a straightforward verification to see that

[TABLE]

As a result, we use the relation

[TABLE]

to get

[TABLE]

This gives

[TABLE]

Moreover, by use of (3.1), we compute

[TABLE]

which implies that for any $\varepsilon>0,$

[TABLE]

the last inequality using (3.4). Consequently,

[TABLE]

Letting $\varepsilon>0$ small enough gives (3.6).

Step III**.** Now we prove that for any $\varepsilon>0$ there exists a constant $C_{\varepsilon}$ such that

[TABLE]

As a preliminary step, we firstly show the following estimate:

[TABLE]

where $\left<y\right>=\left(1+\left|y\right|^{2}\right)^{1\over 2}$. Using (2.3) gives

[TABLE]

This together with (2.3) implies that

[TABLE]

Moreover observe that

[TABLE]

and hence for any $\varepsilon>0,$

[TABLE]

This along with (3.4) and (2.3) gives

[TABLE]

Now combining (3.9) and (3.10), we get (3.8). As a result of (3.8), we have

[TABLE]

Indeed by (2.3) one has

[TABLE]

So (3.11) can be deduced from (3.8) and (3.10). Now we are ready to prove (3.7). Observe

[TABLE]

Then in view of (3.4), we see that the required inequality (3.7) will follow if the following estimate holds:

[TABLE]

Since

[TABLE]

then

[TABLE]

This along with (3.11) gives (3.12), completing the proof (3.7).

Now we combine the inequalities (3.3), (3.4), (3.6) and (3.7), to obtain

[TABLE]

Taking $\varepsilon=\frac{1}{2}$ gives the desired estimate (3.2). This completes the proof of Proposition 3.1. ∎

4. Proof of Theorem 1.1: the second part

This section is devoted to the proof of (1.4), and then the proof of Theorem 1.1 will be completed. As a convention, we use the capital letter $C$ to denote different suitable constants. Let $V$ satisfy the assumption (1.2). In the sequel we use the notation

[TABLE]

Then (1.2) yields

[TABLE]

In view of (2.3), to prove (1.4) one only has to show

\propositionname** 4.1**

If $V(x)$ satisfies the assumption (1.2), then

[TABLE]

where $\delta$ equals to $\frac{2}{3}$ if $s\leq\frac{2}{3}$, $\frac{4}{3}-s$ if ${2\over 3}<s\leq\frac{10}{9},$ and $\frac{2}{3}-{s\over 2}$ if ${{10}\over 9}<s<\frac{4}{3}.$

We will use localization arguments to prove the above proposition. Firstly let’s recall some standard results concerning the partition of unity. For more detail we refer to [6] for instant. Let $g$ be a metric of the following form

[TABLE]

where $s$ is the real number given in (4.1).

\lemmaname** 4.2**

Suppose $f$ satisfies the assumption (4.1). Then the metric $g$ defined by (4.3) is slowly varying, i.e., we can find two constants $C_{*},r>0$ such that if $g_{x}(x-y)\leq r^{2}$ then

[TABLE]

Proof. We only need to show that

[TABLE]

Making use of (3.1) and the fact that $s<\frac{4}{3},$ we have

[TABLE]

with $C$ the constant in (3.1). As a consequence, one can find a constant $\tilde{C}$ depending only on $C$ and the dimension $n,$ such that

[TABLE]

from which we conclude that if $\left|x-y\right|\leq rf(x)^{-{s\over 2}}$ then

[TABLE]

Thus

[TABLE]

This gives (4.4) if we choose $r=\frac{\tilde{C}}{2}$ and $C_{*}=2.$ ∎

Let $g$ be the metric given by (4.3). We denote by $S(1,g)$ the class of smooth real-valued functions $a(x)$ satisfying the following condition:

[TABLE]

The space $S(1,g)$ endowed with the seminorms

[TABLE]

becomes a Fréchet space.

The main feature of a slowly varying metric is that it allows us to introduce some partitions of unity related to the metric. We state it as the following lemma.

\lemmaname** 4.3**** **((Lemma 18.4.4. of [6]))

Let $g$ be a slowly varying metric. We can find a constant $r_{0}>0$ and a sequence $x_{\mu}\in\mathbb{R}^{n},\mu\geq 1,$ such that the union of the balls

[TABLE]

coves the whole space $\mathbb{R}^{n}.$ Moreover there exists a positive integer $N,$ depending only on $r_{0},$ such that the intersection of more than $N$ balls is always empty. One can choose a family of nonnegative functions $\left\{\,\varphi_{\mu}\,\right\}_{\mu\geq 1}$ uniformly bounded in $S(1,g)$ such that

[TABLE]

Here by uniformly bounded in $S(1,g),$ we mean

[TABLE]

\remarkname** 4.4**

If we choose $r_{0}$ small enough such that $r_{0}\leq r$ with $r$ the constant given in Lemma 4.2, then there exists a constant $C,$ such that for any $\mu\geq 1$ one has

[TABLE]

\lemmaname** 4.5**

Let $V(x)$ satisfy the assumption (1.2), and let $\left\{\,\varphi_{\mu}\,\right\}_{\mu\geq 1}$ be the partition of unity given above. Then we have for any u\in C_{0}^{\infty}\big{(}\mathbb{R}^{2n}\big{)},

[TABLE]

and

[TABLE]

Proof. Firstly we show (4.7). Observe

[TABLE]

and by Lemma 4.3, we see that $\sum_{\mu\geq 1}\left|\partial_{x}\varphi_{\mu}\right|^{2}$ is a sum of at most $N$ terms and hence bounded from above by $f^{s}.$ As a result,

[TABLE]

Then (4.7) follows. Next we estimate (4.8). Note that $\left|x-x_{\mu}\right|\leq Cf(x_{\mu})^{-{s\over 2}}$ for any $x\in$ supp $\varphi_{\mu},$ and hence we can deduce from (1.2) and (4.6) that

[TABLE]

This along with the inequality

[TABLE]

implies (4.8). Then the proof is completed. ∎

\lemmaname** 4.6**

Let $\left\{\,\varphi_{\mu}\,\right\}_{\mu\geq 1}$ be the partition given in Lemma 4.3, and let $a\in]0,1/2[$ be a real number. Then there exists a constant $C,$ depending on the integer $N$ given in Lemma 4.3, such that

[TABLE]

In order to prove Lemma 4.6 we need the following technical lemma.

\lemmaname** 4.7**

Let $b\in]0,1[$ be a real number and $\left|D_{x}\right|^{b}$ be the Fourier multiplier defined by, with $u\in C_{0}^{\infty}(\mathbb{R}^{n}),$

[TABLE]

Let $\left\{\,\varphi_{\mu}\,\right\}_{\mu\geq 1}$ be the partition given in Lemma 4.3. Then there exists a constant $C$ such that for any $u\in C_{0}^{\infty}(\mathbb{R}^{n}),$

[TABLE]

and

[TABLE]

Recall here $f(x)=\left(1+\left|\partial_{x}V(x)\right|^{2}\right)^{1/2}$ and $s$ is the real number given in (4.1).

Proof. In the proof we use $C$ to denote different suitable positive constants, and for simplicity we use the notation

[TABLE]

In view of Lemma 4.3 and the estimate (4.1), we have

[TABLE]

Next we will show the following relation

[TABLE]

with $C_{b}\neq 0$ being a complex constant depending only on the real number $b$ and the dimension $n.$ In fact, the inverse Fourier transform implies

[TABLE]

On the other hand, we can verify that

[TABLE]

Observe that $\int_{\mathbb{R}^{n}}\frac{1-e^{-i\,z\cdot\frac{\xi}{\left|\xi\right|}}}{\left|z\right|^{n+b}}\,dz\neq 0$ is a complex constant depending only on $b$ and the dimension $n$, but independent of $\xi.$ Then the above two equalities give (4.13). Now we use (4.13) to get

[TABLE]

which gives

[TABLE]

Let $\rho$ be the characteristic function of the unit ball $\left\{\,x\in\mathbb{R}^{n};\>\left|x\right|\leq 1\,\right\}.$ We compute

[TABLE]

Now we treat the terms $\mathcal{A}_{1}$ and $\mathcal{A}_{2}.$ Cauchy’s inequality yields

[TABLE]

This along with (4.12) gives that for any $x,\tilde{x}\in\mathbb{R}^{n},$ we have $\left|\sum_{\mu\geq 1}\varphi_{\mu}(x-\tilde{x})\left(\omega_{\mu}(x)-\omega_{\mu}(x-\tilde{x})\right)\right|\leq C$ and hence

[TABLE]

Moreover, using the relation

[TABLE]

and the inequality (4.12) yields that for any $x,\tilde{x}\in\mathbb{R}^{n}$ we have $\left(\sum_{\mu\geq 1}\left|\omega_{\mu}(x)-\omega_{\mu}(x-\tilde{x})\right|^{2}\right)^{1\over 2}\leq C\left|\tilde{x}\right|$ and hence

[TABLE]

which implies

[TABLE]

Combining these inequalities gives

[TABLE]

Moreover, for the terms on the right side of the above inequality, we can use Young’s inequality for convolutions and the fact that $\rho$ is the characteristic function of the unit ball, to get

[TABLE]

and

[TABLE]

We combine these inequalities to get the desired estimate (4.10). The estimate (4.11), which is easier to treat, can be obtained via the similar arguments as above. This completes the proof. ∎

Proof of Lemma 4.6. We only need show that, with $b\in]0,1[,$

[TABLE]

By (4.5), we see \big{\|}\left|D_{x}\right|^{b}u\big{\|}_{L^{2}}^{2}=\big{\|}\sum_{\mu\geq 1}\left|D_{x}\right|^{b}\varphi_{\mu}^{2}\,u\big{\|}_{L^{2}}^{2}. Thus

[TABLE]

In view of (4.10) we have

[TABLE]

the last inequality following from (3.2). It remains to handle the second term on the right side of (4.16). For each $\mu\geq 1,$ set

[TABLE]

Then $I_{\mu}$ is a finite set and has at most $N$ elements. Recall $N$ is the integer given in Lemma 4.3 such that the intersection of more than $N$ balls is always empty. Direct calculus give that for any $u\in C_{0}^{\infty}(\mathbb{R}^{2n}),$

[TABLE]

Since $I_{\mu}$ has at most $N$ elements then it follows that

[TABLE]

One the other hand, one can verify that

[TABLE]

the second inequality using (4.11) and the last inequality using (3.2). These inequalities along with (4.18) gives

[TABLE]

This along with (4.16) and (4.17) yields the desired estimate (4.15) , completing the proof of Lemma 4.6. ∎

4.1. End of the proof of Theorem 1.1

In this subsection we prove Proposition 4.1. Let $\left\{\,\varphi_{\mu}\,\right\}_{\mu\geq 1}$ be the partition of unity given in Lemma 4.3. For each $\mu\geq 1,$ define the operator $R_{\mu}$ by

[TABLE]

We associate with each $x_{\mu}\in\mathbb{R}^{n}$ the operator

[TABLE]

Then we have

[TABLE]

with $R_{\mu}$ the operator given in (4.19). This gives

[TABLE]

\propositionname** 4.8**

There is a constant $C$ independent of $x_{\mu},$ such that for any u\in C_{0}^{\infty}\big{(}\mathbb{R}^{2n}\big{)}, one has

[TABLE]

or equivalently,

[TABLE]

where $\tilde{\Lambda}_{x_{\mu}}=\left(1+\frac{1}{2}\left|\partial_{x}V(x_{\mu})\right|^{2}-\triangle_{x}\right)^{1\over 2}.$

The above proposition can be proven in the same way as Proposition 5.22 of [3], by taking Fourier analysis in the x-variable and then reducing the problem to a semi-class problem. We refer to [3] and references therein for more details. For the sake of completeness we present a direct proof in the Appendix at the end of the paper.

\lemmaname** 4.9**

Suppose $V(x)$ satisfies the assumption (1.2). Let $R_{\mu}$ be the operator given in (4.19). Then

[TABLE]

where $\tilde{s}=\frac{2}{3}-\delta$ with $\delta$ given in (1.4), i.e., $\tilde{s}$ equals to [math] if $s\leq{2\over 3}$, $s-{2\over 3}$ if ${2\over 3}<s\leq\frac{10}{9},$ and $\frac{s}{2}$ if ${10\over 9}<s<\frac{4}{3}$.

Proof. As a convention, we use the capital letter $C$ to denote different suitable constants. Since $V(x)$ satisfies (1.2), then (4.1) holds. Observe \sum\limits_{\mu\geq 1}\big{\|}R_{\mu}u\big{\|}_{L^{2}}^{2} is bounded from above by

[TABLE]

Then in view of (4.7) and (4.8), we have

[TABLE]

So we only have to treat the term \big{\|}\Lambda_{y}f(x)^{s\over 2}u\big{\|}_{L^{2}}^{2}. It follows from (2.3) that

[TABLE]

Since ${s\over 2}<\frac{2}{3}$ then by (3.2) we have

[TABLE]

The above two inequalities yield that for any $u\in C_{0}^{\infty}\left(\mathbb{R}^{2n}\right),$

[TABLE]

a)  Firstly let us consider the case when $s\leq\frac{2}{3}.$ In such a case, we have

[TABLE]

the last inequality using (3.2). This along with (4.25) gives

[TABLE]

On the other hand using (4.1) with $s\leq{2\over 3}$ implies, for any $\varepsilon>0,$

[TABLE]

Combining the above two inequalities and taking $\varepsilon\leq{1\over 2}$, we get

[TABLE]

Since \sum\limits_{\mu\geq 1}\big{\|}R_{\mu}u\big{\|}_{L^{2}}^{2}\leq C\big{\|}\Lambda_{y}f(x)^{s\over 2}u\big{\|}_{L^{2}}^{2}, then the above estimate gives the validity of (4.23) for $s\leq{2\over 3}$.

b) Next we shall prove (4.23) for ${2\over 3}<s<{4\over 3}$. If $\frac{10}{9}<s<\frac{4}{3},$ then it follows from (4.25) and (4.24) that

[TABLE]

This gives the validity of (4.23) for $s\in]\frac{10}{9},~{}\frac{4}{3}[.$

Now we focus on the case when $\frac{2}{3}<s\leq\frac{10}{9}.$ Observe that

[TABLE]

Moreover since $f(x)$ satisfies (4.1), then

[TABLE]

and thus

[TABLE]

Combination of the above three inequalities gives

[TABLE]

Moreover since $2s-{5\over 3}\leq\frac{s}{2}$ for $s\leq\frac{10}{9},$ then

[TABLE]

and hence by (3.2) we obtain

[TABLE]

Inserting the above inequality into (4.25) and then taking $\varepsilon$ small enough, we get the desired estimate (4.23) for ${2\over 3}<s<{10\over 9}.$ Thus the proof of Lemma 4.9 is completed. ∎

Now we are ready to prove the main result of this section.

Proof of Proposition 4.1. Now we want to show that

[TABLE]

Recall $\delta$ equals to $\frac{2}{3}$ if $s\leq\frac{2}{3}$, $\frac{4}{3}-s$ if ${2\over 3}<s\leq\frac{10}{9},$ and $\frac{2}{3}-{s\over 2}$ if ${{10}\over 9}<s<\frac{4}{3}.$ Using the estimates (4.20) and (4.23) gives that

[TABLE]

where $\tilde{s}=\frac{2}{3}-\delta.$ We can verify that

[TABLE]

Firstly let us consider the case of $s\leq\frac{2}{3}.$ Then $\tilde{s}=0$ and (4.28) becomes

[TABLE]

On the other hand, using (4.9) with $a=\frac{1}{3}$ and then (4.21), we have

[TABLE]

As a result, we get from these inequalities

[TABLE]

This gives the validity of (4.27) for $s\leq{2\over 3}.$

Now we consider the case when $\frac{2}{3}<s<\frac{4}{3}.$ Note that $\delta=\frac{2}{3}-\tilde{s}.$ Then we use (4.9) with $a=\frac{\delta}{2}$ to get

[TABLE]

Consequently, using (4.22) yields

[TABLE]

Thus (4.27) will follow if we can show that

[TABLE]

To prove (4.30), we write

[TABLE]

Then

[TABLE]

with $(I),(II)$ given by

[TABLE]

and

[TABLE]

By (4.6), we see

[TABLE]

This along with (4.28) gives

[TABLE]

By use of (4.1) and (4.29), we have

[TABLE]

and hence

[TABLE]

This along with (4.31) gives

[TABLE]

Now it remains to treat the term $(II).$ The equality

[TABLE]

gives

[TABLE]

By (4.6), (4.1) and (4.29), we have

[TABLE]

So

[TABLE]

This along with (4.32) gives

[TABLE]

Combining the estimate on the term $(I),$ we get the required inequality (4.30). The proof of Proposition 4.1 is thus completed. ∎

5. Proof of Corollary 1.4

The method is quite similar as that in [3], and the main difference is that we have to use the functional calculus for self-adjoint operators instead of the pseudo-differential calculus used in [3], since in our case the potential $V$ only belongs to C^{2}\big{(}\mathbb{R}^{2n}\big{)}. Firstly let us mention some well-known facts on the functional calculus for positive self-adjoint operators (see for instance Chapter XI of [9]). Consider the Schrödinger operator $1+\left|\partial_{x}V(x)\right|^{2}-\triangle_{x}$ which is defined on $C_{0}^{\infty}(\mathbb{R}^{n}).$ Since

[TABLE]

holds in the sense of operators, then it is well-known that $1+\left|\partial_{x}V(x)\right|^{2}-\triangle_{x}$ admits a unique self-adjoint extension on $L^{2}(\mathbb{R}^{n})$, still denoted by $1+\left|\partial_{x}V(x)\right|^{2}-\triangle_{x}.$ As a result, using the notation

[TABLE]

the self-adjoint operator $A$ admits a spectral representation

[TABLE]

with domain

[TABLE]

Here $\left\{\,E_{\lambda}\,\right\}_{\lambda\geq 0}$ is called a spectral resolution of $1+\left|\partial_{x}V(x)\right|^{2}-\triangle_{x}$. By the spectral representation, we can define the fractional power of the operator $A$ as follows: for each $\theta\geq 0,$

[TABLE]

with domain

[TABLE]

Note that $\left(1+\left|\partial_{x}V(x)\right|^{2}-\triangle_{x}\right)^{0}=I$. Moreover, since $A=1+\left|\partial_{x}V(x)\right|^{2}-\triangle_{x}\geq 1,$ then $E_{\lambda}=0$ for $0\leq\lambda<1.$ This allows us to define the negative fraction power by

[TABLE]

with domain

[TABLE]

Now we list some classical results to be used frequently on the fractional power of the operator $A$. For each $\theta\geq 0,$ the operators $A^{\pm\theta}$ are self-adjoint on L^{2}\big{(}\mathbb{R}^{n}\big{)}, and satisfy the following relation

[TABLE]

that is, $A^{-\theta}$ is the inverse operator of $A^{\theta}$. If $\theta\in[0,1]$ then $u\in D(A)$ if and only if $u\in D(A^{\theta})$ and also $A^{\theta}\in D(A^{1-\theta}).$ For such $u$ we have

[TABLE]

Moreover the negative power $A^{-\theta},\theta>0,$ is bounded on $L^{2}(\mathbb{R}^{n}),$ and for any $\theta_{1},\theta_{2}\geq 0,$ and any u\in L^{2}\big{(}\mathbb{R}^{n}\big{)}, we have

[TABLE]

Since the following inequalities

[TABLE]

hold in the sense of operators, then the monotonicity of operator functional implies for each $\theta\geq 0,$

[TABLE]

and

[TABLE]

As a result, for each $\theta\geq 0$ the following estimates hold: for any u\in L^{2}\big{(}\mathbb{R}^{2n}\big{)},

[TABLE]

and

[TABLE]

The rest of the paper is devoted to showing Corollary 1.4.

Proof of Corollary 1.4. Assume V\in C^{2}\big{(}\mathbb{R}^{2n}\big{)} satisfies (1.2). Then the global hypoelliptic estimates (1.3) and (1.4) hold. Let $\delta$ be the number given in (1.4). For any $u\in C_{0}^{\infty}(\mathbb{R}^{2n}),$ we can verify that

[TABLE]

the last inequality using from (5.1) and (5.2) with $-\theta={\delta\over 2}-1<0$. As a result, applying (1.3) and (1.4) yields

[TABLE]

This implies the operator

[TABLE]

is bounded. Since

[TABLE]

then the compactness of the resolvent $\left(1+P\right)^{-1}$ will follow if the operator

[TABLE]

is compact. Recall that a operator $B$ acting on L^{2}\big{(}\mathbb{R}^{2n}\big{)} is compact if and only if

[TABLE]

Then to get the compactness of $\left(1+\left|\partial_{x}V(x)\right|^{2}-\triangle_{x}\right)^{-{\delta\over 2}}$, we have to show that for any sequence $\left\{\,u_{n}\,\right\}_{n\geq 1}$ converging to [math] weakly in $L^{2}(\mathbb{R}^{2n})$,

[TABLE]

This can be derived from the compactness of resolvent of the Witten Laplacian $\triangle_{V/2}^{(0)}.$ Indeed, since $\triangle_{V/2}^{(0)}$ has a compact resolvent then the operator $\left(1+\left|\partial_{x}V(x)\right|^{2}-\triangle_{x}\right)^{-1}$ is also compact. As a result,

[TABLE]

Moreover since $\left\{\,u_{n}\,\right\}_{n\geq 1}$ is a weakly convergent sequence then it is bounded in L^{2}\big{(}\mathbb{R}^{2n}\big{)}. Hence

[TABLE]

As a result, using the relation

[TABLE]

gives

[TABLE]

and hence

[TABLE]

Repeating the above arguments, we can get

[TABLE]

Then the proof of Corollary 1.4 is completed. ∎

6. Appendix

Here we present another proof of Proposition 4.8. Let’s restate it as

\propositionname** 6.1**

We associate with each fixed $x_{0}\in\mathbb{R}^{n}$ the operator

[TABLE]

Then there is a constant $C$ independent of $x_{0},$ such that for any u\in C_{0}^{\infty}\big{(}\mathbb{R}^{2n}\big{)}, one has

[TABLE]

or equivalently,

[TABLE]

where $\tilde{\Lambda}_{x_{0}}=\left(1+\frac{1}{2}\left|\partial_{x}V(x_{0})\right|^{2}-\triangle_{x}\right)^{1\over 2}.$

Proof. We will prove (6.2) in this proof. To simplify the notation the capital letter $C$ will be used to denote different suitable constants independent of $x_{0}.$ Denoting $Q_{x_{0}}=y\cdot D_{x}-\partial_{x}V(x_{0})\cdot D_{y},$ we can write $P_{x_{0}}$ as

[TABLE]

from which we deduce

[TABLE]

Then

[TABLE]

We will proceed to prove (6.2) in the following three steps.

Step A**.**  We claim, for any $\varepsilon>0,$ there is a constant $C_{\varepsilon},$ depending only on $\varepsilon,$ such that

[TABLE]

To confirm this, we apply (6.3) to get, for any u\in C_{0}^{\infty}\big{(}\mathbb{R}^{2n}\big{)},

[TABLE]

Hence

[TABLE]

To estimate the first term on the right side of the above inequality, we make use of the following commutation relations satisfied by $iQ_{x_{0}},L_{j},L_{k}^{*},j,k=1,2,\cdots,n,$

[TABLE]

this gives, for any $\tilde{\varepsilon}>0,$

[TABLE]

Moreover we use (6.3) again to obtain

[TABLE]

Combining these inequalities, we conclude

[TABLE]

and thus by (6.6)

[TABLE]

Observe

[TABLE]

Then the desired estimate (6.5) follows.

Step B**.**  For $x_{0}\in\mathbb{R}^{n}$ consider the operator

[TABLE]

It’s a straightforward verification to see that

[TABLE]

Next we will prove that, for any $\varepsilon>0,$

[TABLE]

To confirm this, observe

[TABLE]

This along with (6.4) gives that

[TABLE]

Then we make use of Cauchy-Schwarz inequality and the interpolation inequality that

[TABLE]

to obtain the desired estimate (6.8).

Step C**.** Now the equality

[TABLE]

gives

[TABLE]

the last inequality following from (6.5) and (6.8). This along with (6.7) gives at once

[TABLE]

Letting $\varepsilon$ small enough such that $\varepsilon\leq{1\over 2},$ we obtain the desired upper bound of the term on the left of (6.2), completing the proof of Proposition 6.1. ∎

Acknowledgements. The author would like to thank Professor N. Lerner for his fruitful discussions and help during the preparation of this paper.