A direct proof of the Sharp Gårding inequality for symbols with limited smoothness

Abstract

We give a proof of the (possibly optimal) Sharp Gårding inequality for system operators with symbol of limited smoothness directly from the original symmetrization arguments by Friedrichs and Kumano-Go. The fact that only a few derivatives of the regularized symbol are really important was already there.

1 Introduction and main results

The Sharp Gårding inequality is a powerful tool in the study of systems of PDE. Let $P=p(x,D_x)=(P_{jk})$ be an $\ell \times \ell$ matrix of operators $P_{jk}=p_{jk}(x,D_x)$ with matrix symbol $p(x,\xi )=(p_{jk}(x,\xi ))\in S^m_{\rho ,\delta }$, $0\le \delta <\rho \le 1$, that is satisfying

$$\begin{aligned} |\partial _\xi ^\alpha D_x^\beta p(x,\xi )|\le C_{\alpha ,\beta }\langle \xi \rangle ^{m-\rho |\alpha |+\delta |\beta |},\ \ \langle \xi \rangle =\sqrt{1+|\xi |^2}. \end{aligned}$$

(1.1)

Assume that the Hermitian part $p'=(p+p^*)/2$ of $p(x,\xi )$ is positive semidefinite. Then, there exists $C>0$ such that

$$\begin{aligned} \Re (Pu,u)\ge -C\Vert u\Vert ^2_{H^{(m-\mu )/2}},\ \ \mu =\rho -\delta , \end{aligned}$$

(1.2)

for every $u\in {\mathcal S}$. In particular, for $p(x,\xi )\in S^m_{1,0}$ we have

$$\begin{aligned} \Re (Pu,u)\ge -C\Vert u\Vert ^2_{H^{(m-1)/2}}. \end{aligned}$$

(1.3)

Hörmander [4] proved inequality (1.3) for scalar operators and Lax-Nirenberg [7] extended this result to systems. Friedrichs [3], Kumano-Go [5] and others improved it and simplified the proof.

For scalar operators, there is the great strengthening $\mu =2(\rho -\delta )$ in (1.2) due to Fefferman and Phong [2] but for matrix operators with smooth symbol the bound for $\mu$ remains $\mu =\rho -\delta$.

In many applications operators with symbol of limited smoothness are involved. Let us consider $p(x,\xi )$ in the class $C^sS^m_{1,0}$ of symbols with $C^s$ regularity in the space variable x defined by

$$\begin{aligned} \Vert \partial _\xi ^\alpha p(x,\xi )\Vert _{C^s}\le C_{\alpha }\langle \xi \rangle ^{m-\alpha |}. \end{aligned}$$

(1.4)

For any fixed $\delta \in ]0,1[$ one can regularize the symbol obtaining a splitting

$$\begin{aligned} p(x,\xi )=p^\sharp (x,\xi )+p^b(x,\xi ), \ \ p^\sharp (x,\xi )\in S^m_{1,\delta }, \ p^b(x,\xi )\in C^sS^{m-s\delta }_{1,\delta }, \end{aligned}$$

(1.5)

e.g. Taylor [9]. If $p'$ is positive semidefinite, then the Hermitian part of $p^\sharp (x,\xi )$ is positive semidefinite as well. Applying (1.2) to $P^\sharp (x,D_x)$ and using the boundedness

$$\begin{aligned} P^b(x, D_x): H^m\rightarrow H^{m-s\delta }, \end{aligned}$$

the sharp Gårding inequality (1.2) for $P(x,D_x)$ holds true with a order

$$\begin{aligned} \mu \le 1-\delta , \ \mu \le s\delta . \end{aligned}$$

Negotiating on $\delta$ as done in [9], one obtains (1.2) for $p(x,\xi )\in C^sS^m_{1,0}$ with

$$\begin{aligned} \mu = \frac{s}{s+1}. \end{aligned}$$

(1.6)

Taylor’s bound (1.6) gives $\mu \rightarrow 1$ for $s\rightarrow \infty$ but it is not optimal. By means of the paradifferential calculus, Bony [1] proved that the best possible bound $\mu =1$ is achieved already for $s=2$. For $0<s<2$ Bony obtained the bound $\mu =s/2$ which is better than Taylor’s one for $1<s<2$ but it is worse for $0<s<1$.

Conjugating the operator with the FBI transform, Taturu [8] proved a generalization of the Sharp Gårding inequality for regular symbols from which he obtained also inequality (1.2) for symbols $p(x,\xi )\in C^sS^m_{1,0}$ with

$$\begin{aligned} \mu =\mu ^*(s)= {\left\{ \begin{array}{ll} 1,\ s\ge 2,\\ 2s/(s+2), \ 0<s<2. \end{array}\right. } \end{aligned}$$

(1.7)

We believe this one the optimal estimate for $C^s$ symbols, agreeing with Tataru.

Our aim is to show that a generalization of the Sharp Gårding inequality for regular symbols, sufficient to get $\mu =\mu ^*(s)$ in the case of $C^s$ limited smoothness, can be proved directly from Friedrichs symmetrization, that is going back to the original proofs of (1.2) in [3, 5, 6].

As in Tataru’s result, what is really important is the order of $\partial _x^\beta p(x,\xi )$ with $|\beta |=2$, let us denote $m+m_2$ this order. From $p(x,\xi )\in S^m_{\rho ,\delta }$ clearly we have $m_2\le 2\delta$. In case of equality one can not obtain better than $\mu =\rho -\delta$ in (1.2) but we can improve this bound in the case $m_2<2\delta$. As we will see later on, this is exactly what happens for $p^\sharp (x,\xi )\in S^m_{1,\delta }$ in the splitting (1.5) of $p(x,\xi )\in C^sS^m_{1,0}$.

For sake of simplicity, from now on we take $\rho =1$ which is the case of our interest. Here we prove the following generalization of inequality (1.2) for regular symbols.

Theorem 1.1

Let $P=p(x,D_x)=(P_{jk})$ be an $\ell \times \ell$ matrix of operators $P_{jk}=p_{jk}(x,D_x)$ with matrix symbol $p(x,\xi )=(p_{jk}(x,\xi ))\in S^m_{1,\delta }$, $0\le \delta <1$, and such that

$$\begin{aligned} \partial _x^\beta p(x,\xi )\in S^{m+m_1}_{1,\delta }, \ |\beta |=1;\ \ \ \partial _x^\beta p(x,\xi )\in S^{m+m_2}_{1,\delta }, \ |\beta |=2. \end{aligned}$$

(1.8)

Assume that the Hermitian part $p'=(p+p^*)/2$ of $p(x,\xi )$ is positive semidefinite.

Then, there exists $C>0$ such that

$$\begin{aligned} \Re (Pu,u)\ge -C\Vert u\Vert ^2_{(m-\mu ^\sharp )/2} \end{aligned}$$

(1.9)

for every $u\in {\mathcal S}$, with

$$\begin{aligned} \mu ^\sharp = {\left\{ \begin{array}{ll} \min \{1-m_1, 1-m_2/2\}, \ 2\delta -1\le m_2/2,\\ \min \{1-m_1, 2(1-\delta )\},\ \ \ 2\delta -1> m_2/2. \end{array}\right. } \end{aligned}$$

(1.10)

For the largest possible $m_2=2\delta$ of course we have $2\delta -1< m_2/2$ hence the general bound $\mu ^\sharp =1-\delta$. The same we have with $m_1=\delta$ and any $m_2\le 2\delta$.

With $m_2<2\delta$ and $m_1<\delta$ there is a gain. For instance, for $m_1=m_2=0$ we have $\mu ^\sharp =1$ for $0\le \delta \le 1/2$ and $\mu ^\sharp =2(1-\delta )$ for $1/2<\delta <1$. Spending such a gain we can prove the result for symbols of limited smoothness.

Theorem 1.2

Let $P=p(x,D_x)=(P_{jk})$ be an $\ell \times \ell$ matrix of operators with symbol $p(x,\xi )=(p_{jk}(x,\xi ))\in C^sS^m_{1,0}$. Assume that the Hermitian part $p'=(p+p^*)/2$ of $p(x,\xi )$ is positive semidefinite.

Then, there exists $C>0$ such that

$$\begin{aligned} \Re (Pu,u)\ge -C\Vert u\Vert ^2_{(m-\mu ^*(s))/2} \end{aligned}$$

(1.11)

for every $u\in {\mathcal S}$, with

$$\begin{aligned} \mu ^*(s)= {\left\{ \begin{array}{ll} 1,\ s\ge 2,\\ 2s/(s+2), \ 0<s<2. \end{array}\right. } \end{aligned}$$

(1.12)

2 Proof of Theorem 1.1

We follow the proof of Friedrichs [3] and Kumano-go [5, 6].

Let $p(x,\xi )\in S^m_{1,\delta }$ and for $\delta '\ge 2\delta -1$, $\tau =(1+\delta ')/2$ $(\ge \delta )$ let us consider

$$\begin{aligned} p_0(x,\xi )=\int p(x,\xi +\sigma \langle \xi \rangle ^\tau )q(\sigma )^2d\sigma \end{aligned}$$

(2.1)

where $q(\sigma )\ge 0$ is a smooth function of $\sigma \in {\mathbb R}^n$ with support for $|\sigma |<1$, $q(\sigma )=q(-\sigma )$, $\int q(\sigma )^2d\sigma =1$.

In the original proof $\tau =(1+\delta )/2$ that is $\delta '=\delta$ from the beginning. We take some advantage by fixing $\delta '\in [2\delta -1,1[$ related to $m_2$ later on.

Performing a change of variable in the integral (2.1) we have

$$\begin{aligned} p_0(x,\xi )=\int p(x,\zeta )F(\xi ,\zeta )^2d\zeta \end{aligned}$$

(2.2)

with

$$\begin{aligned} F(\xi ,\zeta )= q((\zeta -\xi )\langle \xi \rangle ^{-\tau })\langle \xi \rangle ^{-\tau n/2}. \end{aligned}$$

(2.3)

To obtain a symmetric operator, we introduce the double symbol $p_F(\xi ,x',\xi ')$, such that $p_F(\xi ,x,\xi )=p_0(x,\xi )$, defined by

$$\begin{aligned} p_F(\xi ,x',\xi ')=\int F(\xi ,\zeta )p(x',\zeta )F(\xi ',\zeta )d\zeta . \end{aligned}$$

(2.4)

We denote again $p_F(x,\xi )$ the simplified symbol of the operator $P_F(x, D_x)$.

If the matrix is $p(x,\xi )$ is positive semidefinite, then $P_F$ is a positive operator:

$$\begin{aligned} (P_Fu,u)\ge 0, \ u\in {\mathcal S}, \end{aligned}$$

see Theorem 4.3 in [6].

Taking $\tau =(1+\delta ')/2>\delta$ (this is the case with the original choice $\delta '=\delta$ of [6]), from the proof of Theorem 4.2 in [6] we have that the simplified symbol $p_F(x,\xi )$ of the operator $P_F$ belongs to the class $S^m_{1,\delta }$ and has an asymptotic expansion

$$\begin{aligned}{} & {} p_F(x,\xi )\sim p(x,\xi )+\sum _{|\beta |=1}\psi _\beta (\xi )p_{(\beta )}(x,\xi )+\sum _{|\alpha +\beta |\ge 2}\psi _{\alpha ,\beta }(\xi )p_{(\beta )}^{(\alpha )}(x,\xi ),\nonumber \\{} & {} \qquad \quad \qquad \quad \psi _\beta \in S^{-1},\ \psi _{\alpha ,\beta }\in S^{\tau (|\alpha |-|\beta |)}. \end{aligned}$$

(2.5)

Looking at the orders of $\psi _\beta$ and $\psi _{\alpha ,\beta }$ and at the orders of $\partial _x^\beta p(x,\xi )$ for $|\beta |\le 2$ in (1.8), from the above expansion we get

$$\begin{aligned}{} & {} p(x,\xi )=p_F(x,\xi )+p_1(x,\xi )+p_2(x,\xi ),\nonumber \\{} & {} p_1(x,\xi )\in S_{1,\delta }^{m-(1-m_1)},\ p_2(x,\xi )\in S_{1,\delta }^{m-\mu _2},\nonumber \\{} & {} \mu _2=\mu _2(\delta ')=\min \{1-\delta ', 1+\delta '-m_2\}. \end{aligned}$$

(2.6)

In the limit case $\tau =(1+\delta ')/2=\delta$ the proof of Thoerem 4.2 in [6] still gives (2.6) with the difference $p_2(x,\xi )\in S_{\delta ,\delta }^{m-\mu _2}$ instead of $S_{1,\delta }^{m-\mu _2}$ and what we loose in this case is the complete asymptotic expansion (2.5) which is not essential for our aims.

The positivity of the operator $P_F$ and the orders of $P_1$, $P_2$ in the splitting (2.6) yield inequality (1.2) for $P=P_F+P_1+P_2$ with

$$\begin{aligned} \mu \le \min \{1-m_1,\mu _2\}. \end{aligned}$$

The order of $P_1$ gives the bound $\mu ^\sharp \le 1-m_1$ for $\mu ^\sharp$ in (1.10). Then, we have to maximize $\mu _2=\mu _2(\delta ')$ in (2.6) for $2\delta -1\le \delta '<1$ in order to get the best possible second bound. Since

$$\begin{aligned} \max _{2\delta -1\le \delta '<1}\mu _2(\delta ')= {\left\{ \begin{array}{ll} 1-m_2/2,\ 2\delta -1\le m_2/2,\\ 2-2\delta ,\ 2\delta -1>m_2/2, \end{array}\right. } \end{aligned}$$

we complete the proof of Theorem 1.1.

3 Proof of Theorem 1.2

Let us show how Theorem 1.1 implies Theorem 1.2. Coming back to the splitting (1.5) of $p(x,\xi )\in C^sS^m_{1,0}$, now we have to negotiate between $\mu ^\sharp =\mu ^\sharp (\delta ,m_1,m_2)$ of Theorem 1.1 for $p^\sharp$ and $s\delta$. We obtain the optimal bound $\mu =\mu ^*(s)$ for

$$\begin{aligned} \mu ^\sharp =s\delta . \end{aligned}$$

We use the more precise estimates for the regularized part $p^\sharp$

$$\begin{aligned} \partial _x^\beta p^\sharp \in S^m_{1,\delta }, \ |\beta |\le s; \ \ \partial _x^\beta p^\sharp \in S^{m+\delta (|\beta |-s)}_{1,\delta }, \ |\beta |> s, \end{aligned}$$

(3.1)

given by Proposition 1.3.D in [9]. This means, with our notation,

$$\begin{aligned} m_1=m_1(s)= {\left\{ \begin{array}{ll} 0, \ s\ge 1,\\ \delta (1-s),\ 0< s<1, \end{array}\right. } \end{aligned}$$

(3.2)

and

$$\begin{aligned} m_2=m_2(s)= {\left\{ \begin{array}{ll} 0, \ s\ge 2,\\ \delta (2-s),\ 0< s<2. \end{array}\right. } \end{aligned}$$

(3.3)

We have $m_1\le m_2/2$ in any case. In particular $m_1$ does not influence $\mu ^\sharp$ and (1.10) for $p^\sharp$ reduces to

$$\begin{aligned} \mu ^\sharp = {\left\{ \begin{array}{ll} 1-m_2/2, \ 2\delta -1\le m_2/2,\\ 2(1-\delta ),\ 2\delta -1> m_2/2. \end{array}\right. } \end{aligned}$$

(3.4)

Here the best choice, if it is possible to fix $\delta$ such that $2\delta -1\le m_2/2$, is always $\mu ^\sharp =1-m_2/2$.

For $s\ge 2$ we have $m_2=0$ in (3.3). Choosing $\delta =1/s$, ($2\delta -1\le m_2/2$ reads exactly $s\ge 2$), we have

$$\begin{aligned} \mu ^\sharp =1-m_2/2=1=s\delta \end{aligned}$$

(3.5)

and the best possible bound $\mu =\mu ^*(s)=1$ is achieved in (1.12).

For $0<s< 2$ we have $m_2=\delta (2-s)$ in (3.3). Choosing $\delta =2/(s+2)$ we have $2\delta -1= m_2/2$ and

$$\begin{aligned} \mu ^\sharp =1-m_2/2=2s/(s+2)=s\delta \end{aligned}$$

(3.6)

that leads to $\mu =\mu ^*(s)=2s/(s+2)$ in (1.12).

This completes the proof of Theorem 1.2.