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.
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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
Assume that the Hermitian part \(p'=(p+p^*)/2\) of \(p(x,\xi )\) is positive semidefinite. Then, there exists \(C>0\) such that
for every \(u\in {\mathcal S}\). In particular, for \(p(x,\xi )\in S^m_{1,0}\) we have
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
For any fixed \(\delta \in ]0,1[\) one can regularize the symbol obtaining a splitting
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
the sharp Gårding inequality (1.2) for \(P(x,D_x)\) holds true with a order
Negotiating on \(\delta \) as done in [9], one obtains (1.2) for \(p(x,\xi )\in C^sS^m_{1,0}\) with
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
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
Assume that the Hermitian part \(p'=(p+p^*)/2\) of \(p(x,\xi )\) is positive semidefinite.
Then, there exists \(C>0\) such that
for every \(u\in {\mathcal S}\), with
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
for every \(u\in {\mathcal S}\), with
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
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
with
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
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:
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
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
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
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
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
We use the more precise estimates for the regularized part \(p^\sharp \)
given by Proposition 1.3.D in [9]. This means, with our notation,
and
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
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
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
that leads to \(\mu =\mu ^*(s)=2s/(s+2)\) in (1.12).
This completes the proof of Theorem 1.2.
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Cicognani, M. A direct proof of the Sharp Gårding inequality for symbols with limited smoothness. J. Pseudo-Differ. Oper. Appl. 14, 2 (2023). https://doi.org/10.1007/s11868-022-00498-z
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DOI: https://doi.org/10.1007/s11868-022-00498-z