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

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.


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 × matrix of operators P jk = p jk (x, D x ) with matrix symbol p(x, ξ) = ( p jk (x, ξ)) ∈ S m ρ,δ , 0 ≤ δ < ρ ≤ 1, that is satisfying Assume that the Hermitian part p = ( p + p * )/2 of p(x, ξ) is positive semidefinite. Then, there exists C > 0 such that 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.
In many applications operators with symbol of limited smoothness are involved. Let us consider p(x, ξ) in the class C s S m 1,0 of symbols with C s regularity in the space variable x defined by For any fixed δ ∈]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 (x, ξ) is positive semidefinite as well. Applying (1.2) to P (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 δ as done in [9], one obtains (1.2) for p(x, ξ) ∈ C s S m 1,0 with μ = s s + 1 . (1.6) Taylor's bound (1.6) gives μ → 1 for s → ∞ but it is not optimal. By means of the paradifferential calculus, Bony [1] proved that the best possible bound μ = 1 is achieved already for s = 2. For 0 < s < 2 Bony obtained the bound μ = 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 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 μ = μ * (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
ρ,δ clearly we have m 2 ≤ 2δ. In case of equality one can not obtain better than μ = ρ − δ in (1.2) but we can improve this bound in the case m 2 < 2δ. As we will see later on, this is exactly what happens for p (x, ξ) ∈ S m 1,δ in the splitting (1.5) of p(x, ξ) ∈ C s S m 1,0 . For sake of simplicity, from now on we take ρ = 1 which is the case of our interest. Here we prove the following generalization of inequality (1.2) for regular symbols.
For the largest possible m 2 = 2δ of course we have 2δ − 1 < m 2 /2 hence the general bound μ = 1 − δ. The same we have with m 1 = δ and any m 2 ≤ 2δ.
Performing a change of variable in the integral (2.1) we have To obtain a symmetric operator, we introduce the double symbol p F (ξ, x , ξ ), such that p F (ξ, x, ξ) = p 0 (x, ξ), defined by We denote again p F (x, ξ) the simplified symbol of the operator P F (x, D x ). If the matrix is p(x, ξ) is positive semidefinite, then P F is a positive operator: see Theorem 4.3 in [6]. Taking τ = (1 + δ )/2 > δ (this is the case with the original choice δ = δ of [6]), from the proof of Theorem 4.2 in [6] we have that the simplified symbol p F (x, ξ) of the operator P F belongs to the class S m 1,δ and has an asymptotic expansion Looking at the orders of ψ β and ψ α,β and at the orders of ∂ β x p(x, ξ) for |β| ≤ 2 in (1.8), from the above expansion we get p(x, ξ) = p F (x, ξ) + p 1 (x, ξ) + p 2 (x, ξ), (2.6) In the limit case τ = (1 + δ )/2 = δ the proof of Thoerem 4.2 in [6] still gives (2.6) with the difference p 2 (x, ξ) ∈ S m−μ 2 δ,δ instead of S m−μ 2 1,δ 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 μ ≤ 1 − m 1 for μ in (1.10). Then, we have to maximize μ 2 = μ 2 (δ ) in (2.6) for 2δ − 1 ≤ δ < 1 in order to get the best possible second bound. Since we complete the proof of Theorem 1.1.

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, ξ) ∈ C s S m 1,0 , now we have to negotiate between μ = μ (δ, m 1 , m 2 ) of Theorem 1.1 for p and sδ. We obtain the optimal bound μ = μ * (s) for μ = sδ.
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