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6 JÉRÔME LE ROUSSEAU AND GILLES LEBEAU We shall compose ψDOs in the sequel. Such compositions yield a calculus at the level of operator symbols. Theorem 2.8 (Symbol calculus). Let a ∈ S m and b ∈ S m′ . Then Op(a) ◦ Op(b) = Op(c) for a certain c ∈ S m+m′ that admits the following asymptotic expansion c(x, ξ, h) = (a ♯ b)(x, ξ, h) ∼ ∑ α h |α| i |α| α! ∂α ξ a(x, ξ, h) ∂α xb(x, ξ, h), where α! = α 1 ! · · · α n ! The first term in the expansion, the principal symbol, is ab; the second term is h i It follows that the principal symbol of the commutator [Op(a), Op(b)] is σ([Op(a), Op(b)]) = h i {a, b} ∈ hS m+m′ −1 . Finally, the symbol of the adjoint operator can be obtained as follows. ∑ j ∂ ξ j a(x, ξ, h) ∂ x j b(x, ξ, h). Theorem 2.9. Let a ∈ S m . Then Op(a) ∗ = Op(b) for a certain b ∈ S m that admits the following asymptotic expansion b(x, ξ, h) ∼ ∑ h |α| α i |α| α! ∂α ξ ∂α xa(x, ξ, h). The principal symbol of b is simply a. For references on usual ψDOs the reader can consult [Tay81, Hör85b, AG91, GS94, Shu01]. For references on semi-classical ψDOs the reader can consult [Rob87, DS99, Mar02]. 3. LOCAL CARLEMAN ESTIMATES FOR ELLIPTIC OPERATORS We shall prove a local Carleman estimates for a second-order elliptic operator. To simplify notation we consider the Laplace operator P = −∆ but the method we expose extends to more general second-order elliptic operators with a principal part of the form ∑ i, j ∂ j (a i j (x)∂ i ) with a i j ∈ C ∞ (R n , R), 1 ≤ i, j ≤ n and ∑ i, j a i j (x)ξ i ξ j ≥ C|ξ| 2 , with C > 0, for all x, ξ ∈ R n . In particular, we note that Carleman estimates are insensitive 3 to changes in the operator by zero- or first-order terms. Let ϕ(x) be a real-valued function. We define the following conjugated operator P ϕ = h 2 e ϕ/h Pe −ϕ/h to be considered as a semi-classical differential operator. We have P ϕ = −h 2 ∆ − |ϕ ′ | 2 + 2〈ϕ ′ , h∇〉 + h∆ϕ. Its full symbol is given by |ξ| 2 − |ϕ ′ | 2 + 2i〈ϕ ′ , ξ〉 + h∆ϕ. Its principal symbol is given by p ϕ = σ(P ϕ ) = |ξ| 2 − |ϕ ′ | 2 + 2i〈ϕ ′ , ξ〉 = ∑ (ξ j + iϕ ′ x j ) 2 , j i.e., we have “replaced” ξ j by ξ j + iϕ ′ x j . In fact we note that the symbol of e ϕ/h D j e −ϕ/h is ξ j + iϕ ′ x j . We define the following symmetric operators Q 2 = (P ϕ + P ∗ ϕ)/2, Q 1 = (P ϕ − P ∗ ϕ)/(2i), with respective principal symbols q 2 = |ξ| 2 − |ϕ ′ | 2 , q 1 = 2〈ξ, ϕ ′ 〉. We have p ϕ = q 2 + iq 1 and P ϕ = Q 2 + iQ 1 . We choose ϕ that satisfies the following assumption. Assumption 3.1 (L. Hörmander [Hör63, Hör85a]). Let V be a bounded open set in R n . We say that the weight function ϕ ∈ C ∞ (R n , R) satisfies the sub-ellipticity assumption in V if |ϕ ′ | > 0 in V and ∀(x, ξ) ∈ V × R n , p ϕ (x, ξ) = 0 ⇒ {q 2 , q 1 }(x, ξ) ≥ C > 0. Remark 3.2. We note that p ϕ (x, ξ) = 0 is equivalent to |ξ| = |ϕ ′ | and 〈ξ, ϕ ′ 〉 = 0. In particular, the characteristic set Z = {(x, ξ) ∈ V × R n ; p ϕ (x, ξ) = 0} is compact as illustrated in Figure 1. Assumption 3.1 can be fulfilled as stated in the following lemma whose proof can be found in Appendix A. 3 In the sense that only constants are affected. In Theorem 3.5 below the constants C and h1 change but not the form of the estimate.
CARLEMAN ESTIMATES 7 ϕ ′ (x) 0 p ϕ = 0 Figure 1: Form of the characteristic set Z at the vertical of each point x ∈ V. Lemma 3.3 (L. Hörmander [Hör63, Hör85a]). Let V be a bounded open set in R n and ψ ∈ C ∞ (R n , R) be such that |ψ ′ | > 0 in V. Then ϕ = e λψ fulfills Assumption 3.1 in V for λ > 0 sufficiently large. The proof of the Carleman estimate will make use of the Gårding inequality. In preparation, we have the following result proven in Appendix A that follows from Assumption 3.1. Lemma 3.4. Let µ > 0 and ρ = µ(q 2 2 + q2 1 ) + {q 2, q 1 }. Then, for all (x, ξ) ∈ V × R n , we have ρ(x, ξ) ≥ C〈ξ〉 4 , with C > 0, for µ sufficiently large. We may now prove the following Carleman estimate. Theorem 3.5. Let V be a bounded open set in R n and let ϕ satisfy Assumption 3.1 in V; then, there exist 0 < h 1 < h 0 and C > 0 such that (3.1) h‖e ϕ/h u‖ 2 0 + h 3 ‖e ϕ/h ∇ x u‖ 2 0 ≤ Ch 4 ‖e ϕ/h Pu‖ 2 0, for u ∈ C ∞ c (V) and 0 < h < h 1 . Proof. We set v = e ϕ/h u. Then, Pu = f is equivalent to P ϕ v = g = h 2 e ϕ/h f or rather Q 2 v + iQ 1 v = g. Observing that (Q j w 1 , w 2 ) = (w 1 , Q j w 2 ) for w 1 , w 2 ∈ Cc ∞ (R n ) we then obtain (( (3.2) ‖g‖ 2 0 = ‖Q 1v‖ 2 0 + ‖Q 2v‖ 2 0 + 2 Re(Q 2v, iQ 1 v) = Q 2 1 + Q 2 2 + i[Q 2, Q 1 ] ) v, v) . We choose µ > 0 as given in Lemma 3.4. Then, for h such that hµ ≤ 1 we have (( h µ(Q 2 1 + Q 2 2 ) + i h [Q 2, Q 1 ] ) ) v, v ≤ ‖g‖ 2 0 . }{{} principal symbol = µ(q 2 1 +q2 2 )+{q 2,q 1 } The Gårding inequality and Lemma 3.4 then yield (3.3) h‖v‖ 2 2 ≤ C‖g‖2 0 . We content 4 ourselves with the norm in H 1 here and we obtain h‖e ϕ/h u‖ 2 0 + h 3 ‖∇ x (e ϕ/h u)‖ 2 0 ≤ Ch 4 ‖e ϕ/h f ‖ 2 0. We write ∇ x (e ϕ/h u) = h −1 e ϕ/h (∇ x ϕ)u + e ϕ/h ∇ x u, which yields since |∇ x ϕ| ≤ C. This concludes the proof. h 3 ‖e ϕ/h ∇ x u‖ 2 0 ≤ Ch‖e ϕ/h u‖ 2 0 + Ch 3 ‖∇ x (e ϕ/h u)‖ 2 0, 4 Note that in the elliptic region, e.g. for large |ξ|, we can obtain a better result without the factor h in (3.3). In the neighborhood of the characteristic set Z = {p ϕ = 0} the choice of the norm in H 1 or H 2 matters very little since this region is compact. See Proposition 3.8 for more details.
Page 1 and 2: ON CARLEMAN ESTIMATES FOR ELLIPTIC
Page 3 and 4: CARLEMAN ESTIMATES 3 The approach o
Page 5: CARLEMAN ESTIMATES 5 Lemma 2.2. Let
Page 9 and 10: CARLEMAN ESTIMATES 9 Proposition 3.
Page 11 and 12: CARLEMAN ESTIMATES 11 exists R > 0
Page 13 and 14: CARLEMAN ESTIMATES 13 Proof. We app
Page 15 and 16: CARLEMAN ESTIMATES 15 We start with
Page 17 and 18: CARLEMAN ESTIMATES 17 These conditi
Page 19 and 20: CARLEMAN ESTIMATES 19 where the com
Page 21 and 22: CARLEMAN ESTIMATES 21 The ψDO symb
Page 23 and 24: CARLEMAN ESTIMATES 23 If we denote
Page 25 and 26: CARLEMAN ESTIMATES 25 We note that
Page 27 and 28: CARLEMAN ESTIMATES 27 Next since (A
Page 29 and 30: CARLEMAN ESTIMATES 29 for all φ

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