Step 3: A complex analysis argument

We also need to estimate the terme^τ(ψ−d)P σ_2Rσ_R,λχ_δ,λ(ψ)χe_δ(ψ)u: we have

e^τ(ψ−d)P σ2RσR,λχδ,λ(ψ)χeδ(ψ)u

₀ ≤

e^τ(ψ−d)σ2RσR,λχδ,λ(ψ)χeδ(ψ)P u ₀ +

e^τ(ψ−d)[σ_2Rσ_R,λχ_δ,λ(ψ)χe_δ(ψ), P]u ₀

≤ Ce^−τdλ^1/2e^δτe^τ

2 λ

kP uk_L2(B(x⁰,4R))+kuk_m−1

≤ Cµ^1/2e^−7δτe^Cτ

2 µ

kP uk_L2(B(x⁰,4R))+kuk_m−1

(3.28) where we have used several times Lemma 2.13 toχδ,λ(ψ)or some of its derivatives of order less thanm−1. So, the Carleman estimate (3.12) applied to σ2RσR,λχδ,λ(ψ)χeδ(ψ)u, together with (3.18), (3.19), (3.27) and (3.28) gives for all τ0≤τ,µlarge enough,λsuch that (3.16) holds, the sought estimate (3.26).

Proof. We now follow [Hör97, proposition 2.1]. For any test function f ∈S(Rⁿ), we dene the following distribution (with β >0to be chosen later on)

hh_f, wi_E0(R),C^∞(R):=h(M^βµf)σ_2Rσ_R,λχ_δ,λ(ψ)χe_δ(ψ)u, w(ψ)i_E0(Rⁿ),C^∞(Rⁿ). We choose the particular test functions w=η_δ,λ, and want to estimate the quantity

hhf, ηδ,λi_E0(R),C^∞(R)=h(M^βµf)σ2RσR,λχδ,λ(ψ)χeδ(ψ)u, ηδ,λ(ψ)i_E0(Rⁿ),C^∞(Rⁿ)

=hM^βµσ2RσR,λχδ,λ(ψ)χeδ(ψ)ηδ,λ(ψ)u, fi_S⁰_(Rn),S(Rⁿ),

uniformly with respect tof to nally obtain an estimate onkM^βµσ_2Rσ_R,λχ_δ,λ(ψ)χe_δ(ψ)η_δ,λ(ψ)ukm−1. As the Fourier transform of a compactly supported distribution, ˆh_f is an entire function satisfying

ˆhf(ζ) =h(M^βµf)σ2RσR,λχδ,λ(ψ)χeδ(ψ)u, e^−iζψi_E⁰(Rⁿ),C^∞(Rⁿ)

=hσ2Rσ_R,λχ_δ,λ(ψ)χe_δ(ψ)u, e^−iζψ(M^βµf)i_E⁰(Rⁿ),C^∞(Rⁿ)

=he^−iζψσ_2Rσ_R,λχ_δ,λ(ψ)χe_δ(ψ)u,(M^βµf)i_E0(Rⁿ),C^∞(Rⁿ), ζ∈C. Usingsupp(σ2R)⊂B(x⁰,4R), we have the a priori estimate

|ˆhf(ζ)|=|he^−iζψσ2RσR,λχδ,λ(ψ)χeδ(ψ)u,(M^βµf)i_E⁰(Rⁿ),C^∞(Rⁿ)|

≤ ke^−iζψσ_2Rσ_R,λχ_δ,λ(ψ)χe_δ(ψ)ukm−1k(M^βµf)k1−m

≤Ch|ζ|i^m−1e^{|Im(ζ)|kψkL∞(B(x0,4R))}kukm−1kfk1−m, ζ∈C. (3.29) Next, forζ∈R, we have

|ˆh_f(ζ)|=|he^−iζψσ_2Rσ_R,λχ_δ,λ(ψ)χe_δ(ψ)u,(M^βµf)i_E⁰(Rⁿ),C^∞(Rⁿ)| ≤Chζi^m−1kukm−1kfk1−m, ζ∈R. (3.30) Finally, forζ∈iR⁺,ζ=iτ withτ >0, we have

|ˆhf(iτ)|=|h(M^βµf), e^{τ ψ}σ2RσR,λχδ,λ(ψ)χeδ(ψ)ui_C^∞_(Rn),E⁰(Rⁿ)|

=|he^2τε|Da|2(M^βµf), e^{−2τε|Da|2}e^{τ ψ}σ2RσR,λχδ,λ(ψ)χeδ(ψ)ui_S(Rn),S⁰(Rⁿ)|

≤ ke^2τε|Da|2M^βµfk1−mke^{−2τε|Da|2}e^{τ ψ}σ2RσR,λχδ,λ(ψ)χeδ(ψ)ukm−1

≤e^2τεβ2µ2kfk1−mkQ^ψ_ε,τσ2RσR,λχδ,λ(ψ)χeδ(ψ)ukm−1,

as |ξa| ≤βµonsupp(m^βµ). Using (3.26), we obtain for all τ≥τ₀,µ≥1, ¹_Cµ≤λ≤Cµ,

|ˆh_f(iτ)| ≤ Ce^2τεβ2µ2kfk_1−m µ^1/2e^Cτ

2

µ e^δτkP uk_B(x0,4R)+e^2δτ

M_λ^2µϑ_λu _m−1

+Cµ^1/2τ^N

e^−8δτ+e^−εµ

2

8τ +e^δτ−cµ

e^Cτ

2

µe^δτkuk_m−1

! . Now, we choose

λ= 2c₁µ, and to simplify the notation we write, forκ >0,

D=e^κµ

M_λ^2µϑλu

m−1+kP uk_B(x0,4R)

. With this notation, we have

|ˆh_f(iτ)| ≤ Ce^2τεβ2µ2kfk_1−m µ^1/2e^Cτ

2

µ e^δτe^−κµD+e^2δτe^−κµD

+µ^1/2τ^N

e^−8δτ+e^−εµ

2

8τ +e^δτ−cµ

e^Cτ

2

µe^δτkuk_m−1

!

≤ Cµ^1/2τ^Ne^2τεβ2µ2e^Cτ

2

µ e^2δτ(D+kuk_m−1)kfk1−m

e^−cµ+e^−εµ

2

8τ +e^−9δτ

, (3.31)

where the new constantc >0may depend onκ.

We now come back to the quantity we want to estimate:

hM^βµσ2RσR,λχδ,λ(ψ)χeδ(ψ)ηδ,λ(ψ)u, fi_S⁰_(Rn),S(Rⁿ)=hhf, ηδ,λi_E⁰_(R),C^∞_(R)= Z

R

ˆhf(ζ)ˆηδ,λ(−ζ)dζ.

As η_δ ∈ C₀^∞(−4δ, δ), the function ηˆ_δ is holomorphic in the lower complex half-plane together with the estimate

|ˆηδ(ζ)| ≤Ce^−4δIm(ζ), for Im(ζ)≤0, that is,

|ˆηδ(−ζ)| ≤Ce^4δIm(ζ), for Im(ζ)≥0, (3.32)

|ηˆδ,λ(−ζ)|=|e^−ζ

2

ληˆδ(−ζ)| ≤CeIm(ζ)2−Re(ζ)2

λ e^4δIm(ζ), for Im(ζ)≥0, (3.33) For a constant0< d≤1(beware that thisdis not the same asdappearing in the Carleman estimate) to be chosen later on, we split the integral in three parts according to

Z

R

ˆh_f(ζ)ˆη_δ,λ(−ζ)dζ =I₋+I₀+I₊, with

I₋:=

Z −dµ

−∞

ˆhf(ζ)ˆηδ,λ(−ζ)dζ, I0:=

Z dµ

−dµ

ˆhf(ζ)ˆηδ,λ(−ζ)dζ, I+:=

Z +∞

dµ

ˆhf(ζ)ˆηδ,λ(−ζ)dζ.

According to (3.30) and (3.33), we have, forµ≥1, λ= 2c₁µ,

|I±| ≤ C Z +∞

dµ

e^−|ζ|

2

λ hζi^m−1kukm−1kfk1−mdζ≤Cµ^2me^−d2µ

2

λ kukm−1kfk1−m

≤ Cde^−cd2µkuk_m−1kfk_1−m. (3.34)

So the main problem is to estimateI0. For this, let us dene

H(ζ) =µ^−1/2(ζ+i)^−Ne^i2δζˆhf(ζ).

From (3.31), we have the estimate on the imaginary axis for allτ ≥τ₀, forµ≥1,λ= 2c₁µ,

|H(iτ)| ≤ Ce^2τεβ2µ2e^Cτ

2

µ (D+kuk_m−1)kfk_1−m

e^−cµ+e^−εµ

2

8τ +e^−9δτ

.

Moreover, (3.29) implies (we can assumeN ≥m−1without loss of generality)

|H(ζ)| ≤Ce^|Im(ζ)|(2δ+kψk_L∞(B(x0,4R)))kuk_m−1kfk_1−m, ζ∈C, Im(ζ)≥0.

Next, we deneH := ^H_c

0, with

c0=C(D+kuk_m−1)kfk_1−m, (3.35)

and apply Lemma 3.11 below to the functionH.

This Lemma implies the existence ofd0>0 (depending only on δ, κ,kψk_L^∞_(B(x0,4R)), εand the con- stants C, c appearing in the exponents of the estimates ofH(iτ)) such that for any d < d0, there exists β0 > 0, (depending on the same parameters, together with d) such that for any 0 < β < β0, for all µ≥ ^τ˜_β⁰ := ^{τ0(kψkL∞(B(x0,4R))+11δ)}

1 2

β , we have

|H(ζ)| ≤c0e^−8δIm(ζ), onQ₁∩ {d

4µ≤ |ζ| ≤2dµ},

withQ₁=R^∗++iR^∗+. The same procedure leads to the same estimate ifQ₁is replaced by the setR^∗−+iR^∗+, and hence, by the wholeC⁺={ζ∈C,Im(ζ)≥0}. Coming back toˆhf, we obtain

|hˆf(ζ)| ≤c0µ^1/2h|ζ|i^Ne^−6δIm(ζ)≤c0µ^N+1/2e^−6δIm(ζ), onC+∩ {d

4µ≤ |ζ| ≤2dµ}. (3.36) wherec₀ is dened in (3.35).

We now come back toI0. The functionˆhf(ζ)ˆηδ,λ(−ζ)being holomorphic inC+, we make the following change of contour in the complex plane:

I0= Z

Γ^V₊

ˆhf(ζ)ˆηδ,λ(−ζ)dζ+ Z

Γ^H

ˆhf(ζ)ˆηδ,λ(−ζ)dζ+ Z

Γ^V₋

hˆf(ζ)ˆηδ,λ(−ζ)dζ, where the contours (oriented counterclockwise) are dened by

Γ^V_± ={Re(ζ) =±dµ,0≤Im(ζ)≤dµ/2}, Γ^H ={−dµ≤Re(ζ)≤dµ,Im(ζ) =dµ/2}.

withd∈]0, d0[still to be chosen later on.

dµ 2dµ

Re(ζ)

dµ 2

dµ 4

Im(ζ)

−dµ

Γ^H

Γ^V₊ 0

Γ^V₋

Figure 5: Coutours of integration

SinceΓ^V₊∪Γ^H∪Γ^V₋ ⊂C+∩ {^d₄µ ≤ |ζ| ≤2dµ} and λ=c1µ, estimates (3.33) and (3.36) yields the estimate

|ˆh_f(ζ)ˆη_δ,λ(−ζ)| ≤ c₀µ^N+1/2e^−6δIm(ζ)e

Im(ζ)2−Re(ζ)2

2c1µ e^4δIm(ζ), ζ∈Γ^V₊∪Γ^H∪Γ^V₋

≤ c0µ^N+1/2e^−2δIm(ζ)e

Im(ζ)2−Re(ζ)2

2c1µ , ζ∈Γ^V₊∪Γ^H∪Γ^V₋ Using that ^3d₄²µ²≤Re(ζ)²−Im(ζ)²≤d²µ² forζ∈Γ^V₊∪Γ^V₋ we now obtain

|ˆhf(ζ)ˆηδ,λ(−ζ)| ≤ c0µ^N+1/2e^−2δIm(ζ)e^−3d

2µ

8c1 , ζ∈Γ^V₊∪Γ^V₋. OnΓ^H, we haveIm(ζ) =dµ/2, so , we can estimate

|ˆh_f(ζ)ˆη_δ,λ(−ζ)| ≤ c₀µ^N+1/2e^−δdµe^d

2 8c1µ

, ζ∈Γ^H Now, we can x0< d≤min(4c1δ, d0)so that we havee^−δdµe^d

2 8c1µ

≤Ce^−cµ(for some0< c≤2c1δ² ). As a consequence, we have

|I0|= Z

Γ^V₊∪Γ^H∪Γ^V₋

ˆhf(ζ)ˆηδ,λ(−ζ)dζ

≤ c0µ^N+1/2|Γ^V₊∪Γ^H∪Γ^V₋|e^−cµ

≤ Ce^−cµ(D+kuk_m−1)kfk_1−m, (3.37)

for any0< β < β₀, for allµ≥max(C,^˜τ_β⁰)(as|Γ^V₊∪Γ^H∪Γ^V₋|=Cdµ).

This, together with (3.34) yields, for any0< β < β₀, for allµ≥ ^˜τ_β⁰, hM^βµσ2RσR,λχδ,λ(ψ)χeδ(ψ)ηδ,λ(ψ)u, fi_S⁰(Rⁿ),S(Rⁿ)

=

Z

R

ˆhf(ζ)ˆηδ,λ(−ζ)dζ

≤ Ce^−cµ(D+kuk_m−1)kfk_1−m. The constants being uniform with respect tof ∈S(Rⁿ), this provides by duality the estimate

M^βµσ2RσR,λχδ,λ(ψ)χeδ(ψ)ηδ,λ(ψ)u

m−1≤Ce^−cµ(D+kuk_m−1), which concludes the proof of the lemma.

With Lemma 3.10, we can now conclude the proof of the local estimate of Theorem 3.1. Lemma 3.11 and its proof are postponed to the end of the section.

End of the proof of Theorem 3.1. Using Lemma 2.3 with m(2·)and1−m(·), we get

M

βµ 2

λ (1−M^βµ)

_Hm−1(Rⁿ)→H^m−1(Rⁿ)

≤Ce^−cλ.

Hence, applying Lemma 3.10, we obtain, for any 0< β < β0, for allµ≥ ^˜τ_β⁰ andλ= 2c1µ, M^βµσ2RσR,λχδ,λ(ψ)χeδ(ψ)ηδ,λ(ψ)u

_m−1 ≤

M

βµ 2

λ (1−M^βµ)σ2RσR,λχδ,λ(ψ)χeδ(ψ)ηδ,λ(ψ)u _m−1 +

M

βµ 2

λ M^βµσ2RσR,λχδ,λ(ψ)χeδ(ψ)ηδ,λ(ψ)u _m−1

≤ Ce^−cµ(D+kuk_m−1). (3.38)

Using Lemma 2.11, estimate (3.38) and the denition ofrin Corollary 3.6, we get for any0< β < β₀, for allµ≥ ^˜τ_β⁰ andλ= 2c₁µ,

M

βµ 4

λ σr,λu _m−1

≤

σr,λM

βµ 2

λ u _m−1

+Ce^−cµkuk_m−1

≤

σr,λM

βµ 2

λ σ2RσR,λχδ,λ(ψ)χeδ(ψ)ηδ,λ(ψ)u _m−1 +

σ_r,λM

βµ 2

λ 1−σ_2Rσ_R,λχ_δ,λ(ψ)χe_δ(ψ)η_δ,λ(ψ) u

m−1

+Ce^−cµkuk_m−1

≤ Ce^−cµ(D+kuk_m−1) +

σ_r,λM

βµ 2

λ 1−σ_2Rσ_R,λχ_δ,λ(ψ)η_δ,λ(ψ) u

m−1

. (3.39) We know that σR =χδ(ψ) =χeδ(ψ) =ηδ(ψ) = 1on a neighborhood of supp(σr) according to (3.15) and the properties of χ, χeδ and η. So, we can select Π ∈ C₀^∞(Rⁿ) such that Π = 1 on a neighborhood of supp(σr)and such that σ2R =σR =χδ(ψ) =χeδ(ψ) = ηδ(ψ) = 1on an neighborhood ofsupp(Π). Now, we have

σ_r,λM

βµ 2

λ 1−σ_2Rσ_R,λχ_δ,λ(ψ)χe_δ(ψ)η_δ,λ(ψ) u

m−1

≤

σ_r,λM

βµ 2

λ 1−σ_2Rσ_R,λχ_δ,λ(ψ)χe_δ(ψ)η_δ,λ(ψ)

(1−Π)u m−1

+

σr,λM

βµ 2

λ 1−σ2RσR,λχδ,λ(ψ)χeδ(ψ)ηδ,λ(ψ) Πu

m−1

. (3.40)

To estimate the rst term, we use Lemma 2.10 to obtain

σr,λM

βµ 2

λ (1−Π) _Hm−1

→H^m−1

≤Ce^−cµ. Con- cerning the second term, we have

σ_r,λM

βµ 2

λ 1−σ_2Rσ_R,λχ_δ,λ(ψ)χe_δ(ψ)η_δ,λ(ψ) Πu

m−1

≤C

1−σ2RσR,λχδ,λ(ψ)χeδ(ψ)ηδ,λ(ψ) Πu

m−1≤Ce^−cµkuk_m−1 (3.41) where we have decomposed in the last inequality

1−σ2RσR,λχδ,λ(ψ)χeδ(ψ)ηδ,λ(ψ) = (1−σ2R) +σ2R(1−σR,λ) +σ2RσR,λ(1−χδ,λ(ψ))

+σ2RσR,λχδ,λ(ψ)(1−χeδ(ψ)) +σ2RσR,λχδ,λ(ψ)χeδ(ψ)(1−ηδ,λ(ψ)) and used Lemmata 2.3 and 2.5. These two Lemmata can be applied thanks to the geometric fact that

dist(supp(Π),{x∈Rⁿ;σ2R(x)6= 1})>0,

and the same is true withσ_2Rreplaced byσ_R,χ_δ(ψ),χe_δ(ψ)orη_δ(ψ). We now have the existence ofτ˜₀>0 such that for any κ, c₁ >0, there existβ₀, C, c >0, such that for any0< β < β₀,µ≥ ^τ˜_β⁰ andλ= 2c₁µ, the following estimate holds:

M

βµ 4

λ σr,λu m−1

≤Ce^−cµ(D+kuk_m−1), D=e^κµ

M_λ^2µϑλu

_m−1+kP uk_B(x0,4R)

.

This concludes the proof of Theorem 3.1 withκ⁰=c, when replacingµandµ0byµ/2andµ0/2respectively.

It only remains to prove Lemma 3.11 below.

Lemma 3.11. Let δ, κ, R₀, C₁, ε, τ₀ > 0. Then, there exists d₀ = d₀(δ, κ, R₀, C₁, ε) such that for any d < d0, there existsβ0(δ, κ, R0, c1, ε, d)such that for any0< β < β0 and for allµ≥ ^τ0(R0+9δ)

1 2

β , we have the following statement:

For everyH holomorphic function inQ1=R^∗++iR^∗+, continuous on Q¯1 satisfying

|H(iτ)| ≤e^εβ

2 2τµ²e^C1τ

2

µ max(e^−κµ, e^−εµ

2

8τ , e^−9δτ) forτ∈[τ0,+∞), (3.42)

|H(ζ)| ≤e^R0Im(ζ) onQ¯₁, (3.43)

we have

|H(ζ)| ≤e^−8δIm(ζ) onQ¯1∩ {d

4µ≤ |ζ| ≤2dµ}. (3.44) The proof essentially consists in performing a scaling argument to get rid of the parameterµand then applying Lemma B.2.

Proof of Lemma 3.11. The functionH is holomorphic in Q1 and z7→log|z|is subharmonic on C^∗. As a consequence, the function

g^µ:ζ7→µ⁻¹log|H(µζ)|

is subharmonic onQ₁ (which is invariant by dilations). Assumption (3.42) (used forτ µ∈[τ₀,+∞)) yields g^µ(iτ)≤c1τ²+εβ²

τ + max(−κ,−9δτ,− ε

8τ), forτ∈[τ0

µ,+∞), (3.45)

and Assumption (3.43) yields

g^µ(ζ)≤R₀Im(ζ), onQ¯₁. (3.46)

Now, we set

f₁^µ(y) =R0y1[0,^τ_µ⁰)(y) +1[^τ_µ⁰,+∞)(y) min{R0y,max(−κ,−9δy,− ε

8y) +C1y²+εβ²

y }, y∈R+. (3.47) According to Lemma B.2, there exists d0 = d0(δ, κ, R0, ε, C1) such that for any d < d0, there exists β0(δ, κ, R0, d, ε, C1), such that for any 0< β < β0, and anyµ≥ ^τ0(R0+9δ)

1 2

β , the function f₁^µ is continuous and the associated function f^µ given by Lemma B.1 withf₀= 0andf₁=f₁^µ satises

f^µ∈C⁰( ¯Q₁), ∆f^µ= 0and|f^µ(x, y)| ≤C_µ(1 +|(x, y)|)inQ₁, f^µ=f₁^µ oniR+, f^µ= 0 onR+

together with

f^µ(ζ)≤ −8δIm(ζ) onQ¯1∩ {d

4 ≤ |ζ| ≤2d}.

This is

f^µ(ζ/µ)≤ −8δIm(ζ)/µ onQ¯1∩ {d

4µ≤ |ζ| ≤2dµ}. (3.48) Now, asg^µ is subharmonic and f^µ harmonic, the function

h^µ(ζ) :=g^µ(ζ)−f^µ(ζ)

is subharmonic too. As a consequence of (3.45), (3.46) and (3.47), we have h^µ(ζ)≤0 onR+∪iR+.

Moreover, (3.46) and|f^µ(ζ)| ≤C(1 +|ζ|)also yield

h^µ(ζ)≤C_µ+ (C_µ+R₀)|ζ|.

According to Lemma B.4, this implies

h^µ(ζ)≤0 onQ¯₁, and hence

|H(µζ)|=e^µgµ(ζ)≤e^µfµ(ζ) onQ¯1. Finally, coming back to (3.48), we obtain

|H(ζ)| ≤e^−8δIm(ζ) onQ¯1∩ {d

4µ≤ |ζ| ≤2dµ}, which concludes the proof of the lemma.

4 Semiglobal estimates

4.1 Some tools for propagating the information

The Local Estimate of Theorem 3.1 only provides information on the low frequency part of the function.

Iterating this result alows us to propagate the low frequency information. In this section, we dene some tools that will be useful for this iterative procedure. They are aimed at describing how information on the low frequency part of the solution can be deduced from one subregion to another one.

Denition 4.1. Fix Ωbe an open set of Rⁿ =R^na×R^nb, P a dierential operator of orderm dened in Ω, and (Vj)_j∈J and(Ui)_i∈I two nite collections of bounded open sets of Rⁿ. We say that(Vj)_j∈J is under the dependence of (Ui)_i∈I, denoted

(Vj)j∈JE(Ui)i∈I,

if for any ϑ_i ∈C₀^∞(Rⁿ)such that ϑ_i(x) = 1 on a neighborhood ofU_i, for any ϑe_j ∈C₀^∞(V_j)and for all κ, α >0, there existC, κ⁰, β, µ₀>0such that for all(µ, u)∈[µ₀,+∞)×C₀^∞(Rⁿ), we have

X

j∈J

M_µ^βµϑe_j,µu

_m−1≤Ce^κµ X

i∈I

M_µ^αµϑ_i,µu

m−1+kP uk_L2(Ω)

!

+Ce^−κ0µkuk_m−1. If]I = 1andU1=U, we write(Vj)_j∈JEU, with the same convention forV.

The normk·k_m−1 being taken in Rⁿ.

Remark 4.2. The denition E actually depends on the splitting Rⁿ = R^na×R^nb, the set Ω and the operatorP. However, in the main part of this work,Rⁿ=R^na×R^nb,ΩandP will be xed, so it should not lead to confusion (in particular in the applications). The dependence ofEupon these object will be mentioned when needed.

For the applications, it is important that the functionuis not necessarily supported inΩ.

In the following, we will only need to use this relation E in some appropriate coordinate charts.

However, it will not be a problem for what we want to prove, even on a compact manifold. Indeed, we will x some coordinate chart on an open setΩ⊂Rⁿ close to a point or close to a trajectory. Then, we will use the relation Erelated to Ωto nally obtain some estimates which will be invariant by change of coordinates.

Now, we list some general properties of the relationE, which actually hold without using any asumption on the setΩand the operatorP.

Proposition 4.3. We have the following properties

1. If(Vj)_j∈JE(Ui)_i∈I with Ui=U for alli∈I, then(Vj)_j∈JEU. 2. If(Vj)j∈JE(Ui)i∈I with Ui⊂Wi for all i∈I, then(Vj)j∈JE(Wi)i∈I. 3. IfV ⊂U then, V EU. In particular, we always haveU EU.

4. S

i∈IUiE(Ui)_i∈I.

5. If for anyi∈I,ViEUi, then(Vi)_i∈IE(Ui)_i∈I. In particular, we always have(Ui)_i∈IE(Ui)_i∈I. Proof. Property 1 is obvious from the denition. Property 2 is also immediate since ϑ_i(x) = 1 on a neighborhood ofW_i impliesϑ_i(x) = 1on a neighborhood ofU_i sinceU_i⊂W_i.

Property 3 is a consequence of Lemma 2.11 applied withαµ/2 instead ofµ,λ=µ,f₁=ϑandf =ϑe. The assumptions on ϑand ϑeensures that f₁ = 1on a uniform neighborhood of supp(f). This gives the result withβ=α/2.

Property 4 is a consequence of Lemma 2.12 with the same parameters as before for Property 3, but withb_i=ϑ_i.

Property 5 is almost a consequence of the denition. Actually, the only dierence is that a priori, we have one βi for each i ∈ I. Taking the worst of the constants C, κ⁰, µ0 given by the application of the denition for anyi, it gives

X

i∈I

M_µ^βiµϑe_i,µu

_m−1≤Ce^κµ X

i∈I

M_µ^αµϑ_i,µu

m−1+kP uk_L2(Ω)

!

+Ce^−κ0µkuk_m−1.

withϑ_i= 1onU_i andϑe_i∈C₀^∞(V_i). But taking2β = inf{β_i, i∈I}, we have

M_µ^βµϑe_i,µu

_m−1 ≤

M_µ^βiµM_µ^βµϑe_i,µu

_m−1+

M_µ^βµ(1−M_µ^βiµ)ϑe_i,µu _m−1

≤

M_µ^βiµϑe_i,µu

_m−1+Ce^−cµkuk_m−1,

where we have used Lemma 2.3 and the properties of support ofm(_β^·)and(1−m(_β^·

i))for the last estimate.

The second part comes from the combination withU_iEU_i for alli∈I.

The relation is not clearly transitive but we have the following weaker but sucient property: if (V_j)_j∈J E(Ue_i)_i∈I and Ue_i bU_i with compact inclusion (that is Ue_i ⊂U_i) and(U_i)_i∈I E(W_k)_k∈K, then, (Vj)j∈JE(Wk)k∈K (this is proved by introducing functionsfi∈C₀^∞(Ui)equal to 1onUei: see the proof of Item 6 in Proposition 4.5 below).

For this reason, it is convenient to introduce the following stronger property.

Denition 4.4. GivenΩan open set ofRⁿ =R^na×R^nb,P a dierential operator of order mdened in Ω, and(Vj)j∈Jand(Ui)i∈I two nite collections of bounded open sets ofRⁿ, we say that(Vj)j∈J is under the strong dependence of(Ui)_i∈I if there exists UeibUi such that(Vj)_j∈JE(Uei)_i∈I. In that case, we write.

(Vj)_j∈JC(Ui)_i∈I.

This makes the relation transitive, but it becomes more strict in the sense that we do not always have UCU. We sumarize again the properties of this relation.

Proposition 4.5. We have the following properties 1. (V_j)_j∈JC(U_i)_i∈I implies(V_j)_j∈JE(U_i)_i∈I.

2. If(V_j)_j∈JC(U_i)_i∈I with U_i=U for alli∈I, then(V_j)_j∈JCU. 3. IfV_ibU_i for any i∈I, then, (V_i)_i∈I C(U_i)_i∈I.

4. IfV_ibU_i for any i∈I, thenS

i∈IV_iC(U_i)_i∈I.

5. If for any i ∈ I, V_iCU_i, then (V_i)_i∈I C(U_i)_i∈I. In particular, if for any i ∈ I, U_i CU, then (U_i)_i∈ICU.

6. The relation is transitive, that is

[(V_j)_j∈JC(U_i)_i∈I and (U_i)_i∈IC(W_k)_k∈K] =⇒(V_j)_j∈JC(W_k)_k∈K.

Proof. Property 1 is obvious. For 2, the assumption gives some(Uei)_i∈I with(Vj)_j∈JE(Uei)_i∈I andUeibU for alli∈I. SinceUe_i⊂U for alli∈IandIis nite, we have∪i∈IUe_i=∪i∈IUe_i⊂U. DenoteW =∪i∈IUe_i. We haveUe_i ⊂W for alli∈I, so Property 2 and then Property 1 of the previous Lemma give(V_j)_j∈JEW which implies (V_j)_j∈JCU sinceW bU.

For 3, we use(V_i)_i∈I E(V_i)_i∈I from Property 5 of the previous Lemma andV_ibU_i. For 4, we use Property 4 of the previous Lemma, which givesS

i≤IViE(Vi)_i∈I. This isS

i≤IViC(Ui)_i∈I by the denition ofC.

For 5, assumeViEUei withUei bUi. Then, Property 5 of the previous Lemma gives (Vi)_i∈I E(Uei)_i∈I which gives (Vi)_i∈IC(Ui)_i∈I by denition. The second part is direct by combining with Property 2.

For 6, the assumptions give the existence ofUe_ibU_i andWf_k bW_k such that (V_j)_j∈JE(Ue_i)_i∈I and(U_i)_i∈IE(fW_k)_k∈K

Since Uei bUi, we can pickχi ∈C₀^∞(Ui) such thatχi = 1 in an neighborhood ofUei. Let α >0, κ >0, and take ϑk ∈C₀^∞(Rⁿ)(for allk ∈K) such thatϑk(x) = 1on a neighborhood of Wfk and ϑej ∈C₀^∞(Vj) (for allj∈J). Since we have(Ui)_i∈IE(fWk)_k∈K andχi ∈C₀^∞(Ui), there existC, κ⁰, β, µ0>0, such that we have

X

i∈I

M_µ^βµχi,µu

m−1≤Ce^κ2µ X

k∈K

M_µ^αµϑk,µu

m−1+kP uk_L2(Ω)

!

+Ce^−κ0µkuk_m−1.

Now, we apply the relation given by (V_j)_j∈JE(Ue_i)_i∈I with α replaced by the above β and κ replaced by κ1 = min(κ⁰, κ)/2 > 0. Since χi = 1 in an neighborhood of Uei and ϑej ∈ C₀^∞(Vj), there exist C⁰, κ⁰⁰, β⁰, µ⁰₀>0 such that

X

j∈J

M_µ^β0µϑe_j,µu

_m−1≤C⁰e^κ1µ X

i∈I

M_µ^βµχ_i,µu

m−1+kP uk_L2(Ω)

!

+C⁰e^−κ00µkuk_m−1.

Combining the above two estimates now yields X

j∈J

M_µ^β0µϑe_j,µu

_m−1 ≤ CC⁰e^(κ2+κ1)µ X

k∈K

M_µ^αµϑ_k,µu

m−1+C⁰e^κ1µ 1 +Ce^κ2µ

kP uk_L2(Ω)

+

C⁰e^−κ00µ+CC⁰e^{(κ1−κ0)µ}

kuk_m−1.

Since κ/2 +κ₁ ≤κandκ₁−κ⁰< κ⁰/2−κ⁰=−κ⁰/2<0, it gives (V_j)_j∈JE(fW_k)_k∈K, which implies the result sinceWfkbWk.

Note that in the proofs above, we have omitted to precise each time the restrictionµ≥µ0. Yet, all the estimates have to be taken with that restriction, taking the worst constant µ0 when several restrictions are involved.

Corollary 4.6. Under the assumptions of Theorem 3.1, there existsR0>0such that for any R∈(0, R0), there existsr,ρ >0 so that we have

B(x⁰, r)E

{φ >2ρ} ∩B(x⁰,3R) , B(x⁰, r)C

{φ > ρ} ∩B(x⁰,4R) .

Proof of Corollary 4.6. First, we restrict R₀ so that B(x⁰,4R₀) ⊂Ω. Theorem 3.1 gives some R, r, ρ,

˜ τ₀>0.

Letκ,α >0. We apply the result withµ=αµ⁰,c₁= 1/αandκreplaced byκ/αto obtain, uniformly forµ⁰≥τ˜₀/(αβ),

M_µ^βαµ0 ⁰σ_r,µ⁰u

_m−1≤Ce^κµ0

M_µ^αµ0 ⁰ϑ_µ⁰u

_m−1+kP uk_L2(B(x⁰,4R))

+Ce^−ακ0µ0 kuk_m−1 . Now, letϑe∈C₀^∞(B(x⁰, r)). Sinceσr= 1onB(x⁰, r), Lemma 2.11 gives

M^βαµ

0/2 µ⁰ ϑeµ⁰u

m−1≤

M_µ^βαµ0 ⁰σr,µ⁰u

m−1+Ce^−cµ0kuk_m−1 This gives the result. The second comes from the compact inclusion of

{φ >2ρ} ∩B(x⁰,3R) into B(x⁰, r)C

{φ > ρ} ∩B(x⁰,4R) .

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