Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

such that |ω j i ∗ ,β∗ i(t, τ)| < ˜ρ 3, i = 1, . . .,n and such that |ḡ(τ)| < ˜ρ 3 for allh ∈ H ρ 2,ρ 1M,κ 0 ,ε and for all (t, τ) ∈ M with |t|, |τ| < ρ 3.In conclusion, the relation (6.7) holds for all h ∈ H ρ 2,ρ 1M,κ 0 ,εand for all(t, τ) ∈ M with |t|, |τ| < ρ 3 .Next, using the explicit expressions of the vector fields L k given in (5.2),we may develop the higher order derivatives L γ¯h(τ) as polynomials in the|γ|-jet (∂τ γ′ ¯h(τ)) |γ ′ |≤|γ| of ¯h(τ) with coefficients being certain holomorphicfunctions of (t, τ) obtained as certain polynomials with respect to the partialderivatives of the functions Θ j (ζ, t).To be more explicit in this desired new representation of (6.7), we remindfirst our jet notation. For each i = 1, . . .,n and each α ∈ N n , we introduceda new independent coordinate Jiα corresponding to the partial derivative∂τ α¯h i (τ) (or ∂t αh i(t)). The space of k-jets of holomorphic mappings ¯h(τ) isthen the complex space C n(n+k)! n! k! with coordinates (Ji α) 1≤i≤n, |α|≤k. It will beconvenient to use the abbreviations J k := (Ji α) 1≤i≤n, |α|≤k and J k¯h(τ) :=(∂τ α¯h i (τ)) 1≤i≤n, |α|≤k .So pursuing with (6.7), we argue that for every γ ∈ N m , there existsa polynomial in the jet J |γ|¯h(τ) with holomorphic cooeficients dependingonly on Θ such that(6.8) L γ ¯h(τ) ≡ Pγ (t, τ, J |γ|¯h(τ)).Putting all these expressions in (6.7), we obtain an important relation betweenh and the l 0 -jet of ¯h which we may now summarize. At first, asκ 0 = l 0 (µ 0 + 1) ≥ l 0 , observe that for every h ∈ H ρ 2,ρ 1M,κ 0 ,ε, we have||J l 0h − J l 0Id || ρ 2≤ ||J κ 0h − J κ 0Id || ρ 2≤ ε. Shrinking ε if necessary, wehave proved the following lemma.Lemma 6.1. There exists a complex algebraic C n -valued mappingΠ(t, τ, J l 0) defined for |t|, |τ| < ρ 3 and for |J l 0− J l 0Id| < ε which dependsonly on the defining functions ξ j − Θ j (ζ, t) of M , such that for everylocal holomorphic self-mapping h ∈ H ρ 2,ρ 1M,κ 0 ,εof M (hence satisfying||J l 0h − J l 0Id || ρ 2< ε), the relation(6.9) h(t) = Π(t, τ, J l 0¯h(τ))holds for all (t, τ) ∈ M with |t|, |τ| < ρ 3 .6.2. Reflection identity for arbitrary jets. Let now Υ j and Υ j be the vectorfields tangent to M defined by(6.10)Υ j := ∂∂w j+d∑l=1Θ l,wj (ζ, t) ∂∂ξ l,Υ j := ∂∂ξ j+d∑l=1Θ l,ξj (z, τ) ∂∂w l,125