Nonlinear Equations - UFRJ

134 [CH. 9: THE PSEUDO-NEWTON OPERATOR
Proposition 9.10. Assume that f ∈ H d , Let R 1 , . . . , R s be as above,
and assume the canonical norm in C n+1 . Then, for ‖X‖ = 1,
( ‖D k f(X)‖
) 1/(k−1)
≤ ‖f‖ 1/(k−1) D 2
with D = max d i .
Proof.
k!
D k f i (X) = 〈f i (·), D k K i (·, ¯X)〉.
Thus,
Theorem 9.11 (Higher derivative estimate). Let f ∈ H d and X ∈
C n+1 \ {0}. Then,
γ(f, X) ≤ ‖X‖ (max d i) 3/2
µ(f, x)
2
Proof. Without loss of generality, scale X so that ‖X‖ = 1. For each
k ≥ 2,
( ‖Df(X) † D k )
f(X)‖
1
k−1
≤ ‖Df(X) −1 ‖ 1/(k−1) ‖f‖ 1/(k−1) D k!
|X ⊥ 2
1
≤ ‖L x (f) −1 ‖ 1/(k−1) 1/(k−1) D1+ k−1
‖f‖ .
2
≤ D3/2
µ(f, x)1/(k−1)
2
≤ D3/2
µ(f, x)
2
using that µ(f, x) ≥ √ n ≥ 1.
Exercise 9.4. Show that Proposition 9.10 holds for multi-homogeneous
polynomials, with
D = max d ij .
Exercise 9.5. Let f denote a system of multi-homogeneous equations.
Let X ∈ C n+s \ Ω, scaled such that ‖X i ‖ = 1. Show that,
γ(f, X) ≤ ‖X‖ (max d ij) 3/2
µ(f, x).
2