Nonlinear Equations - UFRJ

62 [CH. 5: REPRODUCING KERNEL SPACES
Denote by dF, dF x the zero-average, unit variance Gaussian probability
distributions. Note that in F x , π1dF ∗ = 1
(2π)
dF n x . The coarea
formula for (V, M, π 2 , F ) (Theorem 4.9) is
E(#(Z(f) ∩ K)) = 1
∫
(2π)
∫K
n dM(x) NJ(f, ix) −2 dF x
F x
with Normal Jacobian NJ(f, x) = det(Dπ 2 (f, x)Dπ 2 (f, x) ∗ ) 1/2 .
The Normal Jacobian can be computed by
⎛ ⎡
⎤ ⎞
K 1 (x, x)
NJ(f, x) 2 ⎜
= det ⎝Df(x) −∗ ⎢
⎣
. ..
⎥
⎦ Df(x) −1 ⎟
⎠
K n (x, x)
=
∏
Ki (x, x)
| det Df(x)| 2
We pick an arbitrary system of coordinates around x.
Lemma 4.3,
Using
| det Df(x)| 2 dM =
Thus,
E(#(Z(f) ∩ K)) =
=
= 1
= 1
1
n∧
(2π) n ∫K
n∑
i=1 j,k=1
i=1
∂
f i (x) ∂ f i (x)
∂x j ∂x k
jk
√ −1
2 dx j ∧ d¯x k
n∧ ∑ 〈Df(x)
∫F ∂
∂x j
, Df(x) ∂
∂x k
〉
ix
n∧ ∑
π
∫K
n i=1 jk
π
∫K
n i=1
using Proposition 5.9.
n∧
ω i (x)
K i (x, x)
√ −1
2 dx j ∧ d¯x k dF ix (f i )
ω i
( ∂
∂x j
, J ∂
∂x k
) √ −1
2 dx j ∧ d¯x k