content_image/317828.jpg content_image/317829.jpg content_image/317830.jpg content_image/317831.jpg content_image/317832.jpg
and E$_{1}$$_{,}$$_{2}$ = ˜ E$_{1}$ ˜ E$_{2}$ where for i = 1 , 2 

For E$_{1}$ integrating we get (dropping the subsript 1 on variables) 

<img src='content_image/317828.jpg'>

<img src='content_image/317829.jpg'>

<img src='content_image/317830.jpg'>

<img src='content_image/317831.jpg'>

By construction of the partition of unity ∣ Σ ψ ′ v $_{1}$( y ) ψ$_{v}$$_{2}$ ( y ) ∣ as well as Σ ψ$_{v}$$_{1}$ ( y ) ψ$_{v}$$_{2}$ ( y ) are uniformly bounded, say both by some ˜ B. We get 

<img src='content_image/317832.jpg'>

Note that ψ 2 ∈ D ( W ) . By Assumption 5 then this contribution to the covariance is bounded. 

