content_image/304294.jpg content_image/304293.jpg content_image/304295.jpg content_image/304296.jpg content_image/304297.jpg
parameter ν = ξ/ | ξ | and work only with the parameter ξ. 

Lemma 4 . In the new coordinate system, the functions 

are bounded on the set G for | ξ | ≤ 1 / 2. Moreover, the functions 

tend to zero in L$_{2}$ ( R 1 ξ ) as $_{ξ}$′ → 0. 

## Proof. 

<img src='content_image/304294.jpg'>

<img src='content_image/304293.jpg'>

First, let us show that if up to the second order derivatives of the functions . z $^{i}$( x, ξ, t ) with respect to ξ are bounded, then it is possible to prove that the functions ∂$_{ξ}$ i ∂$_{ξ}$ 1 u are bounded for 0 < | ξ | ≤ 1 / 2 and 1 ≤ i ≤ n . 

Since supp b ⊂ D and the solution of Problem (3.2)-(3.3) has property (3.4), the integral in (4.2) can be considered on the finite interval [ 0 , d 0 | ξ | ] , where d$_{0}$ is the diameter of D in the metric g = ( g$_{ij}$ ( x )). Hence we can write (4.2) as 

<img src='content_image/304295.jpg'>

Moreover, since the metric g = ( g$_{ij}$ ) is written down in the semi-geodesic coordi- nates, the equality . z k ( x, ξ $^{1}$, 0 , t ) = 0 ( ξ 1 ̸ = 0) is satisfied for each k (2 ≤ k ≤ n ) (see the proof of Lemma 1 in Appendix) and thus ∂$_{ξ}$ 1 z k ( x, ξ $^{1}$, 0 , t ) = 0 . Then, for each fixed x ∈ D and 0 < | ξ $^{1}$|  ≤ 1, by Taylor’s formula with respect to the variable $_{ξ}$′ = ( ξ $^{2}$, ..., ξ $^{n}$) (e.g., [13], p. 285), we obtain the equalities 

<img src='content_image/304296.jpg'>

<img src='content_image/304297.jpg'>

