easy to see that the functions ∂$_{ξ}$ i ∂$_{x}$ j z $^{s}$(0 , ξ, t ) , ∂$_{ξ}$ l ∂$_{ξ}$ i ∂$_{x}$ j z $^{s}$(0 , ξ, t ) , ∂$_{ξ}$ i ∂$_{x}$ j . z $^{k}$(0 , ξ, t ), ∂$_{ξ}$ i ∂$_{ξ}$ s ∂$_{x}$ j . z $^{k}$(0 , ξ, t ) (2 ≤ k ≤ n, 1 ≤ i, j, s, l ≤ n ) are bounded on the set G for | ξ | ≤ 1 / 2 . This implies that the functions ∂$_{ξ}$ i ∂$_{ξ}$ 1 u (0 , ξ ), ∂$_{ξ}$ i ∂$_{ξ}$ 1 ∂$_{x}$ j u (0 , ξ ) (1 ≤ i, j ≤ n ) are bounded on the set G for | ξ | ≤ 1 / 2 . Now, let us prove the second assertion of the lemma. Since the unique solu- tion to Problem (3.2)-(3.3) for ν 0 = (1 , 0 , ..., 0) ∈ R n is given by z ( x, ν $^{0}$, t ) = x + tν $^{0}$, we obtain that . z k ( x, ξ $^{1}$, 0 , t ) = 0 for $_{ξ}$′ = 0, 2 ≤ k ≤ n . Then by (3.1), we have u ( x, ξ $^{1}$, 0) = ∂$_{ξ}$ i u ( x, ξ $^{1}$, 0) = 0 (1 ≤ i ≤ n ) and ∂$_{ξ}$ 1 ∂$_{ξ}$ i u ( x, ξ $^{1}$, 0) = ∂$_{ξ}$ 1 ∂$_{ξ}$ i ∂$_{x}$ j u ( x, ξ $^{1}$, 0) = 0 (1 ≤ i, j ≤ n ). Hence, taking into account the last equal- ities, the first conclusion of the lemma which we proved above and the analogous formula to (4.2), we see that the functions u (0 , ξ 1 $_{, ξ}$′$_{)}$$_{, ∂}$$_{ξ}$ i u (0 , ξ 1 $_{, ξ}$′$_{)}$$_{, ∂}$$_{ξ}$ i ∂$_{x}$ j u (0 , ξ 1 $_{, ξ}$′$_{)}$$_{, ∂}$$_{ξ}$ i ∂$_{ξ}$ 1 u (0 , ξ 1 $_{, ξ}$′$_{)}$$_{,}$ ∂$_{x}$ j ∂$_{ξ}$ i ∂$_{ξ}$ 1 u (0 , ξ 1 $_{, ξ}$′$_{)}$ tend to zero in L$_{2}$ ( R 1 ξ ) as $_{ξ}$′ → 0 . Lemma 5. Let the conditions of Lemmata 1 and 3 be satisfied. We set ̂ u (0 $_{, η, ξ}$′$_{) =}$ p (0 $_{, η, ξ}$′$_{) +}$ √ − 1 q (0 $_{, η, ξ}$′$_{)}$$_{. Then the functions}$ p (0 $_{, η, ξ}$′$_{)}$$_{, ∂}$$_{x}$ j p (0 $_{, η, ξ}$′$_{)}$$_{,}$ ∂$_{ξ}$ k p (0 $_{, η, ξ}$′$_{)}$$_{,}$ ∂$_{x}$ j ∂$_{ξ}$ k p (0 $_{, η, ξ}$′$_{)}$$_{,}$ ηq (0 $_{, η, ξ}$′$_{)}$ tend to zero as $_{ξ}$′ → 0 in L$_{1}$ ( R 1 $_{η}$) for 2 ≤ k ≤ n and 1 ≤ j ≤ n . Proof. Since the Fourier transform is continuous in L$_{2}$ ( R 1 ξ ), by Lemma 4 the functions ̂ u (0 $_{, η, ξ}$′$_{),}$ ∂$_{ξ}$ k ̂ u (0 $_{, η, ξ}$′$_{),}$ $^{η}$̂ u (0 $_{, η, ξ}$′$_{),}$ η∂$_{ξ}$ k ̂ u (0 $_{, η, ξ}$′$_{),}$ η∂$_{x}$ j ̂ u (0 $_{, η, ξ}$′$_{),}$ ∂$_{ξ}$ k ∂$_{x}$ j ̂ u (0 $_{, η, ξ}$′$_{),}$ η $^{2}$∂$_{x}$ j ̂ u (0 $_{, η, ξ}$′$_{),}$ η∂$_{ξ}$ k ∂$_{x}$ j ̂ u (0 $_{, η, ξ}$′$_{), 2}$ ≤ k ≤ n , 1 ≤ j ≤ n , tend to zero in L$_{2}$ ( R 1 $_{η}$) as $_{ξ}$′ → 0. Then, since and ∫ ∞ 1 ∣ ∂$_{ξ}$ k ∂$_{x}$ j p ∣ dη ≤ (∫ ∞ 1 η 2 ( ∂$_{ξ}$ k ∂$_{x}$ j p ) 2 dη ) 1 2 (∫ ∞ 1 1 η $_{2}$dη ) 1 2 , ∫ 1 0 ∣ ∂$_{ξ}$ k ∂$_{x}$ j p ∣ dη ≤ (∫ 1 0 ( ∂$_{ξ}$ k ∂$_{x}$ j p ) 2 dη ) 1 2 ∫ ∞ 1 η 2 ( ∂$_{ξ}$ k ∂$_{x}$ j p ) 2 dη → 0 , ∫ 1 0 ( ∂$_{ξ}$ k ∂$_{x}$ j p ) 2 dη → 0 , 13