we see that the functions ∂$_{ξ}$ k ∂$_{x}$ j p (0 $_{, η, ξ}$′$_{) tend to zero in}$ L$_{1}$ (0 , ∞ ) as $_{ξ}$′ → 0. By the same argument, it can be proved that the function ∂$_{ξ}$ k ∂$_{x}$ j p (0 $_{, η, ξ}$′$_{) tends to}$ zero in L$_{1}$ ( −∞ , 0) as $_{ξ}$′ → 0. Similarly one can prove that the functions p (0 $_{, η, ξ}$′$_{),}$ ∂$_{x}$ j p (0 $_{, η, ξ}$′$_{),}$ ∂$_{ξ}$ k p (0 $_{, η, ξ}$′$_{),}$ ηq (0 $_{, η, ξ}$′$_{) tend to zero in}$ L$_{1}$ ( R 1 $_{η}$) as $_{ξ}$′ → 0 . In this section, we prove Theorem 1. As it was noted in Section 3, the uniqueness of the solution to Problem 1 follows from the uniqueness of the solution to Problem 2 in the class C 5 $_{0}$( D ). Therefore, we consider Problem 2 with homogeneous boundary data below. First we recall that ̂ u = p + √ − 1 q and Γ i $_{jk}$(0) = 0. Then by (4.3) and (4.6), in the new coordinates, we have for x = 0, where ϝ 2 = n ∑ j =2 ξ $^{j}$∂$_{x}$ j p . By (6.1), we obtain for $_{ξ}$′ = $_{ξ}$′ i ∈ G ′ , 2 ≤ i ≤ n , where $_{ξ}$′ i = $_{εξ}$′ $_{i}$(1) $_{, ξ}$′ $_{i}$(1) = (0 , · · · , 0 , ︸ ︷︷ ︸ i − 2 1 , 0 , · · · , 0) ∈ R n − $^{1}$, ε > 0 $_{, ξ}$′ = ( ξ $^{2}$, ξ$^{3}$, · · · , ξ n ) and F$_{2}$$_{i}$ (0 $_{, η, ξ}$′ $_{i}$) = ε∂$_{x}$ i p (0 $_{, η, ξ}$′ $_{i}$). On the other hand, we know (e.g., Theorem 3.1.3 on p. 56 in H¨ ormander [15]) Lemma 6. Let a function U ( y ) defined in an open set Y ⊂ R , belong to space C $^{1}$( Y / { y$_{0}$ } ) for some y$_{0}$ ∈ Y and let a function V ( y ) coincide with dU ( y ) dy for y = y$_{0}$ and be integrable on some neighbourhood of y$_{0}$ . Then the limits 6. THE PROOF OF THE MAIN RESULT Proof of Theorem 1. ∂$_{η}$∂$_{x}$ 1 q = − 2 πδ ( η ) n ∑ k,j =2 a$_{jk}$ (0) ξ $^{k}$ξ j + ϝ $_{2}$, (6.1) ∂$_{η}$ ( ∂$_{x}$ 1 q (0 $_{, η, ξ}$′ $_{i}$) ) = − 2 πδ ( η ) a$_{ii}$ (0) ε 2 + F$_{2}$$_{i}$ (0 $_{, η, ξ}$′ $_{i}$), (6.2) U ( y$_{0}$ ± 0) := lim y → y 0 ± U ( y ) 14