(A.17) by the Fubini-Tonelli theorem, we have ∫ N 0 ( ∂$_{η}$q ( $_{x, η, ξ}$′$_{)}$ ) 2 dη ≤ d$_{0}$ ∫ N 0 ( ∫ x 1 x 1 0 ϝ 2 ( $_{τ , x}$′$_{, η, ζ}$ ′$_{(}$$_{τ , ξ}$ ′$_{)}$ ) dτ ) dη = d$_{0}$ x 1 x 1 0 N 0 ϝ 2 τ , x ′ $_{, η, ζ}$′$_{(}$$_{τ , ξ}$ ′$_{)}$ dη dτ ≤ d$_{0}$M ∫ N 0 ( ∂ β ′ ξ ′ ∂$_{η}$q ( $_{x, η, ξ}$′$_{))}$ $^{2}$dη, ∫ N 0 ( ∂ β ′ ξ ′ ∂$_{x}$ j ∂$_{η}$q ( $_{x, η, ξ}$′$_{))}$ $^{2}$dη ≤ d$_{0}$M$_{1}$ (A.18) ∫ x 1 x 1 0 ( ∫ + ∞ 0 ( ∂ β ′ ξ ′ ϝ $_{2}$( τ , x ′ $_{, η, ζ}$′$_{(}$$_{τ , ξ}$ ′$_{)))}$ $^{2}$dη ) dτ ∫ x 1 x 1 0 ( ∫ + ∞ 0 ( ∂ β ′ ξ ′ ∂$_{x}$ $_{j}$ϝ $_{2}$( τ , x ′ $_{, η, ζ}$′$_{(}$$_{τ , ξ}$ ′$_{)))}$ $^{2}$dη ) dτ ∂ β ′ ξ ′ ∂$_{η}$q ( $_{x, η, ξ}$′$_{)}$$_{,}$ ∂ β ′ ξ ′ ∂$_{x}$ j ∂$_{η}$q ( $_{x, η, ξ}$′$_{)}$ ∈ L$_{1}$ ( Δ 1 η ) ∩ C ( D × Δ 1 η × G ′ ) . ∂ β ′ ξ ′ ∂$_{η}$q ( $_{x, η, ξ}$′$_{)}$$_{,}$ ∂ β ′ ξ ′ ∂$_{x}$ j ∂$_{η}$q ( $_{x, η, ξ}$′$_{)}$ ∈ L$_{1}$ ( Δ − 1 η ) ∩ C ( D × Δ − 1 η × G ′ ) . ∂ β ′ ξ ′ ∂$_{η}$p ( $_{x, η, ξ}$′$_{)}$$_{,}$ ∂ β ′ ξ ′ ∂$_{x}$ j ∂$_{η}$p ( $_{x, η, ξ}$′$_{)}$ ∈ L$_{1}$ ( Δ ρ η ) ∩ L$_{2}$ ( Δ ρ η ) ∩ C ( D × Δ ρ η × G ′ ) for each N > 0. By analogous reasoning, from (A.16), one can prove that for ∣ $_{β}$′ ∣ ≤ 2, where M$_{1}$ > 0 is the maximum of the continuous functions and on D × G ′ . Inequalities (A.18) imply that ∂ β ′ ξ ′ ∂$_{η}$q, ∂ β ′ ξ ′ ∂$_{x}$ j ∂$_{η}$q ∈ L$_{2}$ ( Δ 1 η ) for ∣ $_{β}$′ ∣ ≤ 2. Moreover, taking into account Remark 2 after Lemma 2, from (A.11) we obtain In a similar manner, we can prove that Taking into account (4.5) and using arguments similar to the previous, it can be shown that 25