B.2 Hypoth` ese A2”
I.2 Conditions suffisantes pour que l’hypoth` ese B3 soit satisfaite
Nous allons montrer que l’hypoth`ese B3.1 est satisfaite si (I.7), (I.8), (I.16), (I.19) et (I.22) sont v´erifi´ees.
Nous avons :
A(x1, x2, y) ≤ |x2|(1 + |x2|) (|y|n2+|y|m2) +|x2|(1 + |x1|) (|y|p2+|y|m1) , +|x1|(1 + |x1|) (|y|n1 +|y|p1) .
(I.28)
Par cons´equent :
A(x1, x2, y) ≤ |x2|(1 + |x2| + |x1|) (|y|n2+|y|m2+|y|p2+|y|m1) +|x1|(1 + |x1|) (|y|n1+|y|p1) . (I.29) De (I.16), (I.19) et (I.22) on d´eduit l’existence d’une fonction ˜κ continue et positive telle que :
A(x1, x2, y) ≤ |x2|(1 + |x2| + |x1|)˜κ(y)|y|
q
2 + |x1|(1 + |x1|)˜κ(y)|y|q . (I.30)
D’apr`es les d´efinitions de V , W , Q, S et T , on a donc :
A(x1, x2, y) ≤ c " 1 + Q(x1) + S(x2) # ˜ κ(y) V (y)14 W (y) T (x2) + c " 1 + Q(x1) + S(x2) #2 κ(y)˜ V (y)12 W (y) . (I.31)
Mais, d’apr`es le Lemme E.1 de l’annexe E, il existe une fonction ˜κ telle que :˜ ˜
˜
κ(V (y)) ≥ ˜κ(y)2+ ˜κ(y) . (I.32)
D´efinissons maintenant κ par :
κ(v) = cκ(v)˜˜√ v . (I.33) Alors : A(x1, x2, y) ≤ c " 1 + Q(x1) + S(x2) # κ(V (y)) W (y) T (x2) + c " 1 + Q(x1) + S(x2) #2 κ(V (y))W (y) . (I.34)
L’in´egalit´e (2.10) est donc obtenue. De plus :
κ(V (y))∂V∂y(y) ≤ ˜˜κ(V (y)) . (I.35)
La fonction ˜κ(v) ´˜ etant continue, la propri´et´e (2.12) est v´erifi´ee. On v´erifie de fa¸con imm´ediate que B3.2 est v´erifi´ee si et seulement si
r > 1 (I.36)
ou
q < min{n1, p1, 2n2, 2m2, 2p2, 2m1,} . (I.37)
Les conditions n´ecessaires et suffisantes pour que l’hypoth`ese B3 soit v´erifi´ee avec les fonctions
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