Page 88 - Textos de Matemática Vol. 47

78 P. E. OLIVEIRA
Lemma 4.8. Assume wr(x,·), ws(x,·), (K,x.·) and (K2,x,·) are bounded,
(SB1), (SB2), (K), (3.2) and (4.2), for some ↵ 2 (0, 1 ), hold. Then 6
E ⇣(rb (x) r(x))(fb (x) f (x))(gb (x) r(x)f(x))⌘ nn0n
=o 1 + 1/2(K,x,h) + (K,x,h)+ 1/2(K,x,h)+wr(x,h) nF (x, h) n3/4F 3/4(x, h) n1/2F 1/2(x, h)
+ 3/2(K,x,h)+ 1/2(K,x,h)wr(x,h)+ (K,x,h) n1/4 F 1/4 (x, h)
+ 2(K,x,h)+ 3/2(K,x,h)+ (K,x,h)wr(x,h)!. Proof. Using Cauchy’s inequality we have
E (rbn(x) r(x))(fn(x) f0(x))(gbn(x) r(x)f(x)) ⇣bb⌘
 ⇣ E ( rb n ( x ) r ( x ) ) 2 ( f n ( x ) f 0 ( x ) ) 2 ⌘ 1 / 2 ⇣ E ( gb n ( x ) r ( x ) f 0 ( x ) ) 2 ⌘ 1 / 2 . The last factor above has been characterized in Lemma 4.1. As what regards
the first factor, its square is bounded above by
⇣ 2bb2⌘ 2E (rbn(x) ren(x)) (fn(x) f0(x))
(4.3) +2(ren(x) r(x))2E (fn(x) f0(x))2.
As re (x) ! r(x) and taking into account Lemma 4.1, the second term is an ⇣n⌘
o 1 + 2(K,x,h) .Choose↵n =(nF(x,h))↵,"n =(nF(x,h)) where nF (x,h)
>↵>0and2↵+ < 1.Notethat,as↵< 1,thischoiceisalwayspossible. 26
Decompositing as before, the first term in (4.3) is bounded above by
h⇣2b2⌘ (↵n + ren(x)) E (rbn(x) ren(x))b(fn(x) f0(x)) I{rbn(x)↵n,|rbn(x) ren(x)|"n}
+E ⇣(rb (x) re (x))2(fb (x) f (x))2I ⌘i
n n n
0 {rbn (x)↵n ,|rbn (x) ren (x)|>"n } +E ⇣(rbn (x) ren (x))2 (fn (x) f0 (x))2 I{rbn (x)>↵n } ⌘ .