Page 138 - Textos de Matemática Vol. 47

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we deduce
n 2 m lnn
On the other hand, we have
u 2 exp n,m
M. G. TEMIDO
!!2 1 + ( p , 1 ) l n n Xn Xq
ri,j n i=p+1j=1 ln(ij)
2
k+↵ ln n k+↵ ln n i, j
K(lnm) ↵lnn m1+↵ 2↵lnn sup |r ln(ij) |
i 1,j 1 k+↵ ln n i, j
Klnmm1+↵ 2↵lnn sup |r ln(ij) |!0, n,m!+1. i 1,j 1
u2 nm exp( n,m)
!2 1+ (p,1) Xn Xq
↵ i=p+1j=1 ln(ij) ln(nm )
 K  K1
ln ↵ ↵ ↵lnni=p+1j=1 nm nm
n 2 m 1 + ↵ ↵
u 2 exp( n,m)
!2
1 + ( p , 1 ) Xn Xq i j 1
2
ln(nm ) n2m1+↵
2
u2 exp( n,m)
! 2↵ ln n k+↵ ln n
Z 1 Z 1 ↵lnn0 0
|lnxy|dxdy
lnn (lnn) k
2
k+↵ ln n 2k
lnm
2↵ ln n
1
nk+↵ ln n exp⇣ klnlnn⌘
m 1 ↵+k+↵ ln n lnn
⇣k+↵lnn⌘ lnm 1 !0, n,m!+1. 2↵ ln n
=K
Similarly, for the third sum of the right hand side of (2.4), we obtain
1
exp 2klnn m 1 ↵+k+↵lnn lnn k+↵lnn
u2 nm exp n,m
!!2 1+ (p,1) Xn Xq
↵ i=p+1j=1 ln(nm ) ln(nm)
2
◆lnp ✓
ln(nm) k+lnp lnn + lnn
✓
lnn (nm)2 ln(nm↵) ln(nm)
For the sum concerning A1 ⇥ B2 the desired result is obtained changing n with m.
K n2m1+↵
1 k+↵ ln n

◆
K ,lnmm1+↵ 2↵lnn !0, n,m!+1.