556 T. Ichinose, T. Tsuchida

( 4 . 21 ) Йл-Фv„ = H^^v

mv „ i

HoV „ - hI , v „ - hlimil2ir ) - \ - I^ { r ) - \ - I^ ( r ) ) ,

= mv„(x) - lim J [e-^^^-^^+^^^^r^x -\-y)- v„ix)2nidy)

riO ,y,^^

riO

with

( Ho v„) (x) = wt;„ - lim f (v„ (x-^y)- v„ (x)) n (dy),

»•^^ \y\>r

{ hv „ ) ( x ) = - J ie-'y^^'^^yf''~i)vAx-^y)n(dy),

hir ) = - J {e-'y^-^^^yi^^ - 1 + iyA,(x + y\T))v,(x + y)n{dy),

r^|y|<l

/ 3W = J 1у{А^{хЛ-у\1) - A„{x))v„{x-{-y)nidy),

r< . \y\<i

/ 4 W = J iyA„(x)v„(x + y)n{dy).

r^|y|<l

This decomposition differs from (4.4). Here each lim^^o is taken in the sense of distributions.

Since Й„ v„ — Фv„ is in L^ (R**), we have only to show that /^ v„ is in L^ (R"*) and Ij (r), j = 2, 3,4, are L^ convergent as r | 0. Take a Cq function 1/;(л:) with \p{x) = 1 on Ä'i and suppip cz K2 where АГ= supp<^. Then we can use for I^v„ and /2(r) the same arguments as in (4.6) and (4.7,8) with t;„ in place of cp. For I^(r) and I^(r) the arguments are similar to those used to get (4.9,10) and (4.11) with xpA^ (resp. v„) in place of (p (resp. v;v4). Thus Ê„v„ — Фv„, I^v^ and Xivcir^ol^ir),] = 2, 3, 4 are all in L^(R*'), so that HqV^ is in L2(R0 or i;„ is in ЯЧRO•

To show boundedness of the ||ЯоГ„|| we rewrite (4.21) as

( 4 . 22 ) Ä„i;„ - Фv, = Яо1;„ + I.v, + lim(/^(r) + 1,{г) + /,(r)),

where Яо1^„, /it;„ and /2(r) are the same as in (4.21) and

I^if ) = f iyA,{x + >^/2)(t;„(x + j) - i;„(x))«(rf>;),

rs|y|<l

4 ( r ) = f r>^„(x + j;/2)t;„(x)Az(rf>;).

r : S|y|<l

We estimate the L^-norm of each term on the right of (4.22) except HqV^. By a calculation analogous with (4.6) and (4.8), we have

( 4 . 23 ) \\hv„\\^2nJK\'i^v„\U, and

( 4 . 24 ) ||lim/2(r)|| ^ 3«,/2lkJlool|tp^ll^V/^^^

For /5 (r), we have by the Schwarz and Holder inequalities