We have I(gx)(0,0) Ig(0,0)x| Ig(0,0) llx, and from the one-
dimensional case proved in Dinculeanu [73, we have
Var[O,s](gx)(' ,O) < jx VarFo,s]g(-,O);
Varo,t](gx)(O,*) 5 x Varro,t]g(O,.).
In fact (we prove the first; the proof of the second is identical):
for 0 = s0 < s1 < ... < s = s a partition of [0,s], we have
(gx)(si+ ,O0) (gx)(si,0)I = I(g(si+ ,0) g(si,0))xI
s Ig(si+1,0) g(si,0)|xl
for i=0,1,2,...,n-1. Summing over i, we obtain
n-1 n-1
E 1(gx)(s ,o0) (gx)(si,0) 5 ||xl lg(s + 0) g(si,0)|
i=O i=O0
n-1
= Axi I Ig(sil,o0) g(si,0)l
i=O
5 xjVar[Os]g(-,O).
Taking supremum over partitions of [O,s], we get Var0,s](gx)(-,0)
S|xj[Var[Os]g(.,0). The same proof gives Var0o,t](gx)(0,.)
< Ixf-Var0o,t]g(0,-). We obtain a similar inequality for the
remaining term of jgx|: For any rectangle R = [(p,q),(p',q')]C 2
we have
R(gx)l = I(gx)(p',q') (gx)(p',q) (gx)(p,q') + (gx)(p,q)l