10 Stephane Jaffard and Yves Meyer
Remark. Notice that (1.3) doesn't hold for moduli such as u(h) = h™,
m G IN, or cj(h) =
hm(log h)a.
In such cases, we will see in Section III.l
that the uniform modulus of continuity of / cannot be characterized by
a condition on the LittlewoodPaley decomposition, and we will study
what can happen in that case in some explicit examples.
Proof of Proposition 1,1,
Using the definition of Aj and the cancellation of ^,
A,(/)(x) = \Jf(t)2^^(xt))dt
=
\J(f(t)P(xt))2n^(2i(xt))dt
C f
6(\xt\)2nj\^(2j(xt))\dt

cJe(ij)(i
+
\t\)M
where M can be chosen arbitrarily large. We split the domain of inte
gration into the domain A\ defined by \t\ 1 and the domains A
m
defined by 2
m
\t\
2m+1
for m e IN. The integral on the first domain
is bounded by c6(2~i) because 6 is increasing. The integral over Am is
bounded by
C
J22m m\t\2m+1 x z '
V1
1
dt
\t\2 m
C2
( +
1*1M)
6(2
m+lj
y\)dy
n(m+l) f
7ii»ii (i +
2m+1
M)
M
f e(2\y\)dy
h\y\i
(X
+
2m+l\y\)M
C
{
" 6(23) .
We choose M large enough so that
2M 2n
A and we deduce the
first part of Proposition 1.1
Let us prove the converse result. If (1.2) holds, then, by using Bern
stein inequalities (see [30]), for any multiindex a we have
(1.4)
I^AJCOI
C2Mj0(2j).