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 Littlewood-Paley 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^^(x-t))dt
=
\J(f(t)-P(x-t))2n^(2i(x-t))dt
C f
6(\x-t\)2nj\^(2j(x-t))\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+l-j
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(2-3) .
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(2-j).
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