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) =
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
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
defined by 2
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
J22m m\t\2m+1 x z '
7ii»ii (i +
" 6(2-3) .
We choose M large enough so that
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 ), for any multiindex a we have