12 Stephane Jaffard and Yves Meyer
example of 9(x) = x which we will consider in Chapter III.l shows that,
if (1.3) fails, the uniform modulus of continuity is not characterized by a
decay condition of the Littlewood-Paley decomposition. An interesting
problem would be to determine if (1.3) is the necessary and sufficient
condition for the existence of such a characterization.
We will now study the modulus of continuity at a given point x0-
The following proposition has the well-known structure of a Tauberian
theorem. On the one hand one has a trivial implication: if 9 is a modulus
of continuity of / at XQ, then (1.6) holds. On the other hand, if (1.6)
holds and if / satisfies a rather weak global regularity asumption, then
a weaker form of (1.1) is true.
Proposition 1.2. If 9 is a modulus of continuity of f at XQ, then
(1.6) lAjfix)] C[6(2-i) + 0{\x-xo\)].
Conversely, if (1.6) holds, if UJ is a uniform modulus of continuity of f
and if assumption (1.3) holds for 9 and to, there exists a polynomial P
such that
|
/ ( x
) _
P
(
x
_
X o
) |
C
inf [(j1-j0)9(\x-x0\)+u(2-^)}
h3o
(where jo is defined by (1-5)).
As we will see later, if no global assumption is made, (1.6) may hold
and yet / may have no smoothness at all at x$.
Proof of Proposition 1.2. Suppose that 9 is a modulus of continuity
of / . Then
lAj-CfX*)! - 2^\J (f(t) - P(t-x0))^(x-t))dt
/
2nJ
dt
^ - ^ a
+
|g(t-,)i) "
The hypotheses on 9 imply that there exists C such that
(1.7)
9(a + b) C{6{a) + e(b)),
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