9 Wavelet Methods for Pointwise Regularity
and 0 is a uniform modulus of continuity if (1.1) holds for all XQ €
IRn
and if the constant C is uniform (in #o). The case 6(x) = x
a
corresponds to the Holder regularity.
Our first goal will be to "translate" condition (1.1) into conditions on
the LittlewoodPaley decomposition of / . Of course, if / has a uni
form modulus of continuity, we will obtain a uniform bound on the
LittlewoodPaley decomposition, and if we are interested in the point
wise condition, we will obtain local estimates on the LittlewoodPaley
decomposition. Let us first recall the definition of the LittlewoodPaley
decomposition. Let (p be a function in the Schwartz class such that
0(0 = 1 if K l \
0(0 = o if C i i
and
m =
Mm

p(o
•
Let Sj be the "lowpass filter" which, after performing a Fourier trans
form, is the multiplication by (p(2~^). Define Aj = Sj+i — Sj. Thus
Id = S0 + A0 + Ai + • • • .
The Fourier transform of Aj(/) is supported by the set
271
£
2j+i rpj^ following Proposition deals with uniform modulus of conti
nuity.
Proposition 1.1. If 6 is a uniform modulus of continuity of f, then
(12) I I Aj(/) oo C9(23).
Suppose now that 6 satisfies:
there exists an integer N 0 such that for all integers J 0
• oo
^2Nj 9(2j) C2NJ0(2J)
J j=J
(13*
3
J2
2^N+l)j 9(2~j) C2(N+1)J 0(2~J)
.
If (1.2) and (1.3) hold, 9 is a uniform modulus of continuity of f.