9 Wavelet Methods for Pointwise Regularity
and 0 is a uniform modulus of continuity if (1.1) holds for all XQ €
and if the constant C is uniform (in #o). The case 6(x) = |x|
corresponds to the Holder regularity.
Our first goal will be to "translate" condition (1.1) into conditions on
the Littlewood-Paley decomposition of / . Of course, if / has a uni-
form modulus of continuity, we will obtain a uniform bound on the
Littlewood-Paley decomposition, and if we are interested in the point-
wise condition, we will obtain local estimates on the Littlewood-Paley
decomposition. Let us first recall the definition of the Littlewood-Paley
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
Let Sj be the "low-pass 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
2j+i rpj^ following Proposition deals with uniform modulus of conti-
Proposition 1.1. If 6 is a uniform modulus of continuity of f, then
(1-2) I I Aj(/) ||oo C9(2-3).
Suppose now that 6 satisfies:
there exists an integer N 0 such that for all integers J 0
^2Nj 9(2-j) C2NJ0(2-J)
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.