I. Modulus of continuity and
two-microlocalization
Our purpose in this chapter is to investigate how the modulus of
continuity of a function can be determined using the Littlewood-Paley
decomposition or the wavelet transform.
We will first obtain some general results concerning the modulus of
continuity; then we will carefully study the 2-microlocal spaces. These
spaces are defined by a local condition on the Littlewood-Paley decom-
position; they are closely related to the usual condition of pointwise
regularity and can actually be defined by local Holder conditions (see
Theorem 1.2). Finally, we will give a first application of this study : a
pointwise regularity result for elliptic operators.
1.1 Modulus of continuity.
The modulus of continuity of a continuous function / at #o is defined
by
u{h) = sup \f(x) - P(x-x0)\
\x—xo\h
where P is the unique polynomial of smallest degree which gives the
smallest order of magnitude for u (for h small).
The determination of u at each point is often impossible and one is
interested in obtaining upper bounds for a;. A modulus of continuity
will be for us a function 9 defined on
IR+,
non-decreasing, not identically
vanishing, and such that
0(0) = 0
0(2x) C9(x) .
Such a function 6 is a modulus of continuity of / at xo if there exists a
polynomial P such that
(1.1) \f(x)-P(x-x0)\ C6(x-x0),
8
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