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