4

Stephane Jaffard and Yves Meyer

s G IN; see [29]). We will pay special attention to the "limit cases" where

s = ^. More precisely, we will consider the two following problems :

• Prove similar results for functions in BV(TR)

• Find which Sobolev spaces contain the characteristic function of a

given set Q (depending on the regularity of the boundary 9Q).

In Section II.4, we show how wavelet methods can extend the usual

trace formulas for functions in Sobolev or Besov spaces; we illustrate

the ideas involved with an example. Let / be a function defined on

IR2

and which belongs to the Sobolev space

HS(1R2).

Consider now the one

variable functions fa which are the traces of / on the line x = a; then

fa G iy

s _ 1 / / 2

(IR) (if s 1/2) uniformly in the parameter a; and for

almost every a, fa G

H8(JR).

Our purpose is to obtain intermediate

results. For instance, here, if s — 1/2

sf

s, fa G

Hs

(IR) except

on a set of values of the parameter a of Hausdorff dimension at most

l-2(s-s').

In Chapter III we examine the relation between wavelet expansions

and lacunary trigonometric series, returning first to the problem of the

"almost characterization" of differentiability of a function. In

Chapter I different criteria of differentiability are given; we study here

a specific lacunary trigonometric series whose regularity cannot be ob-

tained through microlocal arguments and we show how direct estimates

give some information on the pointwise behavior of this trigonometric

series.

Up to now we described the singularity of f{x) at x

0

only by the

Holder exponent a(x

0

) so that, if a 1, we looked for the order of

magnitude of \f(x) — /(xo)| when x tends to xo, without taking into

account the oscillations of f(x) — f(xo). One of our purposes is to inves-

tigate some specific local behaviors such as approximate selfsimilarities

and very strong oscillatory behaviors ("chirps" of the form

xa

sin(l/a;^))

and show that the two-microlocal spaces allow us to characterize and eas-

ily distinguish these behaviors. The definition of a chirp requires first

the notion of indefinitely oscillating functions.

A function g(x) (x G IR) is indefinitely oscillating if the iterated prim-

itives of g (defined by #o = 9 and p

n +

i(x ) = — f 9n(t) dt) are 0(1) at

infinity. We can now define the chirps of type (a, (5) and of regularity r.

a is the Holder exponent as above, whereas /3 measures the oscillations

which accelerate when x tends to XQ. Finally, r 0 gives information

on the global smoothness of / .