3 Wavelet Methods for Pointwise Regularity

the wavelet transform and of local Holder conditions. As a first appli-

cation, we prove a pointwise regularity result for elliptic operators.

We will study the following refinements of the notion of "singularity"

at Xo :

• A point Xo is an a-singularity of / if no better estimate than (0.1)

holds, i.e., if, for all C 0 there exist xn — » xo such that for any

polynomial P

\f(xn)-P(xn-x0)\

C\xn-x0\a.

It may happen that (0.1) does not hold because of a very small set

of "bad points" near xo, so that, if modified on a small set, / would

be smooth. Such points XQ are very difficult to detect using averaged

quantities (such as the wavelet coefficients of / ) , so we introduce the

following stronger definition.

• Let \A\ denote the Lebesgue measure of the set A. A point xo is a

strong a-singularity of / if there exist C, C 0 such that, if B£(xo) is

the ball centered at XQ of radius £, there exist two subsets A£ and C£ of

B€ of size comparable to B£ (i.e. \A£\ C\B£\ and \C£\ C\B£\) such

that

(0.3) inf/(x)-sup/(x ) C'ea.

As B£

This means that, at small scales,

\f(x)-f(y)\ C\x-y\a

for a relatively large set of x's and y's close to x$. In this definition a

can take negative values (—n a 1).

This natural definition is also motivated by results in PDEs. For

instance, singularities of PDEs are sometimes known to be strong singu-

larities (as is the case for the singularities of the Navier-Stokes equations

in three dimensions, if they exist; see [8]).

In Chapter II, we give a bound on the dimension of the set where

functions in Ws'p or Lp'5 have singularities. Depending on the type of

singularities we consider, we bound either their Hausdorff or their pack-

ing dimension. Using wavelet expansions, we will show the optimality

of these results. As a limit case, we will deduce the almost everywhere

Cs

regularity of functions in

Lp,s

(thus generalizing a classical result if