2
Stephane Jaffard and Yves Meyer
where the Holder regularity at a point XQ depends on the Diophantine
approximation properties of xo and is thus discontinuous everywhere (see
[23]). The determination of the pointwise regularity of such a function
cannot be directly deduced from the definition, but requires the use of
some powerful tools that were first introduced by J.M. Bony in a different
context [3], [4]: the two-microlocal spaces C^ which are defined by
local conditions on the Littlewood-Paley decomposition of the function.
The definition of pointwise regularity is difficult to handle. For in-
stance, suppose that (0.1) holds for a large a and that V / exists in a
neighbourhood of XQ. In general, V / does not belong to Ca~1(xo) as
might be expected (actually, V / may have no regularity at XQ). Another
drawback is that (0.1) is not stable under some natural operators used
in analysis, such as the Hilbert transform. Hence the need for some
"good substitutes" for the Holder regularity condition; we will see that
the two-microlocal spaces play that role.
Another substitute for condition (0.1) had been introduced by Calde-
ron and Zygmund ([7]) and is defined as follows :
(0.2)
A function / belongs to TP(xo) if, for p small enough,
I/P
Cpu
\ \ j \f(x)-P(x-x0)\Pdx
VP J\x—xo\p
The usual condition
CU(XQ)
corresponds to p = oo and of course, if /
belongs to Cn(xo), / belongs to
T^(XQ)
for any p. These classes of func-
tions have all the stability properties that we mentioned, and Calderon
and Zygmund proved pointwise regularity results for elliptic operators
in terms of conditions. Later, "refined Sobolev embeddings" were
obtained by Ziemer [42] (i.e., estimates of the dimension of the set of
points x where a function in
Ws'v
does not belong to T£(x)).
The main problem when dealing with the T% condition is that one
cannot "come back" and deduce from it any regularity of type (0.1),
even if one accepts a some loss on the value of u. This is a reason for
using two-microlocal conditions instead: they will be very closely related
to the pointwise regularity condition. Another reason that will not be
developed here is its particular efficiency in the study of multifractal
functions (for such applications, we refer to [23], [24] and [26]). In
Chapter I, after some general results on the modulus of continuity of
functions, we recall the definition of two-microlocal spaces, derive their
main properties, and characterize them in terms of decay properties of
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