A real-valued function / defined on IRn belongs to Ca(x0) (a —n)
if there exists a polynomial P of order at most a such that
(0.1) \f(x)-P(x-x0)\ C\x-x0\a .
The supremum of all values of a such that (0.1) holds is called the
Holder exponent of / at XQ and denoted a(a?o). This exponent gives an
idea of the pointwise regularity of the function / at XQ.
The uniform regularity is defined as follows if a is positive (and
a ^ IN): a function / belongs to Ca(IRn) if there exists C 0 such
that (0.1) holds for all xo G K
. The fact that we require the constant
C to be uniform has an important consequence: the uniform Holder
exponent of a function need not be the infimum of the pointwise Holder
exponents. For instance, the function f(x) = xsin(l/x) is a C1 function
at the origin and C°° everywhere else, but its uniform Holder exponent
is only a — 1/2.
Many results in mathematical analysis involve the uniform Holder
regularity. Let us mention an example related to the Sobolev embed-
dings. By definition, a function / belongs to the Sobolev space
/ G Lp and (-A)s/2f G Lp. One of the Sobolev imbeddings states that
if / G Lp s with s J , then / G Ca(IRn) with a = s - J . One of our
purposes will be to obtain similar pointwise results. For instance, we
can pick a larger than s — - and wonder if (0.1) fails on a "large" set.
We will obtain a sharp bound on the dimension of this set.
The problem of determining the exact Holder regularity of a function
is quite easy if this regularity is everywhere the same, because in such
a case it usually coincides with the uniform regularity (the simplest
example is the Weierstrass functions ^2 an sin bnx with a 1 and b 1).
The determination of the pointwise Holder regularity is much harder if
this regularity changes widely from point to point; a good example is
the famous Riemann 's nondifferentiable function
/ 2 \
v--s S m
K n X)
n = l