Introduction

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

n

. 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

Lp,s

if

/ 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

0 0

/ 2 \

v--s S m

\

K n X)

"(*)

= L

n2

n = l

1