TABLE OF CONTENTS This appendix contains the definitions and some fundamental results about Besov spaces and related spaces that are mentioned in Chapter 9 and again prominently in Chapter 11.
We begin by defining the homogeneous H lder spaces because they are simple and they lead naturally to the Besov spaces.
For a given 
is defined to be the set of all continuous functions f such that
If we let
then ? ?? ? is a norm and 
is a Banach space in this norm, modulo the constant functions.
This definition can be reformulated using the modulus of continuity ? ?( f,h), which is defined as follows:
Then f belongs to 
if and only if ? ?( f,h) ? Ch ?. It is easy to see that
If 1 ? ? < 2, the definition is similar, but [ ? y f]( x) = f( x + y) ? f( x) is replaced by [ ? 2 y f]( x) = f( x + 2 y) ? 2 f( x + y) + f( x). The space 
is again defined by the condition ? ?( f,h) ? Ch ?. It is a Banach space, but now the elements are modulo the affine functions. For N ? ? < N + 1, [ ?
