For , an open and bounded subset of and a non-increasing and non-negative function defined in , , we introduce the space of locally integrable functions satisfying
for every , , where denotes the volume of the ball centered in and radius . The constant does not depend on
We also define the atomic space as the set of functions f(x) such that in the sense of distributions where , , and are atoms satisfying a) , b) , c) ,
We have: I) If (t) is non-increasing and is non-decreasing then is a Banach space. can be represented as the dual of .