Periodic Function Spaces

Abstract

Let T _n be the n-torus. As usual, we represent T _n in R _n by the cube

$$T_n = \{ x|x \in R_n ,x = (x_1 ,...,x_n ),|x_j | \le \pi \,{\rm{with}}\,j = 1,...,n\} ,$$

where opposite points are identified. More precisely,x∈T _n and y∈T _n are identified if and only if x–y=2πk,where k=(k ₁,⃛,k _n ( is a vector with the integer-valued components k ₁,⃛,k _n .We denote the lattice of such ve·ctors k with integer-valued components by Z _n .The problem is to introduce and to study spaces of type $B_{p,q}^8$ and $B_{p,q}^8$ on T _n .One can try to develop such a theory parallel to Chapter 2, where the Euclidean n-space R _n is replaced by the n-torus T _n .