POINTWISE MULTIPLICATION OF BESOV AND TRIEBEL-LIZORKIN SPACES

It is shown that para-multiplication applies to a certain product pi(u ,v) defined for appropriate u and v in L' (R(n)). Boundedness of pi(., .) is investigated for the anisotropic Besov and Triebel-Lizorkin spac es - i.e., for B-p,B-q (M,s) and F-p,F-q (M,s) with s is an element of R and p and q in ]0, infinity in the F-case) - with a treatment of th e generic as well as various borderline cases. For max (s(0),s(1)) > 0 the spaces B-p0,B-q0 (M,s0) circle plus B-p1,B-q1 (M,s1) and F-p0,F-s 0 (M,s0) circle plus F-p1 q1,F- (M,s1) to which pi(.,.) applies are de termined. For generic F-p0,F-q0 (s0) circle plus F-p1 q1 (s1) the rece iving F-p,F-q (s) spaces are characterized. It is proved that pi(f,g) = f . g holds for functions f and g when f . g is an element of L(1,lo c), roughly speaking. In addition, pi(f,u) = fu when f is an element o f O-M and u is an element of L'.. Moreover, for an arbitrary open set Omega subset of R(n), a product pi(Omega)(.,.) is defined by lifting t o R(n). Boundedness of pi on R(n) is shown to carry over to pi(Omega) is general.