CAFFARELLI-KOHN-NIRENBERG INEQUALITIES FOR BESOV AND TRIEBEL-LIZORKIN-TYPE SPACES

We present some Caffarelli-Kohn-Nirenberg-type inequalities for Herz-type Besov-Triebel-Lizorkin spaces, Besov-Morrey and Triebel-Lizorkin-Morrey spaces. More precisely, we investigate the inequalities f k 1,r v,  c f 1- K 2, u f  K 3,1 p As and f E p,2,u c f 1- M & mu;f Nqs,,v,with some appropriate assumptions on the parameters, where k 1,r v, are the Herz-type Bessel potential spaces, which are just the Sobolev spaces if a1 = 0,1 < r = v < and a N0, and Kp 3,1As  are Besov or Triebel-Lizorkin spaces if a3 = 0 and 6.1 = p. The usual Littlewood-Paley technique, Sobolev and Franke embeddings are the main tools of this paper. Some remarks on Hardy-Sob olev inequalities are given.