The Equivalent Norms Of F(p,q,s)Space In C~n

Let0<p<∞,0<s<∞,-1<q+s<∞,-1<q+n<∞. This paper is devoted to six equivalent norms of the functions in the F(p,q,s) space on the unit ball in Cn under different conditions. It consists of three chapters.In Chapter1, we summerize the background and present conditions of the equiv-alent norms of F(p,q,s) space.In Chapter2, we already know the original definition of F(p,q,s) space, it uses the complex gradient and logarithmic weight of automophism. Through proving, we get other three equivalent norms which are described by radial derivative, modulus of an automorphism or logarithmic weight of automophism. That is to say four norms are equivalent.In Chapter3, changing the restriction of parameters p,q,s, we get another two equivalent norms which are described by invariant gradient, modulus of an automor-phism or logarithmic weight of automophism. When n>1, altogether six norms are equivalent. It also points out that if the parameters do not satisfy these requirements, the latter two norms are not equivalent of the original norm.