Riemann-Stieltjes Operators And Pointwise Multipliers On F(p,q,s)Spaces In The Unit Ball

Complex harmonic analysis and theory of function space are significant research field of fundamental mathematics.Great achievements have been received in this field since the 1960s such as Corona theorem,H¹-BMO duality theorem,and Real variable theory inH^p spaces.The theory of function spaces of one complex variable has many beautiful results through half-century-long research.Compared to one complex variable,structure and analysis property of several complex variables are immature.Moreover,invariant potential theory associated with the Lapalace-Beltrami operators in several complex variables is essentially different from classical potential theories onRⁿ and the complex unit disc.Thus,this dissertation is devoted to studying the structure and operators of holomorphic function spaces in the unit ball of Cⁿ and it is significant.Some different methods from the unit disc will be used in this dissertation.This dissertation is devoted to characterize the Riemann-Stieltjes operators and pointwise multipliers on F?p,q,s?spaces in the unit ball ofCⁿ.Based on previous research,we will extend the cases of s1=s2 to s1<s2.For convenience,we will write q=p?-n-1,where?>0.We will study the necessary and sufficient conditions for the boundedness and compactness of Riemann-Stieltjes operators and multipliers fromF?p,q,s1?^{spaces to}F?p,q,s2?^{spaces in the three cases}?i??=1?ii??<1?iii??>1.We carried out by an embedding theorem,i.e.F?p,q,s₁?spaces boundedly embedded to the tent type spaceT_p^?,_s2???.The reason we study the properties of F?p,q,s?spaces is that it contains many classical function spaces,such as Besov spaces,the weighted Bergman spaces,the weighted Dirichlet spaces,the?-Bloch spaces,BMOA and the recently introduced _sQ spaces.We could get many important results of other function spaces through studying F?p,q,s?spaces.Firstly,we recall some preliminaries about holomorphic function spaces and operators.Secondly,the boundedness of operators on F?p,q,s?spaces are deduced by an embedding theorem,i.e.F?p,q,s?spaces boundedly embedded to the non-isotropic tent type spacesT_p^?,s???.For the case of s1<s2 and?with?=1 or??1,we study the necessary and sufficient conditions for the boundedness and compactness of Riemann-Stieltjes operators and pointwise multipliers fromF?p,q,s1?spaces to F?p,q,s2?spaces in the unit ball ofCⁿ.Thirdly,we study the necessary and sufficient conditions for the boundedness and compactness of Riemann-Stieltjes operators fromF?p,q,s1?spaces to F?p,q,s2?spaces in the unit ball for the case of s1<s2 by using the previous embedding theorem.Finally,we use the previous conclutions to study the necessary and sufficient conditions for the boundedness and compactness of pointwise multipliers from F?p,q,s1?spaces toF?p,q,s2?spaces in the unit ball for the case of s1<s2.