Pointwise Multiplication Operators From Hardy Spaces To Weighted Bergman Spaces In The Unit Ball Of C~n

Complex harmonic analysis and function space theory are significant research field of fundamental mathematics.Great achievements have been received in this field since the 1960s.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.Pointwise multiplication operators not only be closely related to Toeplitz operators and Hankel operators,but also it can be used to solve the important corona type problems.Thus,it is of great significance to studying the pointwise multiplication operators from Hardy spaces to weighted Bergman spaces in the unit ball ofCⁿ.In the dissertation,some new methods have been used,which are different from the case of one variable.The specific research work are as follows:Firstly,we describe the boundedness of multiplication operators from Hardy spaces to weighted Bergman spaces in the unit ball ofCⁿ.The boundedness of multiplication operatorsM_g:H^p→A_β^q are fully characterized in the unit ball of Cⁿ for the values of p,q in the five cases:（1）0<p<q<∞,γ>0,（2）γ=0,0<p<q<∞,（3）0<p<q<∞,γ<0,（4）0<q<p<∞,（5）0<p=q<∞,It is an extension of to the complex ball.Secondly,we further describe the compactness of multiplication operators from Hardy spaces to weighted Bergman spaces in the unit ball ofCⁿ.In the case of the unit disk,the boundedness is characterized by Zhao,but the compactness is not given.The methods of the proof of compactness are inspired by Pau.The compactness of multiplication operatorsM_g:H^p→A_β^q are characterized in the unit ball ofCⁿ for the values of p,q in the five cases:（1）0<p<q<∞,γ>0,（2）0<p<q<∞,γ=0,（3）0<p<q<∞,γ<0,（4）0<q<p<∞,（5）p=q=2.Finally,we need the characterization of higher order radial derivatives for the VMOA_p^αspace when we describe the compactness of multiplication operators M_g:H^p→A_β^q in the unit ball ofCⁿ.Therefore,the characterization of higher order radial derivatives for theBMOA₂^αspaces andVMOA₂^αspaces are given in detail.