Some Characterizations Of Bloch-Type Space On The Unit Ball

The dissertation collects the main results obtained by the author during the period when she has applied for the M.D.The author studies some characterizations of the Bloch-type space(including the classic Bloch,theα-Bloch space,theμ-Bloch space)on the n-dimensional unit ball and generalizes the results of some known results on the unit disk and on the unit ball.In the first chapter,research background of the Bloch-type space,some notations, definitions,and theorems are given,at the same time,research significance of this paper is introduced.In the second chapter,firstly,the characterizations of theα-Bloch space on the unit ball are studied.Forα-Bloch space,the earliest work is the coefficient characterization and gradient or radial derivative characterization ofα-Bloch function.Recently,the characterizations ofα-Bloch functions on the unit ball without use of derivative are given.Based on the above idea,this chapter gives a new characterization ofα-Bloch functions on the unit ball without use of derivative,then expands the research method ofα-Bloch functions space.At the same time,the corresponding results of the littleα-Bloch space are given.Secondly,we gain three new necessary and sufficient conditions about the characterization of the classic Bloch space. In the third chapter,theμ-Bloch space on the unit ball is studied.Theμ-Bloch space includes the classic Bloch space,α-Bloch space,logarithmic Bloch space,we can say,the Bloch-type spaces we studied before are all the special cases of theμ-Bloch space.The earlyμ-Bloch space is defined by the simple gradient and radial derivative.Based on the various characterizations of theα-Bloch space,this chapter gives the characterizations of theμ-Bloch space and littleμ-Bloch space without use of derivative.In the fourth chapter,the research methods are different from the ones of the second chapter and the third chapter,by integral characterization,the author obtains some necessary and sufficient conditions about logarithmic Bloch space on the unit ball,at the same time,we obtain the new integral characterization of the classic Bloch space, and generalize the results before.