Biharmonic Bloch Spaces,Bergman Projection And BMO Spaces

Bloch spaces,Bergman projections and BMO spaces are three research objects of spaces of analytic functions,their theories of functions enrich the theoretical basis of complex analysis.Bergman projection builds a relation between the set of bounded functions and Bloch spaces.There is an inclusion relation between a Bloch spaces and a BMO spaces in the case of univalent functions.BMO spaces are closely related to Carleson measures.In this paper,we will study Bloch spaces in the sense of biharmonic mappings,Bergman projection and Carleson type measure.Main results of this paper are exhibited as followed:Part one,we study biharmonic Bloch spaces.By the linearity and a composition invariant of biharmonic mappings,a criterion on biharmonic Bloch functions is given.Basing on the theory of representation and estimate of Pre-Schwarz derivative for biharmonic mappings,we obtain a univalent criterion and coefficient estimate for biharmonic Bloch functions.Part two,we estimate the norm of weighted Bergman projection on the upper half planeΠ.It is shown that a weighted Bergman projection_αP maps the spaceL^∞（Π）into a Bloch space B（Π）,satisfying that|P_αf||_B（Π）≤C||f||_L∞（Π）,here C is a sharp constant.Moreover,we construct a new Bergman projection operator,and obtain an norm estimate of it.Part three,we study a criterion of BMO spaces.We first generalize a criterion of BMO spaces to the case of a general exponent.Next,we obtain a sufficient and necessary condition to determine a positive measureλgiven by_αT to be a Carleson measure.Last,we define a type of Carleson measure and then give its criterion.