The Pointwise Multipliers And Composition Operators On Holomorphic Function Spaces In C~n

In this thesis we mainly consider some problems on holomorphic function spaces on Cn. The thesis consists of four chapters.In Chapter1,we introduce the background of our studying problems and state our main results.In Chapter2, we characterize some multiplier spaces M(βμ) and M(βμ, βv) on polydisc for general normal functions μ and v. The main results as followTheorem2.3.1Let n>1and μ be a normal function on [0,1),then(1)when(?);(2)when(?)if and only if ψ is the constant function.Theorem2.3.2Let μand v be normal functions on[0,1), then ψ∈M(βμ(Dn),βv(Dn)) if and only if ψ∈H(Dn) and hold at the same time.In Chapter3, we discuss the boundedness and compactness conditions of the weighted composition operator from BMOA space to Bloch type spaces on the unit ball, and give the necessary and sufficient conditions by means of Finsler measure.Theorem3.3.1Let0<α<∞and φ be holomorphic self-map on B, ψ∈H(B), then Tψ,φ is a bounded operator from BMOA to βα if and only if hold at the same time, where R<φ(Z)=(Rφ1(Z),…,Rφn(Z)).Theorem3.3.2Let O<α<∞and φ be holomorphic self-map on B, ψ∈H(B), then Tψ,φ is a compact operator from BMOA toβα if and only if: ψ∈βα,ψφ1∈βα(l=1,2,…,n)and hold at the same time, where Rψ(Z)=(Rψ1(Z),…Rψn(Z)).In Chapter4, we discuss the boundedness and compactness conditions to a kind of integral operator Tψ,g from Dirichlet type space Dp to μ-Bloch space βμ on the unit ball, and give the necessary and sufficient conditionsTheorem4.3.1Let p be any real number,μ, and v be normal functions on[0,1), ψ be holomorphic self-map on B,ψ∈H(B), then Tψ,φ is a bounded operator from Dp to βμ if and only if(ii) when p>n,g∈βμ.Theorem4.3.2Let p be any real number, μ and v be normal func-tions on [0,1), ψ be holomorphic self-map on B, ψ∈H(B). If||ψ||∞=then Tψ,g is a compact operator from Dp to βμ if and only if g∈βμ.If||ψ||∞=1>,then Tψ,g is a compact operator from Dp to βμ if and only if(ii) when p>n,g∈βμ.Key words:Polydisc; Bloch type space; Unit ball; Boundedness; Com-...