Products Of Extended Cesaro Operator And Composition Operator On Function Spaces

This thesis studies the boundedness and compactness of products of ex-tended Cesaro operator and composition operator between different function spaces,mainly including,first,products of extended Cesaro operator and com-position operator from generalized Besov spaces to Zygmund spaces on the unit disk;and second,products of extended Cesaro operator and composition operator from Bloch-type spaces to Zygmund-type spaces on the unit ball.Supposing that D be the unit disk in the complex plane C,H(D) be the class of all holomorphic functions on D.For 0<p<00,-1<q<∞,the generalized Bcsov space B(p,q)can be defined as where dA(z)is the normalized Lebesgue area measure on D.The Zygmund space Z defined by Z={f∈H(D)∩C(D):‖f‖z<∞}, where It is easy to see that‖f‖z(?)(1-│z│2)│f"(z)│. Z is a Banach space under the norm‖f‖=│f(0)│+│f'(0)│+(?)(1-│z│2)│f"(z)│.Let Bn be the unit ball of Cn,H(Bn)be the family of all holomorphic functions on Bn.The Bloch-type spaces and little Bloch-type spaces defined by Bw,0={f∈H(Bn):(?) w(z)│▽f(z)│=0}, where▽f(z)=((?)) and w is a norm function.Zygmund-type space Z on the unit ball defined by Zμ={f∈H(Bn):(?)μ(│z│)│R2f(z)│<∞}, where R2f(z)=R(Rf(z)),Rf(z)=(?)zj be the radial derivative of f. What's more,we say f in little Zygmund-type spaces if f∈Zμand satisfies (?)μ(│z│)│R2f(z)│=0. The Zμ(Zμ,0)be Banach space under the norm‖f‖=│f(0)│+│f'(0)│+(?)μ(│z│)│R"f(z)│.Let g∈H(Bn),φis a holomorphic self-mapping,the products of extended Cesaro operator and composition operator defined by Products of extended Cesaro operator and composition operator on H(Bn)defined by Ifφis identically mapping,the integral-type operator be extended Cesaro oper-ator defined on H(D) and H(Bn) as and The extended Cesaro operator is significant in the operator theory of holomorphic function spaces.Therefore,it is necessary to study this operator TgCφon the holomorphic function spaces.We characterize the boundedness and compactness of the operator TgCφfrom generalized Besov spaces to Zygmund spaces on the unit disk;what's more,we characterize the boundedness and compactness of the operator TgCφfrom Bloch-type spaces to Zygmund-type spaces on the unit ball. Our study not only enlarge the research field of operators, but also gives a deeper clarification of this operator.