We investigate the adjoints of linear fractional composition operators C ? acting on classical Dirichlet space D(B N ) in the unit ball B N of ? N , and characterize the normality and essential normality of C ? on D(B N ) and the Dirichlet space modulo constant function D 0(B N ), where ? is a linear fractional map ? of B N . In addition, we also show that for any non-elliptic linear fractional map ? of B N , the composition maps σ o ? and ? o σ are elliptic or parabolic linear fractional maps of B N .
Let V be a hypersurface with an isolated singularity at the origin in Cn+1. It is a natural question to ask when V is defined by weighted homogeneous polynomial or homogeneous polynomial up to biholomorphic change of coordinates. In 1971, a beautiful theorem of Saito gives a necessary and sufficient condition for V to be defined by a weighted homogeneous polynomial. For a two-dimensional isolated hypersurface singularity V, Xu and Yau found a coordinate free characterization for V to be defined by a homogeneous polynomial. Recently Lin and Yau gave necessary and sufficient conditions for a 3-dimensional isolated hypersurface singularity with geometric genus bigger than zero to be defined by a homogeneous polynomial. The purpose of this paper is to prove that Lin-Yau's theorem remains true for singularities with geometric genus equal to zero.
Motivated by Ru and Stoll's accomplishment of the second main theorem in higher dimension with moving targets, many authors studied the moving target problems in value distribution theory and related topics. But thereafter up to the present, all of researches about normality criteria for families of meromorphic mappings of several complex variables into PN(C) have been still restricted to the hyperplane case. In this paper, we prove some normality criteria for families of meromorphic mappings of several complex variables into PN(C) for moving hyperplanes, related to Nochka's Picard-type theorems.The new normality criteria greatly extend earlier related results.
In this paper we first look upon some known results on the composition operator as bounded or compact on the Bloch-type space in polydisk and ball, and then give a sufficient and necessary condition for the composition operator to be compact on the Bloch space in a bounded symmetric domain.
ZHOU Zehua & CHE Renyu Department of Mathematics, Tianjin University, Tianjin 300072, China
This article gives a normal criterion for families of holomorphic mappings of several complex variables into P N(C)for moving hypersurfaces in pointwise general position,related to an Eremenko’s theorem.
This article proves the existence of singular directions of value distribution theory for some transcendental holomorphic curves in the n-dimensional complex projective space P^n（C）.. An example is given to complement these results.
