1 Main resultA connected open set Ω C<sup>n</sup> is called a domain of holomorphy, if there do not exist nonemptyopen sets Ω<sub>1</sub>, Ω<sub>2</sub> with Ω<sub>2</s...1 Main resultA connected open set Ω C<sup>n</sup> is called a domain of holomorphy, if there do not exist nonemptyopen sets Ω<sub>1</sub>, Ω<sub>2</sub> with Ω<sub>2</sub> connected, Ω<sub>1</sub> Ω<sub>2</sub>∩Ω, Ω<sub>2</sub> Ω, so that for every u that is holo-morphic on Ω there is a u<sub>2</sub> holomorphic on Ω<sub>2</sub> with u=u<sub>2</sub> on Ω<sub>1</sub>. This definition is compli-cated. Generally speaking, a domain of holomorphy is a domain of definition of holomorphicfunctions in the sense that there exists a holomorphic function on Ω that cannot be holomor-phically continued to any slightly large open set. Domains of holomorphy play a very importantrole in the theory of several complex variables. There are many different characterizations fordomains of holomorphy--geometrical, analytical or algebraical, each takes its different effi-cient functions in treating various concrete problems of several complex variables （see ref.[1], for example）. In this note, we use the skills of sheaf theory to give a characterization ofdomains of holomorphy by mealls of some interpolation property.展开更多
We study solutions to convolution equations for functions with discrete support in R^n, a special case being functions with support in the integer points. The Fourier transform of a solution can be extended to a holom...We study solutions to convolution equations for functions with discrete support in R^n, a special case being functions with support in the integer points. The Fourier transform of a solution can be extended to a holomorphic function in some domains in C^n, and we determine possible domains in terms of the properties of the convolution operator.展开更多
This paper studied the connection between normal family and unicity, and proved some results on unicity of entire functions. Mostly, it was proved： Let f be a nonconstant entire function, and let a, c be two nonzero ...This paper studied the connection between normal family and unicity, and proved some results on unicity of entire functions. Mostly, it was proved： Let f be a nonconstant entire function, and let a, c be two nonzero complex numbers. If E（a,f）=E（a,f＇ ）, and f＂（z）=c whenever f＇ （z）=a, then f（z）=Ae^（cz）/u ＋（ac-a^2）/c.The proof uses the theory of normal families in an essential way.展开更多
For a tuple A = （A1, A2,..., An） of elements in a unital algebra/3 over C, its projective spectrum P（A） or p（A） is the collection of z ∈ Cn, or respectively z ∈ pn-1 such that A（z） = z1A1＋z2A2＋…＋znAn is ...For a tuple A = （A1, A2,..., An） of elements in a unital algebra/3 over C, its projective spectrum P（A） or p（A） is the collection of z ∈ Cn, or respectively z ∈ pn-1 such that A（z） = z1A1＋z2A2＋…＋znAn is not invertible in/3. The first half of this paper proves that if/3 is Banach then the resolvent set PC（A） consists of domains of holomorphy. The second half computes the projective spectrum for the generating vectors of a Clifford algebra. The Chern character of an associated kernel bundle is shown to be nontrivial.展开更多
We study the class of functions called monodiffric of the second kind by Isaacs. They are discrete analogues of holomorphic functions of one or two complex variables. Discrete analogues of the Cauchy-Riemann operator,...We study the class of functions called monodiffric of the second kind by Isaacs. They are discrete analogues of holomorphic functions of one or two complex variables. Discrete analogues of the Cauchy-Riemann operator, of domains of holomorphy in one discrete variable, and of the Hartogs phenomenon in two discrete variables are investigated. Two fundamental solutions to the discrete Cauchy-Riemann equation are studied: one with support in a quadrant, the other with decay at infinity. The first is easy to construct by induction; the second is accessed via its Fourier transform.展开更多
We study discrete analogues of holomorphic functions of one and two variables, especially those that were called monodiffric functions of the first kind by Rufus Isaacs. Discrete analogues of the Cauchy-Riemann operat...We study discrete analogues of holomorphic functions of one and two variables, especially those that were called monodiffric functions of the first kind by Rufus Isaacs. Discrete analogues of the Cauchy-Riemann operators, domains of holomorphy in one discrete variable, and the Hartogs phenomenon in two discrete variables are investigated.展开更多
Let P_n={Z=(Z_1,…,Z_n)|Z_iZ_i^n<1, Z_i are 2×2 complex matrices},H_n={W=(W_1,…, W_n)|W_i=Z_iB,(Z_1,…,Z_n)∈P_n,B∈SL(2,C)}, D_n={W=(W_1,…,W_n)| W_i=AZ_iB,(Z_1,…,Z_n)∈P_n,A, B∈SL(2, C)}. Are H_n,D_n doma...Let P_n={Z=(Z_1,…,Z_n)|Z_iZ_i^n<1, Z_i are 2×2 complex matrices},H_n={W=(W_1,…, W_n)|W_i=Z_iB,(Z_1,…,Z_n)∈P_n,B∈SL(2,C)}, D_n={W=(W_1,…,W_n)| W_i=AZ_iB,(Z_1,…,Z_n)∈P_n,A, B∈SL(2, C)}. Are H_n,D_n domains of holomorphy? In the present paper, we prove that H_2, D_2 are domains of holomorphy by using the follow-ing proposition: H_2={W∈C^2[2×2]|W_1W_2~*∈P_1, |detW_1|<1, |detW_2|<1}.展开更多
In this survey article, we present some known results and also propose some open questions related to the analytic and geometric aspects of Bishop submanifolds in a complex space. We mainly focus on those problems tha...In this survey article, we present some known results and also propose some open questions related to the analytic and geometric aspects of Bishop submanifolds in a complex space. We mainly focus on those problems that the author and his coauthors have recently worked on. The article also contains an example of a Bishop submanifold in C^3 of real codimension two, which cannot be quadratically flattened at a CR singular point but is CR non-minimal at any CR point. This provides a counter-example to a question asked in a private communication by Zaistev(2013).展开更多
In the study of the holonomic modules over D_n(■_n) and ■p, it is claimed and used that gr(D_n)(gr(■_n)) and gr(■p) are regular Noetherian rings with pure dimension 2n, where D_n is the stalk of the sheaf of diffe...In the study of the holonomic modules over D_n(■_n) and ■p, it is claimed and used that gr(D_n)(gr(■_n)) and gr(■p) are regular Noetherian rings with pure dimension 2n, where D_n is the stalk of the sheaf of differential operators withholomorphic coefficients, and ■p is the stalk of the sheaf ■ of microlocal differential operators. This property is used to prove j(M)+d(M)=2n for any finitely generated modules over D_n(■_n) and ■p by using the generalized Roos Theorem. In [1], it was proved that gr(D_n)(gr(■_n)) and gr(■p) do not have pure dimension, so we cannot apply the generalized Roos Theorem directly. In this paper, we reestablish the formula j(M)+d(M)=2n for any finitely generated modules over D_n(■_n) and■p.展开更多
文摘1 Main resultA connected open set Ω C<sup>n</sup> is called a domain of holomorphy, if there do not exist nonemptyopen sets Ω<sub>1</sub>, Ω<sub>2</sub> with Ω<sub>2</sub> connected, Ω<sub>1</sub> Ω<sub>2</sub>∩Ω, Ω<sub>2</sub> Ω, so that for every u that is holo-morphic on Ω there is a u<sub>2</sub> holomorphic on Ω<sub>2</sub> with u=u<sub>2</sub> on Ω<sub>1</sub>. This definition is compli-cated. Generally speaking, a domain of holomorphy is a domain of definition of holomorphicfunctions in the sense that there exists a holomorphic function on Ω that cannot be holomor-phically continued to any slightly large open set. Domains of holomorphy play a very importantrole in the theory of several complex variables. There are many different characterizations fordomains of holomorphy--geometrical, analytical or algebraical, each takes its different effi-cient functions in treating various concrete problems of several complex variables （see ref.[1], for example）. In this note, we use the skills of sheaf theory to give a characterization ofdomains of holomorphy by mealls of some interpolation property.
文摘We study solutions to convolution equations for functions with discrete support in R^n, a special case being functions with support in the integer points. The Fourier transform of a solution can be extended to a holomorphic function in some domains in C^n, and we determine possible domains in terms of the properties of the convolution operator.
文摘This paper studied the connection between normal family and unicity, and proved some results on unicity of entire functions. Mostly, it was proved： Let f be a nonconstant entire function, and let a, c be two nonzero complex numbers. If E（a,f）=E（a,f＇ ）, and f＂（z）=c whenever f＇ （z）=a, then f（z）=Ae^（cz）/u ＋（ac-a^2）/c.The proof uses the theory of normal families in an essential way.
基金supported by National Natural Science Foundation of China(Grant No.11101079)and China Scholarship Council
文摘For a tuple A = （A1, A2,..., An） of elements in a unital algebra/3 over C, its projective spectrum P（A） or p（A） is the collection of z ∈ Cn, or respectively z ∈ pn-1 such that A（z） = z1A1＋z2A2＋…＋znAn is not invertible in/3. The first half of this paper proves that if/3 is Banach then the resolvent set PC（A） consists of domains of holomorphy. The second half computes the projective spectrum for the generating vectors of a Clifford algebra. The Chern character of an associated kernel bundle is shown to be nontrivial.
文摘We study the class of functions called monodiffric of the second kind by Isaacs. They are discrete analogues of holomorphic functions of one or two complex variables. Discrete analogues of the Cauchy-Riemann operator, of domains of holomorphy in one discrete variable, and of the Hartogs phenomenon in two discrete variables are investigated. Two fundamental solutions to the discrete Cauchy-Riemann equation are studied: one with support in a quadrant, the other with decay at infinity. The first is easy to construct by induction; the second is accessed via its Fourier transform.
文摘We study discrete analogues of holomorphic functions of one and two variables, especially those that were called monodiffric functions of the first kind by Rufus Isaacs. Discrete analogues of the Cauchy-Riemann operators, domains of holomorphy in one discrete variable, and the Hartogs phenomenon in two discrete variables are investigated.
文摘Let P_n={Z=(Z_1,…,Z_n)|Z_iZ_i^n<1, Z_i are 2×2 complex matrices},H_n={W=(W_1,…, W_n)|W_i=Z_iB,(Z_1,…,Z_n)∈P_n,B∈SL(2,C)}, D_n={W=(W_1,…,W_n)| W_i=AZ_iB,(Z_1,…,Z_n)∈P_n,A, B∈SL(2, C)}. Are H_n,D_n domains of holomorphy? In the present paper, we prove that H_2, D_2 are domains of holomorphy by using the follow-ing proposition: H_2={W∈C^2[2×2]|W_1W_2~*∈P_1, |detW_1|<1, |detW_2|<1}.
基金supported by National Science Foundation of USA(Grant No.NSF-1363418)
文摘In this survey article, we present some known results and also propose some open questions related to the analytic and geometric aspects of Bishop submanifolds in a complex space. We mainly focus on those problems that the author and his coauthors have recently worked on. The article also contains an example of a Bishop submanifold in C^3 of real codimension two, which cannot be quadratically flattened at a CR singular point but is CR non-minimal at any CR point. This provides a counter-example to a question asked in a private communication by Zaistev(2013).
基金Project supported by the National Natural Science Foundation of China
文摘In the study of the holonomic modules over D_n(■_n) and ■p, it is claimed and used that gr(D_n)(gr(■_n)) and gr(■p) are regular Noetherian rings with pure dimension 2n, where D_n is the stalk of the sheaf of differential operators withholomorphic coefficients, and ■p is the stalk of the sheaf ■ of microlocal differential operators. This property is used to prove j(M)+d(M)=2n for any finitely generated modules over D_n(■_n) and ■p by using the generalized Roos Theorem. In [1], it was proved that gr(D_n)(gr(■_n)) and gr(■p) do not have pure dimension, so we cannot apply the generalized Roos Theorem directly. In this paper, we reestablish the formula j(M)+d(M)=2n for any finitely generated modules over D_n(■_n) and■p.
