Complete Reinhardt Domains Research Articles

Hardy spaces in Reinhardt domains, and Hausdorff operators

We give criteria for a function to be in the Hardy space on a bounded complete Reinhardt domain. Using these and known one-dimensional results, we obtain boundedness conditions for Hausdorff operators on Hardy spaces in Reinhardt domains. The only known earlier result for the polydisk is a paticular case of the obtained results.

Continuous Reinhardt domains from a Jordan viewpoint

As a natural extension of bounded complete Reinhardt domains in ${{\mathbb C}}^N$ to spaces of continuous functions, continuous Reinhardt domains (CRD) are bounded open connected solid sets in commutative $C^*\! $-algebras with respect to the natural orde

The generalized Roper-Suffridge extension operator on bounded complete Reinhardt domains

The generalized Roper-Suffridge extension operator Φ(f) on the bounded complete Reinhardt domain Ω in C n with n ⩾2 is defined by $$\Phi _{n, \beta _2 , \gamma _2 , ..., \beta _n , \gamma _n }^r (f)(z) = \left( {rf\left( {\frac{{z_1 }}{r}} \right), \left( {\frac{{rf(\tfrac{{z_1 }}{r})}}{{z_1 }}} \right)^{\beta _2 } \left( {f'\left( {\frac{{z_1 }}{r}} \right)} \right)^{\gamma _2 } z_2 ,...,\left( {\frac{{rf(\tfrac{{z_1 }}{r})}}{{z_1 }}} \right)^{\beta _n } \left( {f'\left( {\frac{{z_1 }}{r}} \right)} \right)^{\gamma _n } z_n } \right)$$ for (z 1, z 2, ..., z n ) ∈ Ω, where r = r(Ω) = sup{|z 1|: (z 1, z 2, ..., z n ) ∈ Ω}, 0 ≼ γ j ≼ 1 − β j , 0 ≼ β j ≼1, and we choose the brach of the power functions such that \((\tfrac{{f(z_1 )}}{{z_1 }})^{\beta _j } |_{z_1 = 0} = 1\) and \((f'(z_1 ))^{\gamma _j } |_{z_1 = 0} = 1\), j = 2, ..., n. In this paper, we prove that the operator \(\Phi _{n, \beta _2 , \gamma _2 , ..., \beta _n , \gamma _n }^r \) (f) is from the subset of S α * (U) to S α * (Ω) (0 ≼ α 0, j = 1,2, ..., n), U is the unit disc in the complex plane C, and S α * (Ω) is the class of all normalized starlike mappings of order α on Ω. We also obtain that \(\Phi _{n, \beta _2 , \gamma _2 , ..., \beta _n , \gamma _n }^r \) (f) ∈ S α * (D p ) if and only if f ∈ f ∈ S α * (U) for 0 ≼ α < 1 and some suitable constants β j , γ j , p j .

Commutative Algebras of Toeplitz Operators on the Reinhardt Domains

Let $D$ be a bounded logarithmically convex complete Reinhardt domain in $\mathbb{C}^n$ centered at the origin. Generalizing a result for the one-dimensional case of the unit disk, we prove that the $C^*$-algebra generated by Toeplitz operators with bounded measurable separately radial symbols (i.e., symbols depending only on $|z_1|$, $|z_2|$, ..., $|z_n|$) is commutative. We show that the natural action of the $n$-dimensional torus $\mathbb{T}^n$ defines (on a certain open full measure subset of $D$) a foliation which carries a transverse Riemannian structure having distinguished geometric features. Its leaves are equidistant with respect to the Bergman metric, and the orthogonal complement to the tangent bundle of such leaves is integrable to a totally geodesic foliation. Furthermore, these two foliations are proved to be Lagrangian. We specify then the obtained results for the unit ball.

On propagation of boundary continuity of holomorphic functions of several variables

We consider continuity properties of analytic functions in bounded domains in complex space, focusing on the following two questions:Question 1. Given a bounded domain G in complex affine n-space. For which subsets S of the boundary of G does the following hold? If f is a bounded function which is analytic in G and extends continuously to the union of G and S then f extends continously to the closure of G.Question 2. Given a domain G as above and a function f analytic in G and continuous in the closure of G. For which subsets S of the boundary is the rate of continuity of f on the closure of G determined already by its rate of continuity along S?Previously it was known that in some cases, the set S may be taken to be the Shilov boundary of the algebra of analytic functions on the domain which are continuous on the closure of the domain, but examples when this is not sufficient were also known.In Paper I we prove that for a bounded, smoothly bounded pseudoconvex domain G in 2-dimensional complex affine space one may take for S an open neighbourhood of the Shilov boundary (in the boundary of the domain G). This is joint work with B. Joricke.Paper II considers smoothly bounded Reinhardt domains in 2 dimensions and complete Reinhardt domains in complex affine n-space. We prove that in many cases one may let S be the Shilov boundary. Moreover, the respective continuity properties propagate to the closure of the envelope of holomorphy of the domain.Paper III considers pseudoconvex Hartogs domains with connected vertical sections in two dimensions. We prove that in some cases S may be taken to be the Shilov boundary and give examples for when this is not enough.

Global symplectic coordinates on complex domains

In this paper we deal with complex domains M ⊂ C n equipped with a Kähler form ω = i 2 ∂ ∂ ¯ f , where f : M → R only depends on | z j | 2 , j = 1 , … , n for the complex coordinates ( z 1 , … , z n ) in C n . We give an explicit symplectic immersion Φ of ( M , ω ) into R 2 n in Section 2. In Section 3 we study when the map Φ is a global symplectomorphism for the case of complete Reinhardt domains in C 2 .

Roper–Suffridge extension operator and the lower bound for the distortion

Liczberski–Starkov gave a sharp lower bound for ‖ D Φ n ( f ) ( z ) ‖ near the origin, where Φ n is the Roper–Suffridge extension operator and f is a normalized convex mapping on the unit disk in C . They gave a conjecture that the sharp lower bound holds on the Euclidean unit ball B n in C n . In this paper, we will give a sharp lower bound on B n for a more general extension operator and for normalized univalent mappings f or normalized convex mappings f. We will give a lower bound for mappings f in a linear invariant family. We will also give a similar sharp lower bound on bounded convex complete Reinhardt domains in C n .

Regular quantizations of Kähler manifolds and constant scalar curvature metrics

In this paper we prove that if a Kähler manifold ( M, ω ) admits a regular quantization then its scalar curvature is constant. Moreover, we apply this result to the two-dimensional complete Reinhardt domains in C 2 to show that such domains admit a regular quantization iff they are biholomorphically isometric to the 2-ball in C 2 endowed with the hyperbolic metric.

Estimates for the first and second Bohr radii of Reinhardt domains

We obtain general lower and upper estimates for the first and the second Bohr radii of bounded complete Reinhardt domains in C n .

On the order and type of basic and composite sets of polynomials in complete Reinhardt domains

In this paper we will define the order and type of basic and composite sets of polynomials of several complex variables in complete Reinhardt domains. Also, the property \(T_\rho\) of basic and composite sets of polynomials of several complex variables in complete Reinhardt domains is discussed.

Holomorphic mappings into bounded complete reinhardt domains of holomorphy which are kobayashi isometries at one point

Let M be a connected taut complex manifold of dimension n such that the set H(M) of holomorphic functions on M separates points of M and let D be a bounded complete Reinhardt domain of holomorphy in Let f:M→D be a holomorphic mapping. Let p be a point of M. Assume that f(p)=0 and that df p is an isometry for the infinitesimal Kobayashi metric, in this case, we will show that f is a biholomorphic mapping

Proper holomorphic maps between complete reinhardt domains in

Let D and D′ be two bounded complete Reinhardt domains in and let F: D → D′ be a proper holomorphic mapping. We prove that if D or D′ is not a generalized pseudoellipsoid, then there exist two bidiscs δ ⊆ D and δ′ ⊇ D′ such that F restricts to a proper holomorphic map F δ → δ ′. From this it is deduced that, if D ≠ δ F is always of the form F(z1,z2)= (Az m π(1), Bzn π(2)), where π is a permutation of {1,2}. This fact brings to the complete classification of all proper mans between bounded complete: Reinhardt domains in .

Proper holomorphic mappings between bounded complete Reinhardt domains in C2

Proper holomorphic mappings between bounded complete Reinhardt domains in C2

The Bergman kernel function and proper holomorphic mappings

It is proved that a proper holomorphic mapping f f between bounded complete Reinhardt domains extends holomorphically past the boundary and that if, in addition, f − 1 ( 0 ) = { 0 } {f^{ - 1}}(0) = \{ 0\} , then f f is a polynomial mapping. The proof is accomplished via a transformation rule for the Bergman kernel function under proper holomorphic mappings.