Domains Of Finite Type Research Articles

Pinchuk scaling method on domains with non-compact automorphism groups

In this paper, we characterize weakly pseudoconvex domains of finite type in [Formula: see text] in terms of the boundary behavior of automorphism orbits by using the scaling method.

The Gehring–Hayman type theorem on pseudoconvex domains of finite type in $$\mathbb {C}^2$$

The Gehring–Hayman type theorem on pseudoconvex domains of finite type in $$\mathbb {C}^2$$

Sobolev and Hölder estimates for homotopy operators of the [formula omitted]-equation on convex domains of finite multitype

We construct homotopy formulas for the ∂‾-equation on convex domains of finite type that have optimal Sobolev and Hölder estimates. For a bounded smooth finite type convex domain Ω⊂Cn that has q-type mq for 1≤q≤n, our ∂‾ solution operator Hq on (0,q)-forms has (fractional) Sobolev boundedness Hq:Hs,p→Hs+1/mq,p and Hölder–Zygmund boundedness Hq:Cs→Cs+1/mq for all s∈R and 1<p<∞. We also demonstrate the Lp-boundedness Hq:Hs,p→Hs,prq/(rq−p) for all s∈R and 1<p<rq, where rq:=(n−q+1)mq+2q.

The Kobayashi Metric and Gromov Hyperbolicity on Pseudoconvex Domains of Finite Type in $${\mathbb {C}}^2$$

The Kobayashi Metric and Gromov Hyperbolicity on Pseudoconvex Domains of Finite Type in $${\mathbb {C}}^2$$

Composition operators between Bergman spaces on infinite type convex domains in ℂ2

ABSTRACT We provide characterizations of boundedness and compactness for composition operators between Bergman spaces of smoothly bounded convex domains in C 2 admitting the so-called geometric F-type. Such geometric type contains all domains of finite type and many cases of infinite type in the sense of Range.

Bi-Hölder Extensions of Quasi-isometries on Pseudoconvex Domains of Finite Type in $${\mathbb {C}}^2$$

In this paper, we prove that the identity map for the smoothly bounded pseudoconvex domain of finite type in $${\mathbb {C}}^2$$ extends to a bi-Hölder map between the Euclidean boundary and Gromov boundary. As an application, we show the bi-Hölder boundary extensions for quasi-isometries between these domains. Moreover, we get a more accurate index of the Gehring–Hayman type theorem for the bounded m-convex domains with Dini-smooth boundary.

Characterization of Pseudoconvex Domains of Finite Type with Locally Diagonalizable Levi Form by Their Automorphism Groups

Characterization of Pseudoconvex Domains of Finite Type with Locally Diagonalizable Levi Form by Their Automorphism Groups

Lattice points in stretched finite type domains

We study an optimal stretching problem, which is a variant lattice point problem, for convex domains in Rd (d≥2) with smooth boundary of finite type that are symmetric with respect to each coordinate hyperplane/axis. We prove that optimal domains which contain the most positive (or least nonnegative) lattice points are asymptotically balanced.

Dyadic decomposition of convex domains of finite type and applications

In this paper, we introduce a dyadic structure on convex domains of finite type via the so-called dyadic flow tents. This dyadic structure allows us to establish weighted norm estimates for the Bergman projection P on such domains with respect to Muckenhoupt weights. In particular, this result gives an alternative proof of the $$L^p$$ boundedness of P. Moreover, using extrapolation, we are also able to derive weighted vector-valued estimates and weighted modular inequalities for the Bergman projection.

Hilbert-Kunz density function for graded domains

We prove the existence of HK density function for a graded pair ( R , I ) , where R is an N -graded domain of finite type over a perfect field and I ⊂ R is a graded ideal of finite colength. This generalizes our earlier result where one proves the existence of such a function for a pair ( R , I ) , where, in addition R is standard graded. Other properties of the HK density functions also hold for the graded pairs: for example, it is a multiplicative function for Segre products, its maximum support is the F -threshold of an m -primary ideal provided Proj R is smooth, it has a closed formula when either I is generated by a system of parameters or R is of dimension two. As one of the consequences we show that if G is a finite group scheme acting linearly on a polynomial ring R of dimension d then the HK density function f R G , m G , of the pair ( R G , m G ) , is a piecewise polynomial function of degree d − 1 . We also compute the HK density functions for ( R G , m G ) , where G ⊂ S L 2 ( k ) is a finite group acting linearly on the ring k [ X , Y ] .

Formula omitted]-corona problem and convex domains of finite type

We prove that the Hp-corona problem has a solution for two generators in convex domains of finite type in Cn, n≥2.

Triangular resolutions and effectiveness for holomorphic subelliptic multipliers

A solution to the effectiveness problem in Kohn's algorithm for generating holomorphic subelliptic multipliers is provided for general classes of domains of finite type in Cn, that include the so-called special domains given by finite and infinite sums of squares of absolute values of holomorphic functions. Also included is a more general class of domains recently discovered by M. Fassina [23]. More generally, for any smoothly bounded pseudoconvex domain we introduce an invariantly defined associated sheaf S of C-subalgebras of holomorphic function germs, that combined with a result of Fassina, reduces the existence of effective subelliptic estimates at p to a purely algebraic geometric question of controlling the multiplicity of S.Our main new tool, a triangular resolution, is the construction of subelliptic multipliers decomposable as Q∘Γ, where Γ is constructed from pre-multipliers and Q is part of a triangular system. The effectiveness is proved via a sequence of newly proposed procedures, called here meta-procedures, built on top of the holomorphic Kohn's procedures, where the order of subellipticity can be effectively tracked. Important sources of inspiration are algebraic geometric techniques by Y.-T. Siu [54,55] and procedures for triangular systems by D.W. Catlin and J.P. D'Angelo [16,8].The proposed procedures are purely algebraic and as such can be of wider interest for geometric and computational problems involving Jacobian determinants, such as resolving singularities of holomorphic maps.

Bekollé-Bonami estimates on some pseudoconvex domains

We establish a weighted Lp norm estimate for the Bergman projection for a class of pseudoconvex domains. We obtain an upper bound for the weighted Lp norm when the domain is, for example, a bounded smooth strictly pseudoconvex domain, a pseudoconvex domain of finite type in C2, a convex domain of finite type in Cn, or a decoupled domain of finite type in Cn. The upper bound is related to the Bekollé-Bonami constant and is sharp. When the domain is smooth, bounded, and strictly pseudoconvex, we also obtain a lower bound for the weighted norm. As an additional application of the method of proof, we obtain the result that the Bergman projection is weak-type (1,1) on these domains.

Variance of lattice point counting in some special shells in [formula omitted

We study the variance of the random variable that counts the number of lattice points in some shells generated by a special class of finite type domains in Rd. The proof relies on estimates of the Fourier transform of indicator functions of convex domains.

A remark on boundary Schwarz lemma for the convex domain of finite type

Let Ω be a bounded convex domain of finite type with smooth boundary. If f is a holomorphic self-mapping of Ω which fixes a boundary point P, we obtain some estimates of the eigenvalues of the complex Jacobian matrix of f at P. These estimates can be thought of as a generalization of the boundary Schwarz lemma on strongly pseudoconvex domains.

Compactness of the [formula omitted]-Neumann problem on domains with bounded intrinsic geometry

By considering intrinsic geometric conditions, we introduce a new class of domains in complex Euclidean space. This class is invariant under biholomorphism and includes strongly pseudoconvex domains, finite type domains in dimension two, convex domains, C-convex domains, and homogeneous domains. For this class of domains, we show that compactness of the ∂¯-Neumann operator on (0,q)-forms is equivalent to the boundary not containing any q-dimensional analytic varieties (assuming only that the boundary is a topological submanifold). We also prove, for this class of domains, that the Bergman metric is equivalent to the Kobayashi metric and that the pluricomplex Green function satisfies certain local estimates in terms of the Bergman metric.

Commutators of Cauchy–Fantappiè Type Integrals on Generalized Morrey Spaces on Complex Ellipsoids

Let $$\Omega $$ be a domain which belongs to a class of bounded weakly pseudoconvex domains of finite type in $${\mathbb {C}}^n$$ , let $$d\lambda $$ be the Monge–Ampere boundary measure on $$b\Omega $$ and $$\varrho \ge 0$$ be a non-decreasing function. The aim of this paper is to establish the characterizations of boundedness and compactness for the commutator operators of Cauchy–Fantappie type integrals with $$L^1(b\Omega ,d\lambda )$$ functions on the generalized Morrey spaces $$L^{p}_\varrho (b\Omega ,d\lambda )$$ , with $$p\in (1, \infty )$$ .

Lorentz boundedness and compactness characterization of integral commutators on spaces of homogeneous type

We study the Lorentz boundedness properties of Calderón-type commutators and relate those properties to the spaces of bounded mean oscillation and vanishing mean oscillation. In application, we obtain Lorentz boundedness and compactness characterization of integral commutators with singular kernel of Calderón–Zygmund type on strictly pseudoconvex domains in ℂn, or convex domain of finite type in ℂn.

Estimates of Kähler-Einstein metrics on pseudoconvex domains of finite type with locally diagonalizable Levi form

In this short note, we use the weights introduced by Ph. Charpentier and Y. Dupain on pseudoconvex domains of finite type with locally diagonalizable Levi form to prove that on these domains, the Kähler-Einstein metrics are comparable with the Carathéodory, Kobayashi and Bergman metrics.

Boundary behavior of the Carathéodory and Kobayashi–Eisenman volume elements

We study the boundary limits of the Carath\'eodory and Kobayashi-Eisenman volume elements on smoothly bounded convex finite type domains and also on Levi corank one domains.