Uniform estimates with weights for the\bar \partial - equation

This paper concernsL ^∞-variants of Hörmanders weightedL ²-estimates for the $\bar \partial - equation$ . In particular, we discuss a conjecture concerning suchL ^∞-estimates which is related to the corona problem in the ball, and show a weaker version of this conjecture. The proof uses a refinedL ²-estimate for the canonical solution to the $\bar \partial - equation$ . An alternative approach based on von Neumann’s Minimax theorem is also given.     

Invariant metrics and the boundary behavior of holomorphic functions on domains in\mathbb{C}^n

Invariant metrics are used to provide a unified approach to the study of holomorphic functions in Hardy classes on domains in one and several complex variables. Both approach regions and boundary measures are constructed from the metric. Examples are provided to show how diverse theories can be unified with this approach. The Hartogs extension phenomenon and Fatou’s theorem are seen to be two aspects of the same circle of ideas. Author supported in part by a grant from the National Science Foundation     

Topological Components of the Set of Composition Operators on $$H^\infty (B_N)$$

This paper characterizes the component structure of the space of composition operators acting on , both in the operator norm topology and in the topology induced by the essential norm.     

Local existence theorems with estimates for\bar \partial _b on weakly pseudo-convex CR manifolds

Partially supported by NSF grants DMS 89-01455 and DMS 91-01161     

\lambda-presentable Morphisms, Injectivity and (Weak) Factorization Systems

We show that in a locally -presentable category, every -injectivity class (i.e., the class of all the objects injective with respect to some class of -presentable morphisms) is a weakly reflective subcategory determined by a functorial weak factorization system cofibrantly generated by a class of -presentable morphisms. This was known for small-injectivity classes, and referred to as the ‘small object argument.’ An analogous result is obtained for orthogonality classes and factorization systems, where -filtered colimits play the role of the transfinite compositions in the injectivity case. -presentable morphisms are also used to organize and clarify some related results (and their proofs), in particular on the existence of enough injectives (resp. pure-injectives). Finally, locally -presentable categories are shown to be cellularly generated by the set of morphisms between -presentable objects.     

Integral Operators Induced by the Fock Kernel

We study the L ^p boundedness and find the norm of a class of integral operators induced by the reproducing kernel of Fock spaces over . Dostanić is partialy supported by MNZZS Grant N°ON144010, Zhu is partially supported by the US National Science Foundation.     

The Poincaré-Bertrand Formula for the Bochner-Martinelli Integral

The Poincaré-Bertrand formula and the composition formula for the Bochner-Martinelli integral on piecewise smooth manifolds are obtained. As an application, the regularization problem for linear singular integral equation with Bochner-Martinelli kernel and variable coefficients is discussed.     

Atomic Decomposition of a Weighted Inductive Limit in \mathbb{C}^{n}

We construct an atomic decomposition of the space H_V^∞ consisting of analytic functions on the open unit disc. The research of the second named author was supported by the Academy of Finland project “Functional Analysis and Applications”.     

The Plemelj Formula of Higher Order Partial Derivatives of the Bochner-Martinelli Type Integral

In this paper, by using the technique of integral transformation, we obtain the Plemelj formulas with the Cauchy principal value and the Hadamard principal value of mixed higher order partial derivatives for integral of the Bochner-Martinelli type on a closed smooth manifold ∂D in Cⁿ. From the Plemelj formulas and using the theory of complex partial differential equation, we prove that the problem of higher order boundary value D^κΦ⁺(t) = D^κΦ⁻(t) + f(t) is equivalent to a complex linear higher order partial differential equation. Moreover, given a proper condition of the Cauchy boundary value problem, the problem of higher order boundary value has a unique branch complex harmonic solution satisfying Φ⁻(∞) = 0 in Cⁿ\∂D.     

On some isometries on the Bergman-Privalov class on the unit ball

Bergman-Privalov class AN_α(B) consists of all holomorphic functions on the unit ball B⊂Cⁿ such that

‖f‖_ANα:=∫_Bln(1+∣f(z)∣)dV_α(z)<∞,     

Multipliers in Hardy-Sobolev Spaces

For 1 < p < ∞ and 0 ≤ n − sp < 1, we give characterizations of the space of pointwise multipliers of the holomorphic Hardy-Sobolev spaces H_s^p on the unit ball B of As an application of these results we obtain a corona theorem for these spaces.     

The singular integral equation on a closed piecewise smooth manifold inCn

In this paper, we obtain the general permutation formulas and composition formulas of singular integral of the Bochner-Martinelli type on a closed piecewiseC ⁽¹⁾ smooth manifold. As an application, we consider the corresponding singular integral equation of linear variable coefficients, and prove that the singular integral equation can be transformed to an equivalent Fredholm equation, whose characteristic equation has a unique solution in , here denotes the function set which satisfies the Hölder condition on D and holomorphically expands to domainD.Corresponding author. Project supported in part by the Mathematical Tian Yuan Foundation of China (Grant No. TY10126033).     

Characterisation of homogeneous polynomials which are constant on complex-tangential curves in the boundary of the unit ball of

We prove that a complex-tangential curve γ in the boundary of the unit ball of having the property that there exists a homogeneous polynomial P such that P=1 on γ has constant curvature. This implies that a homogeneous polynomial P having the property that there exists a closed complex-tangential curve γ (respectively a totally real 2-dimensional submanifold) in the boundary of the unit ball of such that P=1 on γ (respectively |P|=1 on γ) reduces to a monomial by a unitary chage of variables. These results represent a positive answer to conjectures of H. O. Kim.     

Singular Berezin Transforms

We give examples of pseudoconvex Reinhardt domains where the Berezin transform has integral kernel with singularities and, hence, fails to be a smoothing map. On the other hand, we show that this can never happen for a plane domain – in fact, then the Bergman kernel is always either identically zero or strictly positive everywhere on the diagonal – and also prove that, in contrast to the example by Wiegerinck from 1984, on any pseudoconvex Reinhardt domain the Bergman space can be finite-dimensional only if it reduces to the constant zero. Received: February 02, 2007. Accepted: May 28, 2007.     

The Bergman kernel of the symmetrized polydisc in higher dimensions has zeros

We prove that the Bergman kernel of the symmetrized polydisc in dimension greater than two has zeros. Received: 15 November 2005; revised: 10 January 2006     

Residue currents and ideals of holomorphic functions

We define a residue current of a holomorphic mapping, or more generally of a holomorphic section of a holomorphic vector bundle, by means of Cauchy-Fantappie-Leray type formulas, and prove that a holomorphic function that annihilates this current belongs to the corresponding ideal locally. We also prove that the residue current coincides with the Coleff-Herrera current in the case of a complete intersection. The residue current is globally defined and this is used in some geometric applications. By means of the residue current we also construct, for an arbitrary ideal, an integral formula for interpolation and division.     

On some estimates and Carleson type measure for multifunctional holomorphic spaces in the unit ball

The aim of this paper is to obtain some new estimates for multifunctional holomorphic expressions by using properties of Bergman metric ball. Also we obtain some characterizations of Carleson type measure for some multifunctional holomorphic spaces defined with Bergman metric ball.     

Problèmes d'extension dans les domaines convexes de type fini

Résumé Soient donnés D un domaine borné de , convexe, de type fini, X une hypersurface complexe telle que soit non vide, connexe et transverse et . Nous nous intéressons au problème suivant. Sous quelle(s) condition(s) existe-t-il telle que Φ|X = φ. Nous allons donner une condition nécessaire à l'existence de l'extension Φ puis sous une condition un peu plus forte nous montrerons la continuité d'un opérateur d'extension de type Berndtsson-Andersson.      

Sur la régularitéC ∞ du\bar \partial au bord d'un domaine de ℂ n dont la forme de Levi a exactements valeurs propres strictement négatives

Sans résumé

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