THE -PROBLEM FOR HOLOMORPHIC (0,2)-FORMS ON PSEUDOCONVEX DOMAINS IN SEPARABLE HILBERT SPACES AND D.F.N. SPACES

51. IntroductionL. H5.m.nd..l3] solved the 0problem by using the L2-estimates for partial differentialoperators in C'.. J. Kajiwara[4] studied infinite dimensional generalizations of the poten-tial kernel. Concerning the 0-problem in infinite dimensional spaces, P. ffeb.i.lll] investi-gated the a-equation for coc (o, 1)-forms in arbitrary pseudoconvex open subsets of separableHilbert spaces without growth condition. J. F. Colombeau and B. Perr.t[l1 showed that aCoc solution u of 0u = w ca…     

THE δ^－－PROBLEM FOR HOLOMORPHIC （0,2）－FORMS ON PSEUDOCONVEX DOMAINS IN SEPARABLE HILBERT SPACES AND D.F.N. SPACES

This paper shows that the δ^--problem for holomorphic (0, 2)-forms on Hilbert spaces is solv-able on pseudoconvex open subsets. By using this result, the authors investigate the existence ofthe solution of the δ^--equation for holomorphic(0, 2)-forms on pseudoconvex domains in D.F.N.spaces.     

A remark on a paper by S.R. Bell

It is shown that unbranched proper holomorphic maps between pseudoconvex domains with smooth C boundaries, one of which satisfies subelliptic estimates for the -Neumann problem on (0,1)-forms, extend to unbranched C-coverings between the closures of the domains.     

Holomorphic Deformations of the Ricci-flat ?-manifolds

We present a construction of globally convergent power series of integrable Beltrami differentials on the Ricci-flat ??-manifolds and also a construction of global canonical family of holomorphic(n,0)-forms on the deformation spaces of the Ricci-flat ??-manifolds.     

Hermitian-Einstein metrics for vector bundles on complete Kähler manifolds

In this paper, we prove the existence of Hermitian-Einstein metrics for holomorphic vector bundles on a class of complete Kähler manifolds which include Hermitian symmetric spaces of noncompact type without Euclidean factor, strictly pseudoconvex domains with Bergman metrics and the universal cover of Gromov hyperbolic manifolds etc. We also solve the Dirichlet problem at infinity for the Hermitian-Einstein equations on holomorphic vector bundles over strictly pseudoconvex domains.

     

Division theorems for exact sequences

Under certain integrability and geometric conditions, we prove division theorems for the exact sequences of holomorphic vector bundles and improve the results in the case of Koszul complex. By introducing a singular Hermitian structure on the trivial bundle, our results recover Skoda’s division theorem for holomorphic functions on pseudoconvex domains in complex Euclidean spaces.     

Holomorphic deformations of the Ricci-flat $$\partial \overline \partial $$-manifolds

We present a construction of globally convergent power series of integrable Beltrami differentials on the Ricci-flat \(\partial \overline \partial \)-manifolds and also a construction of global canonical family of holomorphic (n, 0)-forms on the deformation spaces of the Ricci-flat \(\partial \overline \partial \)-manifolds.     

COHOMOLOGY VANISHING IN HILBERT SPACES

§1. Introduction Oka [40] proved that any additive Cousin problem is solvable in a domain of holomorphyin Cn. Oka [41]-Cartan [2, 3]-Serre [46] generalized this as the theorem B for analyticcoherent sheaves over Stein spaces. J. P. Serre [47] proved…     

Extension of CR maps of positive codimension

We study the holomorphic extendability of smooth CR maps between real analytic strictly pseudoconvex hypersurfaces in complex affine spaces of different dimensions.     

Characterizations of Hardy-Orlicz spaces on strictly pseudoconvex domains of C<Superscript>n</Superscript>

In this paper, we characterize Hardy-Orlicz spaces on strictly pseudoconvex domains of Cⁿ using real variable methods. Concretely, we prove the basic properties for a boundary behavior of functions in Hardy-Orlicz spaces, an approximation of functions in Hardy-Orlicz spaces by some functions holomorphic up to a neighborhood of the boundary, and a boundedness of the Szegö projection.     

DYNAMICS ON WEAKLY PSEUDOCONVEX DOMAINS

ＤＹＮＡＭＩＣＳＯＮＷＥＡＫＬＹＰＳＥＵＤＯＣＯＮＶＥＸＤＯＭＡＩＮＳ￥ＺＨＡＮＧＷＥＮＪＵＮ；ＲＥＮＦＹＵＡＯ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨｅｎａｎＵｎｉｖｅｒｓｉｔｙｌＫａｉｆｅｎｇ４７５００１，Ｃｈｉｎａ．）（Ｉｎｓｔｉｔ...     

BERGMAN TYPE PROJECTIONS ON L~p SPACES

The paper is a survey on the action of Bergman type projections on various Lp spaces. The focus is on three types of holomorphic function spaces: weighted Bergman spaces, the Block space, and diagonal Besov spaces.     

ON THE CONVEXITY OF SOME BANACH SPACES

51. IntroductionThe convexity of Banach spaces is an important topic in functional analysis and playsan importals role in indnite dimensional holomorphy. In order to study the geometricproperties of a Banach spacet Clarksonl3] and Krein (independently) introduced the veryimportallt class of strict convex spaces. In the same paper [3], Clarkson also introduced thestronger notion of uniform convexity. Since Clarkson's paper many authors have definedand studied the classes of Banach spaces lyi…     

Fredholm composition operators on spaces of holomorphic functions

Composition operators on vector spaces of holomorphic functions are considered. Necessary conditions that range of the operator is of a finite codimension are given. As a corollary of the result it is shown that a composition operatorC on a certain Banach space of holomorphic functions on a strictly pseudoconvex domain withC ² boundary or a polydisc or a compact bordered Riemann surface or a bounded domainD such that intD = D is invertible if and only if it is a Fredholm operator if and only if is a holomorphic automorphism.     

On the -equation¶over pseudoconvex Kähler manifolds

The Picard variety Pic⁰(? ⁿ ) of a complex n-dimensional torus? ⁿ is the group of all holomorphic equivalence classes of topologically trivial holomorphic (principal) line bundles on ? ⁿ . The total space of a topologically trivial holomorphic (principal) line bundle on a compact K?hler manifold is weakly pseudoconvex. Thus we can regard Pic⁰(? ⁿ ) as a family of weakly pseudoconvex K?hler manifolds. We consider a problem whether the Kodaira's -Lemma holds on a total space of holomorphic line bundle belonging to Pic⁰(? ⁿ ). We get a criterion for this problem using a dynamical system of translations on Pic⁰(? ⁿ ). We also discuss the problem whether the -Lemma holds on strongly pseudoconvex K?hler manifolds or not. Using the result of ColColţoiu, we find a 1-convex complete K?hler manifold on which the -Lemma does not hold. Received: 11 June 1999 / Revised version: 22 November 1999     

Toeplitz operators and weighted Bergman kernels

For a smoothly bounded strictly pseudoconvex domain, we describe the boundary singularity of weighted Bergman kernels with respect to weights behaving like a power (possibly fractional) of a defining function, and, more generally, of the reproducing kernels of Sobolev spaces of holomorphic functions of any real order. This generalizes the classical result of Fefferman for the unweighted Bergman kernel. Finally, we also exhibit a holomorphic continuation of the kernels with respect to the Sobolev parameter to the entire complex plane. Our main tool are the generalized Toeplitz operators of Boutet de Monvel and Guillemin.     

赋范线性空间中的广义凸集

文[1]拓广凸集的概念,在 R~(?) 中引入了伪凸、拟凸等广义凸集的概念,获得了它们的一些性质,因而可使得优化理论的研究更为深入.熟知,逼近理论在优化中的应用是非常广泛的(见[2]),本文试图把广义凸集引入赋范线性空间中,并侧重探究其逼近性质.自然,文[1]在 R~(?) 中得到的广义凸集的一些性质,大多数在赋范空间中都是成立的,且证明     

Super Toeplitz operators on line bundles

Let L^k be a high power of a hermitian holomorphic line bundle over a complex manifold X. Given a differential form f on X, we define a super Toeplitz operator T_f acting on the space of harmonic (0, q)-forms with values in L^k, with symbol f. The asymptotic distribution of its eigenvalues, when k tends to infinity, is obtained in terms of the symbol of the operator and the curvature of the line bundle L, given certain conditions on the curvature. For example, already when q=0, i.e., the case of holomorphic sections, this generalizes a result of Boutet de Monvel and Guillemin to semi-positive line bundles. The asympotics are obtained from the asymptotics of the Bergman kernels of the corresponding harmonic spaces, which have independent interest. Applications to sampling are also given.     

Harmonic and holomorphic 1-forms on compact balanced Hermitian manifolds

On compact balanced Hermitian manifolds we obtain obstructions to the existence of harmonic 1-forms, ∂-harmonic (1,0)-forms and holomorphic (1,0)-forms in terms of the Ricci tensors with respect to the Riemannian curvature and the Hermitian curvature. Necessary and sufficient conditions the (1,0)-part of a harmonic 1-form to be holomorphic and vice versa, a real 1-form with a holomorphic (1,0)-part to be harmonic are found. The vanishing of the first Dolbeault cohomology groups of the twistor space of a compact irreducible hyper-Kähler manifold is shown.     

CLASSES OF CONNECTIONS ASSOCIATED WITH DISTRIBUTIONS ON PRODUCT SPACES

Abstract

It is assumed that an n-dimensional distribution is given on an (n+M)-dimensional product space. The latter is endowed with a connection, by means of which the covariant exterior derivatives of the functions that specify the distribution are defined. It is postulated that the connection be such that these derivatives vanish identically. This gives rise to an analysis of the integrability conditions associated with the distribution in terms of appropriate torsion and curvature 2-forms. A further specialization of the connection leads to Edelen's theory [1] of distributions on spaces of fibres.