The Dolbeault complex in infinite dimensions II

We study the equation on a ball , and prove that it is solvable if is a Lipschitz continuous, closed -form.

     

The Dolbeault complex in infinite dimensions III. Sheaf cohomology in Banach spaces

We prove that the sheaf cohomology groups H ^q (Ω,?) vanish if Ω is a pseudoconvex open subset of a Banach space with unconditional basis, and q≥1. Oblatum 23-XII-1999 & 12-V-2000?Published online: 16 August 2000     

The Hoffman-Wielandt inequality in infinite dimensions

The Hoffman-Wielandt inequality for the distance between the eigenvalues of two normal matrices is extended to Hilbert-Schmidt operators. Analogues for other norms are obtained in a special case.     

The steiner point in infinite dimensions

It is shown that the Steiner point cannot be extended continuously to all convex bodies in infinite dimensional Hilbert spaces. This follows as a corollary of a result on the local behavior of the point.     

The infinity Laplacian in infinite dimensions

We study three properties of real-valued functions defined on a Banach space: The absolutely minimizing Lipschitz functions, the viscosity solutions of the infinity Laplacian partial differential equation, and the functions which satisfy comparison with cones. We prove that these notions are equivalent, and we show the existence of such functions. These results are new in the infinite-dimensional case.Received: 14 May 2003, Accepted: 4 August 2003, Published online: 2 April 2004Mathematics Subject Classification (2000): 49L25, 35J60, 54C20     

The Dolbeault complex with weights according to normal crossings

In the present paper, we define a Dolbeault complex with weights according to normal crossings, which is a useful tool for studying the -equation on singular complex spaces by resolution of singularities (where normal crossings appear naturally). The major difficulty is to prove that this complex is locally exact. We do that by constructing a local -solution operator which involves only Cauchy’s Integral Formula (in one complex variable) and behaves well for L ^p -forms with weights according to normal crossings.      

Dolbeault cohomology of compact complex homogeneous manifolds

We show that ifM is the total space of a holomorphic bundle with base space a simply connected homogeneous projective variety and fibre and structure group a compact complex torus, then the identity component of the automorphism group ofM acts trivially on the Dolbeault cohomology ofM. We consider a class of compact complex homogeneous spacesW, which we call generalized Hopf manifolds, which are diffeomorphic to S¹ ×K/L whereK is a compact connected simple Lie group andL is the semisimple part of the centralizer of a one dimensional torus inK. We compute the Dolbeault cohomology ofW. We compute the Picard group of any generalized Hopf manifold and show that every line bundle over a generalized Hopf manifold arises from a representation of its fundamental group.     

The correspondence formula of Dolbeault complex on pair deformation

Geometriae Dedicata - Given a holomorphic family of pairs $$\{(X_t,E_t)\}$$ where each $$E_t$$ is a holomorphic vector bundle over a compact complex manifold $$X_t$$ , we get a correspondence...     

The convergence-extension theorem of Noguchi in infinite dimensions

In this note we give generalizations of Noguchi's convergence-extension theorem to the case of infinite dimension.

     

Sobolev embeddings in infinite dimensions

In this paper, we study Sobolev spaces in infinite dimensions and the corresponding embedding theorems. Our underlying spaces are ?^r for r ∈ [1, ∞), equipped with corresponding probability measures.For the weighted Sobolev space W_b^1,p(?^r, γ_a) with a weight a ∈ ?^r of the Gaussian measure γa and a gradient weight b ∈ ?^∞, we characterize the relation between the weights(a and b) and the continuous(resp. compact)log-Sobolev...     

Semi-linear problems in infinite dimensions

We prove existence, smoothness and ergodicity results for semilinear parabolic problems on infinite dimensional spaces assuming the Logarithmic Sobolev inequality is satisfied. As a consequence we construct a class of nonlinear Markov semigroup which are hypercontractive.     

Center manifolds of infinite dimensions I: Main results and applications

In this paper, the first of a bipartite work, we consider an abstract, nonautonomous system of evolution equations of hyperbolic type, related to semilinear wave equations. Theorem 1 states that under certain assumptions the system admits a global center manifold, or equivalently a global decoupling function which is continuously differentiable with respect to its arguments, among which timet occurs. The difficult proof is presented in part II, i.e. the continuation of the present paper. For purposes of applications a local version of Theorem 1 is proved, i.e. the local center manifold Theorem 2. We obtain a series of applications both to abstract, nonautonomous wave equations and to concrete nonautonomous, semilinear wave equations subject to Neumann and Dirichlet boundary conditions.     

Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology

We consider a special class of compact complex nilmanifolds, which we call compact nilmanifolds with nilpotent complex structure. It is shown that if is a compact nilmanifold with nilpotent complex structure, then the Dolbeault cohomology is canonically isomorphic to the -cohomology of the bigraded complex of complex valued left invariant differential forms on the nilpotent Lie group .

     

Coverings and dimensions in infinite profinite groups

Answering a question of Miklós Abért, we prove that an infinite profinite group cannot be the union of less than continuum many translates of a compact subset of box dimension less than 1. Furthermore, we show that it is consistent with the axioms of set theory that in any infinite profinite group there exists a compact subset of Hausdorff dimension 0 such that one can cover the group by less than continuum many translates of it.     

On analytic Ornstein-Uhlenbeck semigroups in infinite dimensions

We extend to infinite dimensions an explicit formula of Chill, Fašangová, Metafune, and Pallara for the optimal angle of analyticity of analytic Ornstein-Uhlenbeck semigroups. The main ingredient is an abstract representation of the Ornstein-Uhlenbeck operator in divergence form. The authors are supported by the ‘VIDI subsidie’ 639.032.201 of the Netherlands Organization for Scientific Research (NWO) and by the Research Training Network HPRN-CT-2002-00281. Received: 28 June 2006 Revised: 5 January 2007