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Institute
Given a system of entire functions in Cn with at most countable set of common zeros, we introduce the concept of zeta-function associated with the system. Under reasonable assumptions on the system, the zeta-function is well defined for all s ∈ Zn with sufficiently large components. Using residue theory we get an integral representation for the zeta-function which allows us to construct an analytic extension of the zeta-function to an infinite cone in Cn.
We study the Neumann problem for the de Rham complex in a bounded domain of Rn with singularities on the boundary. The singularities may be general enough, varying from Lipschitz domains to domains with cuspidal edges on the boundary. Following Lopatinskii we reduce the Neumann problem to a singular integral equation of the boundary. The Fredholm solvability of this equation is then equivalent to the Fredholm property of the Neumann problem in suitable function spaces. The boundary integral equation is explicitly written and may be treated in diverse methods. This way we obtain, in particular, asymptotic expansions of harmonic forms near singularities of the boundary.
For each compact subset K of the complex plane C which does not surround zero, the Riemann surface Sζ of the Riemann zeta function restricted to the critical half-strip 0 < Rs < 1/2 contains infinitely many schlicht copies of K lying ‘over’ K. If Sζ also contains at least one such copy, for some K which surrounds zero, then the Riemann hypothesis fails.
We show a Lefschetz fixed point formula for holomorphic functions in a bounded domain D with smooth boundary in the complex plane. To introduce the Lefschetz number for a holomorphic map of D, we make use of the Bergman kernal of this domain. The Lefschetz number is proved to be the sum of usual contributions of fixed points of the map in D and contributions of boundary fixed points, these latter being different for attracting and repulsing fixed points.
Anisotropic edge problems
(2002)
We investigate elliptic pseudodifferential operators which degenerate in an anisotropic way on a submanifold of arbitrary codimension. To find Fredholm problems for such operators we adjoint to them boundary and coboundary conditions on the submanifold.The algebra obtained this way is a far reaching generalisation of Boutet de Monvel's algebra of boundary value problems with transmission property. We construct left and right regularisers and prove theorems on hypoellipticity and local solvability.
The classical Lefschetz fixed point formula expresses the number of fixed points of a continuous map f : M -> M in terms of the transformation induced by f on the cohomology of M. In 1966 Atiyah and Bott extended this formula to elliptic complexes over a compact closed manifold. In particular, they presented a holomorphic Lefschtz formula for compact complex manifolds without boundary, a result, in the framework of algebraic geometry due to Eichler (1957) for holomorphic curves. On compact complex manifolds with boundary the Dolbeault complex is not elliptic, hence the Atiyah-Bott theory is no longer applicable. To get rid of the difficulties related to the boundary behaviour of the Dolbeault cohomology, Donelli and Fefferman (1986) derived a fixed point formula for the Bergman metric. The purpose of this paper is to present a holomorphic Lefschtz formula on a compact complex manifold with boundary
Function spaces with asymptotics is a usual tool in the analysis on manifolds with singularities. The asymptotics are singular ingredients of the kernels of pseudodifferential operators in the calculus. They correspond to potentials supported by the singularities of the manifold, and in this form asymptotics can be treated already on smooth configurations. This paper is aimed at describing refined asymptotics in the Dirichlet problem in a ball. The beauty of explicit formulas highlights the structure of asymptotic expansions in the calculi on singular varieties.
The inhomogeneous ∂-equations is an inexhaustible source of locally unsolvable equations, subelliptic estimates and other phenomena in partial differential equations. Loosely speaking, for the anaysis on complex manifolds with boundary nonelliptic problems are typical rather than elliptic ones. Using explicit integral representations we assign a Fredholm complex to the Dolbeault complex over an arbitrary bounded domain in C up(n).
Let A be a determined or overdetermined elliptic differential operator on a smooth compact manifold X. Write Ssub(A)(D) for the space of solutions to thesystem Au = 0 in a domain D ⊂ X. Using reproducing kernels related to various Hilbert structures on subspaces of Ssub(A)(D) we show explicit identifications of the dual spaces. To prove the "regularity" of reproducing kernels up to the boundary of D we specify them as resolution operators of abstract Neumann problems. The matter thus reduces to a regularity theorem for the Neumann problem, a well-known example being the ∂-Neumann problem. The duality itself takes place only for those domains D which possess certain convexity properties with respect to A.
The problem of analytic representation of integrable CR functions on hypersurfaces with singularities is treated. The nature o singularities does not matter while the set of singularities has surface measure zero. For simple singularities like cuspidal points, edges, corners, etc., also the behaviour of representing analytic functions near singular points is studied.
For a domain D subset of IRn with singular points on the boundary and a weight function ω infinitely differentiable away from the singularpoints in D, we consider a C*-algebra G (D; ω) of operators acting in the weighted space L² (D, ω). It is generated by the operators XD F-¹ σ F XD where σ is a homogeneous function. We show that the techniques of limit operators apply to define a symbol algebra for G (D; ω). When combined with the local principle, this leads to describing the Fredholm operators in G (D; ω).
We prove the existence of a limit in Hm(D) of iterations of a double layer potential constructed from the Hodge parametrix on a smooth compact manifold with boundary, X, and a crack S ⊂ ∂D, D being a domain in X. Using this result we obtain formulas for Sobolev solutions to the Cauchy problem in D with data on S, for an elliptic operator A of order m ≥ 1, whenever these solutions exist. This representation involves the sum of a series whose terms are iterations of the double layer potential. A similar regularisation is constructed also for a mixed problem in D.
We prove a theorem on analytic representation of integrable CR functions on hypersurfaces with singular points. Moreover, the behaviour of representing analytic functions near singular points is investigated. We are aimed at explaining the new effect caused by the presence of a singularity rather than at treating the problem in full generality.