Formulars for the L...-minimal solutions of the ...-equation in the unit ball of C...
The Corona Theorem for Matrices.
Cauchy-Fantappiè-Leray formulas with local sections and the inverse Fantappiè transform
Coleff-Herrera currents, duality, and noetherian operators
Let be a coherent subsheaf of a locally free sheaf and suppose that has pure codimension. Starting with a residue current obtained from a locally free resolution of we construct a vector-valued Coleff-Herrera current with support on the variety associated to such that is in if and only if . Such a current can also be derived algebraically from a fundamental theorem of Roos about the bidualizing functor, and the relation between these two approaches is discussed. By a construction...
Values in the interior of the L-minimal solutions of the ∂∂-equation in the unit ball of C.
We derive formulas for the values in the interior of the L-minimal solutions of the ∂∂-equation in the unit ball of C. These formulas generalize previously known formulas for the boundary values of the same solutions. We estimate the solution and obtain a (known) result concerning weighted Nevanlinna classes.
(Ultra)differentiable functional calculus and current extension of the resolvent mapping
Let be a tuple of commuting operators on a Banach space . We discuss various conditions equivalent to that the holomorphic (Taylor) functional calculus has an extension to the real-analytic functions or various ultradifferentiable classes. In particular, we discuss the possible existence of a functional calculus for smooth functions. We relate the existence of a possible extension to existence of a certain (ultra)current extension of the resolvent mapping over the (Taylor) spectrum of . If ...
Uniqueness and factorization of Coleff-Herrera currents
We prove a uniqueness result for Coleff-Herrera currents which in particular means that if defines a complete intersection, then the classical Coleff-Herrera product associated to is the unique Coleff-Herrera current that is cohomologous to with respect to the operator , where is interior multiplication with . From the uniqueness result we deduce that any Coleff-Herrera current on a variety is a finite sum of products of residue currents with support on and holomorphic forms.
The membership problem for polynomial ideals in terms of residue currents
We find a relation between the vanishing of a globally defined residue current on and solution of the membership problem with control of the polynomial degrees. Several classical results appear as special cases, such as Max Nöther’s theorem, for which we also obtain a generalization. Furthermore there are some connections to effective versions of the Nullstellensatz. We also provide explicit integral representations of the solutions.
Complex Kergin interpolation and the Fantappiè transform.
Boundary Convergence in Non-Tangential and Nonadmissible Approach Regions.
A constructive proof of the composition rule for Taylor's functional calculus
We give a new constructive proof of the composition rule for Taylor's functional calculus for commuting operators on a Banach space.
The essential spectrum of holomorphic Toeplitz operators on spaces
We compute the essential Taylor spectrum of a tuple of analytic Toeplitz operators on , where D is a strictly pseudoconvex domain. We also provide specific formulas for the index of provided that is a compact subset of D.
Non-holomorphic functional calculus for commuting operators with real spectrum
We consider -tuples of commuting operators on a Banach space with real spectra. The holomorphic functional calculus for is extended to algebras of ultra-differentiable functions on , depending on the growth of , , when . In the non-quasi-analytic case we use the usual Fourier transform, whereas for the quasi-analytic case we introduce a variant of the FBI transform, adapted to ultradifferentiable classes.
Residue currents with prescribed annihilator ideals
Formulas for approximate solutions of the ∂∂`-equation in a strictly pseudoconvex domain.
Let D be a bounded strictly pseudoconvex domain in C. We construct approximative solution formulas for the equation i∂∂`u = θ, θ being an exact (1,1)-form in D. We show that our formulas give simple proofs of known estimates and indicate further applications.
Green functions, Segre numbers, and King’s formula
Let be a coherent ideal sheaf on a complex manifold with zero set , and let be a plurisubharmonic function such that locally at , where is a tuple of holomorphic functions that defines . We give a meaning to the Monge-Ampère products for , and prove that the Lelong numbers of the currents at coincide with the so-called Segre numbers of at , introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we show that satisfy a certain generalization...
Henkin-Ramirez formulas with weight factors
We construct a generalization of the Henkin-Ramírez (or Cauchy-Leray) kernels for the -equation. The generalization consists in multiplication by a weight factor and addition of suitable lower order terms, and is found via a representation as an “oscillating integral”. As special cases we consider weights which behave like a power of the distance to the boundary, like exp- with convex, and weights of polynomial decrease in . We also briefly consider kernels with singularities on subvarieties...
On the Briançon-Skoda theorem on a singular variety
Let be a germ of a reduced analytic space of pure dimension. We provide an analytic proof of the uniform Briançon-Skoda theorem for the local ring ; a result which was previously proved by Huneke by algebraic methods. For ideals with few generators we also get much sharper results.
Das Flaviussche Sieb
Two examples of subspaces in spanned by characters of finite order
By a Fourier multiplier technique on Cantor-like Abelian groups with characters of finite order, the norms from L² into of certain embeddings of character sums are computed. It turns out that the orders of the characters are immaterial as soon as they are at least four.
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