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A counter-example in singular integral theory

Lawrence B. Difiore, Victor L. Shapiro (2012)

Studia Mathematica

An improvement of a lemma of Calderón and Zygmund involving singular spherical harmonic kernels is obtained and a counter-example is given to show that this result is best possible. In a particular case when the singularity is O(|log r|), let f ∈ C ¹ ( ℝ N ∖ 0 ) and suppose f vanishes outside of a compact subset of ℝ N , N ≥ 2. Also, let k(x) be a Calderón-Zygmund kernel of spherical harmonic type. Suppose f(x) = O(|log r|) as r → 0 in the L p -sense. Set F ( x ) = ∫ ℝ N k ( x - y ) f ( y ) d y ∀ x ∈ ℝ N ∖ 0 . Then F(x) = O(log²r) as r → 0 in the L p -sense, 1 < p < ∞....

A decomposition of a set definable in an o-minimal structure into perfectly situated sets

Wiesław Pawłucki (2002)

Annales Polonici Mathematici

A definable subset of a Euclidean space X is called perfectly situated if it can be represented in some linear system of coordinates as a finite union of (graphs of) definable 𝓒¹-maps with bounded derivatives. Two subsets of X are called simply separated if they satisfy the Łojasiewicz inequality with exponent 1. We show that every closed definable subset of X of dimension k can be decomposed into a finite family of closed definable subsets each of which is perfectly situated and such that any...

A deformation of commutative polynomial algebras in even numbers of variables

Wenhua Zhao (2010)

Open Mathematics

We introduce and study a deformation of commutative polynomial algebras in even numbers of variables. We also discuss some connections and applications of this deformation to the generalized Laguerre orthogonal polynomials and the interchanges of right and left total symbols of differential operators of polynomial algebras. Furthermore, a more conceptual re-formulation for the image conjecture [18] is also given in terms of the deformed algebras. Consequently, the well-known Jacobian conjecture...

A description based on Schubert classes of cohomology of flag manifolds

Masaki Nakagawa (2008)

Fundamenta Mathematicae

We describe the integral cohomology rings of the flag manifolds of types Bₙ, Dₙ, G₂ and F₄ in terms of their Schubert classes. The main tool is the divided difference operators of Bernstein-Gelfand-Gelfand and Demazure. As an application, we compute the Chow rings of the corresponding complex algebraic groups, recovering thereby the results of R. Marlin.

A differential geometric characterization of invariant domains of holomorphy

Gregor Fels (1995)

Annales de l'institut Fourier

Let G = K ℂ be a complex reductive group. We give a description both of domains Ω ⊂ G and plurisubharmonic functions, which are invariant by the compact group, K , acting on G by (right) translation. This is done in terms of curvature of the associated Riemannian symmetric space M : = G / K . Such an invariant domain Ω with a smooth boundary is Stein if and only if the corresponding domain Ω M ⊂ M is geodesically convex and the sectional curvature of its boundary S : = ∂ Ω M fulfills the condition K S ( E ) ≥ K M ( E ) + k ( E , n ) . The term k ( E , n ) is explicitly computable...

A differential-geometric approach to deformations of pairs (X, E)

Kwokwai Chan, Yat-Hin Suen (2016)

Complex Manifolds

This article gives an exposition of the deformation theory for pairs (X, E), where X is a compact complex manifold and E is a holomorphic vector bundle over X, adapting an analytic viewpoint `a la Kodaira- Spencer. By introducing and exploiting an auxiliary differential operator, we derive the Maurer–Cartan equation and differential graded Lie algebra (DGLA) governing the deformation problem, and express them in terms of differential-geometric notions such as the connection and curvature of E, obtaining...

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