Gevrey函数类和超分布中的Paley—Wiener型定理及其应用

众所周知,Paley-Wiener定理深刻地刻画了具紧支集的无穷可微函数及具紧支集的分布同它们的Fourier-Laplace变换之间的关系,建立了具紧支集的无穷可微函数或分布同指数型整函数之间的关系。正因为如此,Paley-Wiener定理在数学中、特别是在偏微分方程的C~∞理论中起着相当重要的作用。本文在具紧支集的Gevrey函数类及具紧支集的超分布中考虑同样类型的定理,称之为Paley-Wiener型定理。     

The cauchy problem for a class of evolution systems of the parabolic type with unbounded coefficients

For a class of evolution systems of the parabolic type with unbounded coefficients, we study the properties of the fundamental solution matrices and establish the well-posed solvability of the Cauchy problem for these systems in spaces of distributions similar to Gevrey ultradistributions. For a subclass of such systems, we describe the maximal classes of well-posed solvability of the Cauchy problem.     

Regularity and solvability of linear differential operators in Gevrey spaces

We are interested in the following question: when regularity properties of a linear differential operator imply solvability of its transpose in the sense of Gevrey ultradistributions? This question is studied for a class of abstract operators that contains the usual differential operators with real‐analytic coefficients. We obtain a new proof of a global result on compact manifolds (global Gevrey hypoellipticity implying global solvability of the transpose), as well as some results in the non‐compact case by means of the so‐called property of non‐confinement of singularities. We provide applications to Hörmander operators, to operators of constant strength and to locally integrable systems of vector fields. We also analyze a conjecture stated in a recent paper of Malaspina and Nicola, which asserts that, in differential complexes naturally arising from locally integrable structures, local solvability in the sense of ultradistributions implies local solvability in the sense of distributions. We establish the validity of the conjecture when the cotangent structure bundle is spanned by the differential of a single first integral.     

On the well-posedness of weakly hyperbolic equations with time-dependent coefficients

In this paper we analyse the Gevrey well-posedness of the Cauchy problem for weakly hyperbolic equations of general form with time-dependent coefficients. The results involve the order of lower order terms and the number of multiple roots. We also derive the corresponding well-posedness results in the space of Gevrey Beurling ultradistributions.     

Ultra‐differentiable classes and intersection theorems

The aim of the paper is to study the relation between ultra‐differentiable classes of functions defined in terms of estimates on derivatives on one hand and in terms of growth properties of Fourier transforms of suitably localized functions in the class on the other hand. We establish this relation for the ultra‐differentiable classes introduced in 6 , 16 , and show that the classes of 6 , 16 , can be regarded as inhomogeneous Gevrey classes in the sense of 22 . We also discuss a number of properties of the weight functions used to define the respective classes and of their Young conjugates.     

Cauchy problem for parabolic systems with convolution operators in periodic spaces

For a class of periodic systems of parabolic type with pseudodifferential operators containing $\{ \vec p,\vec h\} $ -parabolic systems of partial differential equations, we study the properties of the fundamental matrices of the solutions and establish the well-posed solvability of the Cauchy problem for these systems in the spaces of generalized periodic functions of the type of Gevrey ultradistributions. For a particular subclass of systems, we describe the maximal classes of well-posed solvability of the Cauchy problem.     

Gevrey local solvability in locally integrable structures

We consider a locally integrable real-analytic structure, and we investigate the local solvability in the category of Gevrey functions and ultradistributions of the complex \(\mathrm{d}^{\prime }\) naturally induced by the de Rham complex. We prove that the so-called condition \(Y(q)\) on the signature of the Levi form, for local solvability of \(\mathrm{d}^{\prime }u=f\) , is still necessary even if we take \(f\) in the classes of Gevrey functions and look for solutions \(u\) in the corresponding spaces of ultradistributions.     

Gevrey hypoellipticity and solvability on the multidimensional torus of some classes of linear partial differential operators

Abstract In this paper we consider the problem of global analytic and Gevrey hypoellipticity and solvability for a class of partial differential operators on a torus. We prove that global analytic and Gevrey hypoellipticity and solvability on the torus is equivalent to certain Diophantine approximation properties. Keywords: Global hypoellipticity, Global solvability, Gevrey classes, Diophantine approximation property Mathematics Subject Classification (2000): 35D05, 46E10, 46F05, 58J99     

Fourier integral operators and inhomogeneous Gevrey classes

Summary Fourier integral operators with inhomogeneous amplitude and phase junction are studied in the frame of Gevrey classes. Applications are given to propagation of singularities for a pseudodifferential equation.     

On topological filtrations of groups

The (weak) geometric simple connectivity and the quasi-simple filtration are topological notions of manifolds, which may be defined for discrete groups too. It turns out that they are equivalent for finitely presented groups, but the main problem is the absence of examples of groups which do not satisfy them. In this note we study some algebraic classes of groups with respect to these properties.     

Cauchy problem for a class of degenerate kolmogorov-type parabolic equations with nonpositive genus

We study the properties of the fundamental solution and establish the correct solvability of the Cauchy problem for a class of degenerate Kolmogorov-type equations with { p^?,h^? } \left\{ {\overrightarrow p, \overrightarrow h } \right\} -parabolic part with respect to the main group of variables and nonpositive vector genus in the case where the solutions are infinitely differentiable functions and their initial values are generalized functions in the form of Gevrey ultradistributions.     

Whittaker vectors and conical vectors

A holomorphic family of differential operators of infinite order is constructed that transforms conical vectors for principal series representations of quasi-split, linear, semi-simple Lie groups into Whittaker vectors. Using this transform, it is shown that algebraic Whittaker vectors (as studied by Kostant) extend to ultradistributions of Gevrey type on principal series representations. For each element of the small Weyl group, a meromorphic family of Whittaker vectors is constructed from this transform and the Kunze-Stein intertwining integrals. An explict formula is derived for the smooth Whittaker vector (discovered by Jacquet), in terms of these families of ultradistribution Whittaker vectors. In particular, this gives new proofs of Jacquet's analytic continuation of the smooth Whittaker vector and its functional equation (Jacquet and Schiffman). Applications of the transform are also given to the theory of Verma modules.     

向量优化中集合的一些相对代数性质和相对拓扑性质

基于 Flores-Baz′an 等人的思想,提出了假设 B1和假设 B2,证明了集合和的相对代数内部等于相对代数内部的和;集合代数闭包与相对代数内部的和等于和的相对代数内部;集合和的相对拓扑内部等于相对拓扑内部的和;集合拓扑闭包与相对拓扑内部的和等于和的相对拓扑内部,建立了集合代数闭包相等与代数内部相等,拓扑闭包相等与拓扑内部相等之间的一些等价关系。     

Weakly Linearly Compact Topological Abelian Groups

In this paper, we study linearly topological groups. We introduce the notion of a weakly linearly compact group, which generalizes the notion of a weakly separable group, and examine the main properties of such groups. For weakly linearly compact groups, we construct the character theory and present an algebraic characterization of some classes of such groups. Some well-known theorems for periodic Abelian groups are generalized for the case of linearly discrete, topological Abelian groups; for linearly compact and linearly discrete topological Abelian groups, we also construct the character theory and study some important properties of linearly discrete groups. For linearly discrete, topological Abelian groups, we analyze the splittability condition (Theorem 3.12) and present the characteristic condition of decomposability of a discrete group G into the direct sum of rank-1 groups. We also present an algebraic characterization of linearly compact groups. We introduce the notion of a weakly linearly compact, topological Abelian group, which generalizes the notion of a weakly separable Abelian group, and examine some properties of such groups. These groups are a generalization of fibrous Abelian groups introduced by Vilenkin. We give an algebraic characterization of divisible, weakly locally compact Abelian groups that do not contain nonzero elements of finite order (Proposition 7.9). For weakly locally compact Abelian groups, we construct universal groups.     

Propagation of singularities for operators with constant coefficient hyperbolic-elliptic principal part

Summary In this paper we consider partial differential operators of the type P(x, D)= P_m(D)+Q(x, D), where the constant coefficient principal part P_m is supposed to be hyperbolic-elliptic. We study the propagation of Gevrey singularities for solutions u of the equation P(x, D) u=f, for ultradistributions f, finding exactly to which spaces of ultradistribuiions u microlocally belongs. The results are obtained by constructing a fundamental solution for P when the lower order part Q is with constant coefficients, and a parametrix otherwise.     

FACTORIZATIONS OF CONES ALONG A FUNCTOR

Abstract

First a general Galois correspondence is established, which generalizes at the same time the correspondence between classes of monomorphisms and injective objects and the correspondence between classes of epimorphisms and monomorphisms in a category. This correspondence arises naturally if one tries to generalize some concepts of “topological” or also of “algebraic” functors. Both kinds of functors admit certain factorizations of cones, and just this fact implies some of their common nice properties: lifting limits, continuity and faithfulness, for instance. These properties can be shown without having a left adjoint. Therefore the theory yields also applications to functors which are neither “topological” nor “algebraic”.     

Microlocal analysis of quasianalytic Gelfand-Shilov type ultradistributions

We introduce a global wave front set suitable for the analysis of tempered ultradistributions of quasi-analytic Gelfand–Shilov type. We study the transformation properties of the wave front set and use them to give microlocal existence results for pullbacks and products. We further study quasi-analytic microlocality for classes of localization and ultradifferential operators, and prove microellipticity for differential operators with polynomial coefficients.     

Ultradifferential operators in the study of Gevrey solvability and regularity

In this work we present a new representation formula for ultradistributions using the so‐called ultradifferential operators. The main difference between our representation result and other works is that here we do not break the duality of Gevrey functions of other s and their ultradistributions, i.e., we locally represent an element of by an infinite order operator acting on a function of class . Our main application was in the local solvability of the differential complex associated to a locally integrable structure in a Gevrey environment.     

Global analytic and Gevrey solvability of sublaplacians under Diophantine conditions

In this paper we consider the problem of global analytic and Gevrey solvability for a class of partial differential operators on a torus in the form of squares of vector fields. We prove that global analytic and Gevrey solvability on the torus is equivalent to certain Diophantine approximation properties. Mathematics Subject Classification (2000) 35D05, 46E10, 46F05, 58J99     

Variants of openness

Algebraic conditions on frame homomorphisms representing various types of openness requirements on continuous maps are investigated. It turns out that several of these can be expressed in terms of formulas involving pseudocomplements. A full classification of the latter is presented which shows that they group into five equivalence classes and establishes the logical connections between them. Among the relation of our algebraic conditions to continuous maps between topological spaces, we establish that the coincidence of the algebraic and topological notion of openness is equivalent to the separation axiomT _D for the domain space.In honour of Dieter Pumplün on the occassion of his 60th birthday