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     

Global Hypoellipticity for a Class of Pseudo-differential Operators on the Torus

We show that an obstruction of number-theoretical nature appears as a necessary condition for the global hypoellipticity of the pseudo-differential operator \(L=D_t+(a+ib)(t)P(D_x)\) on \(\mathbb {T}^1_t\times \mathbb {T}_x^{N}\). This condition is also sufficient when the symbol \(p(\xi )\) of \(P(D_x)\) has at most logarithmic growth. If \(p(\xi )\) has super-logarithmic growth, we show that the global hypoellipticity of L depends on the change of sign of certain interactions of the coefficients with the symbol \(p(\xi ).\) Moreover, the interplay between the order of vanishing of coefficients with the order of growth of \(p(\xi )\) plays a crucial role in the global hypoellipticity of L. We also describe completely the global hypoellipticity of L in the case where \(P(D_x)\) is homogeneous. Additionally, we explore the influence of irrational approximations of a real number in the global hypoellipticity.

     

On global hypoellipticity of degenerate elliptic operators

We consider a class of degenerate elliptic operators on a torus and prove that global hypoellipticity is equivalent to an algebraic condition involving Liouville vectors and simultaneous approximability. For another class of operators we show that the zero order term may influence global hypoellipticity. Received August 13, 1997     

On rotationally invariant vector fields in the plane

This work is concerned with global properties of a class of ℂ-valued vector fields in the plane which are rotationally invariant. It is shown that the finite type rotationally invariant vector fields have global first integrals. We also study the global hypoellipticity and global solvability properties of these vector fields.     

Globally analytic hypoelliptic vector fields on compact surfaces

We present a characterization of the global analytic hypoellipticity of a complex, non-singular, real analytic vector field defined on a compact, connected, orientable, two-dimensional, real analytic manifold.

In particular, we show that such vector fields exist only on the torus.

     

Global analytic regularity for sums of squares of vector fields

We consider a class of operators in the form of a sum of squares of vector fields with real analytic coefficients on the torus and we show that the zero order term may influence their global analytic hypoellipticity. Also we extend a result of Cordaro-Himonas.

     

Global s-solvability, global s-hypoellipticity and Diophantine phenomena

In this paper we consider the problem of global Gevrey solvability for a class of sublaplacians on a toruswith coefficients in the Gevrey class Gs(TN). For this class of operators we show that global Gevrey solvability and global Gevrey hypoellipticity are both equivalent to the condition that the coefficients satisfy a Diophantine condition.     

Pseudo-Differential Operators in Algebras of Generalized Functions and Global Hypoellipticity

The aim of this work is to develop a global calculus for pseudo-differential operators acting on suitable algebras of generalized functions. In particular, a condition of global hypoellipticity of the symbols gives a result of regularity for the corresponding pseudo-differential equations. This calculus and this frame are proposed as tools for the study in Colombeau algebras of partial differential equations globally defined on R ⁿ .     

Existence and Regularity of Periodic Solutions to Certain First-Order Partial Differential Equations

We present conditions on the coefficients of a class of vector fields on the torus which yield a characterization of global solvability as well as global hypoellipticity, in other words, the existence and regularity of periodic solutions. Diophantine conditions and connectedness of certain sublevel sets appear in a natural way in our results.     

Perturbations by lower order terms do not destroy the global hypoellipticity of certain systems of pseudodifferential operators defined on torus

We introduce a new class of smooth pseudodifferential operators on the torus whose calculus allows us to show that global hypoellipticity with a finite loss of derivatives of certain systems of pseudodifferential operators is stable under perturbations by lower order systems of pseudodifferential operators whose order depends on the loss of derivatives. We also present some applications.     

Simultaneous reduction of a family of commuting real vector fields and global hypoellipticity

In this paper we consider a family of commuting real vector fields on then-dimensional torus and show that it can be transformed into a family of constant vector fields provided that there is one of them which its transposed is globally hypoelliptic. We apply this result to prove global hypoellipticity for certain classes of sublaplacians. The author was partially supported by CNPq.     

On global analytic and Gevrey hypoellipticity on the torus and the Métivier inequality

We obtain a global version in the N-dimensional torus of the Métivier inequality for analytic and Gevrey hypoellipticity, and based on it we introduce a class of globally analytic hypoelliptic operators which remain so after suitable lower order perturbations. We also introduce a new class of analytic (pseudodifferential) operators on the torus whose calculus allows us to study the corresponding perturbation problem in a far more general context.     

Spectral properties of the twisted bi-Laplacian

The eigenvalues and eigenfunctions of the twisted bi-Laplacian are studied in the context of analytic number theory. The essential self-adjointness and the global hypoellipticity in terms of a new family of Sobolev spaces are also studied.     

On Global Hypoellipticity

We consider a first order linear partial differential operator of principal type on a closed connected orientable two-dimensional manifold sending sections of one complex line bundle to sections of another. We prove that the assumption of global hypoellipticity of the operator implies a relation between the degrees of the line bundles and the Euler characteristic of the manifold.     

Hörmander Class of Pseudo-Differential Operators on Compact Lie Groups and Global Hypoellipticity

In this paper we give several global characterisations of the Hörmander class \(\Psi ^m(G)\) of pseudo-differential operators on compact Lie groups in terms of the representation theory of the group. The result is applied to give criteria for the ellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several examples of the first and second order globally hypoelliptic differential operators are given, in particular of operators that are locally not invertible nor hypoelliptic but globally are. Where the global hypoelliptiticy fails, one can construct explicit examples based on the analysis of the global symbols.     

微分形式的亚椭圆性

本文研究了光滑流形上微分形式的亚椭圆性,利用遍历平均的方法,得到了二维环面上变系数1-形式是亚椭圆的充要条件.     

n级三角矩阵环上的模范畴和同调特征

本文给出了n级三角矩阵环Гn的定义．证明了n级三角矩阵代数Гn上的有限生成模范畴mod Гn与范畴Гn(?)等价,得到了诸如Гn的Jacobson根,Гn(?)的不可分解投射对象的形式及Гn的整体维数等性质．     

Global Analytic Regularity for Structures of Co-Rank One

We consider real analytic involutive structures 𝒱, of co-rank one, defined on a real analytic paracompact orientable manifold M. To each such structure we associate certain connected subsets of M which we call the level sets of 𝒱. We prove that analytic regularity propagates along them. With a further assumption on the level sets of 𝒱 we characterize the global analytic hypoellipticity of a differential operator naturally associated to 𝒱.

As an application we study a case of tube structures.     

Gorenstein global dimension of recollements of abelian categories

Abstract

In this article, we investigate the Gorenstein global dimension with respect to the recollements of abelian categories. With the invariants spli and silp of the categories, we give some upper bounds of Gorenstein global dimensions of the categories involved in a recollement of abelian categories. We apply our results to some rings and artin algebras, especially to the triangular matrix artin algebras.     

Remarks about global analytic hypoellipticity

We present a characterization of the operators

which are globally analytic hypoelliptic on the torus. We give information about the global analytic hypoellipticity of certain overdetermined systems and of sums of squares.