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     

Globally hypoelliptic triangularizable systems of periodic pseudo-differential operators

This paper presents an investigation on the global hypoellipticity problem for a class of systems of pseudo-differential operators on the torus. The approach consists in establishing conditions on the matrix symbol of the system such that it can be transformed into a suitable triangular form involving a nilpotent upper triangular matrix. Hence, the global hypoellipticity is studied by analyzing the behavior of the eigenvalues and their averages.     

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.     

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.     

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.     

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.

     

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.     

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.

     

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.     

Nonlinear eigenvalues and analytic hypoellipticity

Motivated by the problem of analytic hypoellipticity, we show that a special family of compact non-self-adjoint operators has a nonzero eigenvalue. We recover old results obtained by ordinary differential equations techniques and show how it can be applied to the higher dimensional case. This gives in particular a new class of hypoelliptic, but not analytic hypoelliptic operators.     

Critical Gevrey index for hypoellipticity of parabolic operators and Newton polygones

Summary In this paper we give geometrical expressions of the (non) hypoellipticity in Gevrey spaces of parabolic operators via Newton polygones. We also determine the critical Gevrey class for which the hypoellipticity holds.Partially supported by GNAFA, CNR, Italy.Partially supported by JSPS, Japan and a grant MM-410/94 with MES, Bulgaria.Partially supported by Chuo University special research fund.     

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.     

Hypoellipticity for a class of operators with multiple characteristics

We study C ^∞ and analytic hypoellipticity for an invariant class of operators with multiple characteristics, which generalize the Gilioli-Treves model.     

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 .     

The Hypoellipticity and Parametrices for a Class of Nonprincipal Type LPDOs

In this paper, we employ the method of increment operators, by means of the left invariant operators £^{v,s} on nilpotent Lie group G^{d_1+d_2} and the calculus of v- &PsiDOs, to discuss the hypoellipticity for a class of nonprincipal type LPDOs £ on a d_1 + d_2 dimensional manifold M. For those hypoelliptic £ we also construct their parametrices.     

A CLASS OF HOMOGENEOUS LEFT INVARIANT OPERATORS ON THE NILPOTENT LIE GROUP G^{d+2}

This paper is devoted to a class of homogeneous left invariant operators L\ on the nilpotent Lie group G^{d+2} of the form $L-\lambda=-\sum\limits_{j=1}^d X_j^2-i\sum\limits_{m=1}^2 \lambda _m T_m,\lambda=\lambda_1,\lambda_2)\in C^2$ where {X_1,\cdots ,X_d,T_1, T_2} is a base of left invariant vector fields on G^{d+2}. With aid of harmonic analysis on nilpotent Lie groups and the method of increment operators, for all admissible L_\lambda, subelliptic estimate and an explicit inverse axe given and the hypoellipticity and the global solvability are obtained. Also, the structure of the set of admissible points \lambda is described exhaustively.     

A class of sums of squares with a given Poisson-Treves stratification

We study a class of sum of squares exhibiting the same Poisson-Treves stratification as the Oleinik-Radkevič operator. We find three types of operators having distinct microlocal structures. For one of these we prove a Gevrey hypoellipticity theorem analogous to our recent result for the corresponding Oleinik-Radkevič operator.     

Pointwise estimates for a class of non-homogeneous Kolmogorov equations

We consider a class of ultraparabolic differential equations that satisfy the Hörmander’s hypoellipticity condition and we prove that the weak solutions to the equation with measurable coefficients are locally bounded functions. The method extends the Moser’s iteration procedure and has previously been employed in the case of operators verifying a further homogeneity assumption. Here we remove that assumption by proving some potential estimates and some ad hoc Sobolev type inequalities for solutions.     

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.     

Parametrix constructions for right invariant differential operators on nilpotent groups

Algebras of right invariant pseudo-differential operators are constructed on any graded nilpotent group G . We obtain parametrices in such algebras for right invariant differential operators P such that P and its adjoint satisfy the hypoellipticity condition of Rockland .