Global Hypoellipticity Research Articles

Solvability of Vekua-type periodic operators and applications to classical equations

In this note, we investigate Vekua-type periodic operators of the form Pu=Lu−Au−Bū, where L is a constant coefficient partial differential operator. We provide a complete characterization of the necessary and sufficient conditions for the solvability and global hypoellipticity of P. As an application, we provide a comprehensive characterization of Vekua-type operators associated with classical wave, heat, and Laplace equations.

GLOBAL HYPOELLIPTICITY OF SUMS OF SQUARES ON COMPACT MANIFOLDS

AbstractWe present necessary and sufficient conditions for an operator of the type sum of squares to be globally hypoelliptic on $T \times G$ , where T is a compact Riemannian manifold and G is a compact Lie group. These conditions involve the global hypoellipticity of a system of vector fields on G and are weaker than Hörmander’s condition, while generalizing the well known Diophantine conditions on the torus. Examples of operators satisfying these conditions in the general setting are provided.

Global Operator Calculus on Spin Groups

In this paper, we use the representation theory of the group textrm{Spin}(m) to develop aspects of the global symbolic calculus of pseudo-differential operators on textrm{Spin}(3) and textrm{Spin}(4) in the sense of Ruzhansky–Turunen–Wirth. A detailed study of textrm{Spin}(3) and textrm{Spin}(4)-representations is made including recurrence relations and natural differential operators acting on matrix coefficients. We establish the calculus of left-invariant differential operators and of difference operators on the group textrm{Spin}(4) and apply this to give criteria for the subellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several examples of first and second order globally hypoelliptic differential operators are given, including some that are locally neither invertible nor hypoelliptic. The paper presents a particular case study for higher dimensional spin groups.

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

AbstractWe 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.

Globally hypoelliptic triangularizable systems of periodic pseudo‐differential operators

AbstractThis 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.

Global solvability and global hypoellipticity of complex vector fields on surfaces

We consider smooth, nonvanishing complex (not essentially real) vector fields L=X+iY that satisfy the Nirenberg-Treves condition (P) and are allowed to possess some closed one-dimensional orbits in the sense of Sussmann that we call clean orbits. Concerning global solvability, we deal with two different cases —depending on whether the surface is compact or not— and prove two positive results which extend and unify several known results on the subject.In our positive results, it is assumed that X∧Y vanishes of order greater than 1 on all closed one-orbits, on the other hand, we prove that global solvability cannot hold if there exists at least one closed one-orbit on which X∧Y vanishes of order 1.We also give a characterization of globally hypoelliptic complex vector fields L in terms of the topological type of the equivalence classes determined by a standard equivalence relation defined on the set of non elliptic points of L. Some of the consequences are that globally hypoelliptic vector fields are globally solvable and that if a surface M carries a globally hypoelliptic vector field, it must be parallelizable.

Global analytic hypoellipticity and solvability of certain operators subject to group actions

On T × G T \times G , where T T is a compact real-analytic manifold and G G is a compact Lie group, we consider differential operators P P which are invariant by left translations on G G and are elliptic in T T . Under a mild technical condition, we prove that global hypoellipticity of P P implies its global analytic-hypoellipticity (actually Gevrey of any order s ≥ 1 s \geq 1 ). We also study the connection between the latter property and the notion of global analytic (resp. Gevrey) solvability, but in a much more general setup.

Quantizations and Global Hypoellipticity for Pseudodifferential Operators of Infinite Order in Classes of Ultradifferentiable Functions

We study the change of quantization for a class of global pseudodifferential operators of infinite order in the setting of ultradifferentiable functions of Beurling type. The composition of different quantizations as well as the transpose of a quantization are also analysed, with applications to the Weyl calculus. We also compare global \(\omega \)-hypoellipticity and global \(\omega \)-regularity of these classes of pseudodifferential operators.

Global ultradifferentiable hypoellipticity on compact manifolds

We study the global hypoellipticity problem for certain linear operators in Komatsu classes of Roumieu and Beurling type on compact manifolds. We present an approach by combining a characterization of these spaces via eigenfuction expansions, generated by an elliptic operator, and the analysis of matrix-symbols obtained by these expansions.

Time-periodic Gelfand-Shilov spaces and global hypoellipticity on [formula omitted

We introduce a class of time-periodic Gelfand-Shilov spaces of functions on T×Rn, where T∼R/2πZ is the one-dimensional torus. We develop a Fourier analysis inspired by the characterization of the Gelfand-Shilov spaces in terms of the eigenfunction expansions given by a fixed normal, globally elliptic differential operator on Rn. In this setting, as an application, we characterize the global hypoellipticity for a class of linear differential evolution operators on T×Rn.

Global Properties of Vector Fields on Compact Lie Groups in Komatsu Classes

In this paper, we characterize completely the global hypoellipticity and global solvability in the sense of Komatsu (of Roumieu and Beurling types) of constant-coefficient vector fields on compact Lie groups. We also analyze the influence of perturbations by lower-order terms in the preservation of these properties.

Regularity of solutions to a Vekua-type equation on compact Lie groups

We present sufficient conditions to have global hypoellipticity for a class of Vekua-type operators defined on a compact Lie group. When the group has the property that every non-trivial representation is not self-dual we show that these sufficient conditions are also necessary. We also present results about the global solvability for this class of operators.

Global hypoellipticity for a class of overdetermined systems of pseudo-differential operators on the torus

This article studies the global hypoellipticity of a class of overdetermined systems of pseudo-differential operators defined on the torus. The main goal consists in establishing connections between the global hypoellipticity of the system and the global hypoellipticity of its normal form. It is proved that an obstruction of number-theoretical nature appears as a necessary condition to the global hypoellipticity. Conversely, the sufficiency is approached by analyzing three types of hypotheses: a Hormander condition, logarithmic growth and super-logarithmic growth.

Fourier analysis for Denjoy–Carleman classes on the torus

In the present article, we develop Fourier series for a family of classes of Romieu type of ultradifferentiable functions and ultradistributions on the torus, usually known as Denjoy-Carleman classes. In this setting, as applications, we extend the Greenfield-Wallach Theorem and, through a conjugation, we characterize global hypoellipticity for a class of systems of real vector fields of tube type.

Global $${\mathcal {M}}-$$Hypoellipticity, Global $${\mathcal {M}}-$$Solvability and Perturbations by Lower Order Ultradifferential Pseudodifferential Operators

We introduce a new class of ultradifferentiable pseudodifferential operators on the torus whose calculus allows us to show that global hypoellipticity, in ultradifferentiable classes, with a finite loss of derivatives of a system of pseudodifferential operators, is stable under perturbations by lower order pseudodifferential operators whose order depends on the loss of derivatives. The key point in our study is our definition of loss of derivatives. We also give an easy proof of the fact that if a system of pseudodifferential operators is globally $${\mathcal {M}}$$ -hypoelliptic then its transpose is globally solvable in $$D'_{\mathcal {M}}\left( {\mathbb {T}}^N\right) $$ . Finally we present an application of our results.

Global hypoellipticity and global solvability for vector fields on compact Lie groups

We present necessary and sufficient conditions to have global hypoellipticity and global solvability for a class of vector fields defined on a product of compact Lie groups. In view of Greenfield's and Wallach's conjecture, about the non-existence of globally hypoelliptic vector fields on compact manifolds different from tori, we also investigate different notions of regularity weaker than global hypoellipticity and describe completely the global hypoellipticity and global solvability of zero-order perturbations of our vector fields. We also present a class of vector fields with variable coefficients whose operators can be reduced to a normal form, and we prove that the study of the global properties of such operators is equivalent to the study of the respective properties for their normal forms.

Tube Structures of Co-rank 1 with Forms Defined on Compact Surfaces

We study the global solvability of a locally integrable structure of tube type and co-rank 1 by considering a linear partial differential operator $${\mathbb {L}}$$ associated to a general complex smooth closed 1-form c defined on a smooth closed n-manifold. The main result characterizes the global solvability of $${\mathbb {L}}$$ when $$n=2$$ in terms of geometric properties of a primitive of a convenient exact pullback of the form $$\mathfrak {Im}(c)$$ as well as in terms of homological properties of $$\mathfrak {Re}(c)$$ related to small divisors phenomena. Although the full characterization is restricted to orientable surfaces, some partial results hold true for compact manifolds of any dimension, in particular, the necessity of the conditions, and the equivalence when $$\mathfrak {Im}(c)$$ is exact. We also obtain informations on the global hypoellipticity of $${\mathbb {L}}$$ and the global solvability of $${\mathbb {L}}^{n-1}$$ —the last non-trivial operator of the complex when M is orientable.

Global hypoellipticity for strongly invariant operators

In this note, by analyzing the behavior at infinity of the matrix symbol of an invariant operator P with respect to a fixed elliptic operator, we obtain a necessary and sufficient condition to guarantee that P is globally hypoelliptic. As an application, we obtain the characterization of global hypoellipticity on compact Lie groups and examples on the sphere and the torus. We also investigate relations between the global hypoellipticity of P and global subelliptic estimates.

Global hypoellipticity for a class of periodic Cauchy operators

This note presents an investigation on the global hypoellipticity problem for Cauchy operators on Tn+1 belonging to the class L=∏j=1m(Dt+cj(t)Pj(Dx)), where Pj(Dx) are pseudo-differential operators on Tn and cj(t) are smooth complex valued functions on T. The main goal of this investigation consists in establishing connections between the global hypoellipticity of the operators L and its normal form L0=∏j=1m(Dt+c0,jPj(Dx)). In order to do so, the problem is approached by combining Hörmander's and Siegel's conditions on the symbols of the operators Lj=Dt+cj(t)Pj(Dx).

Global hypoellipticity of planar complex vector fields

We study the global hypoellipticity property for non-singular planar complex vector fields. The results obtained are related to condition (P) of Nirenberg-Treves and the boundedness of certain subsets of the plane where the real and the imaginary parts of the vector field under consideration are linearly dependent. We also prove the existence of a globally defined first integral when the vector field is globally hypoelliptic and has only one orbit.