arXiv subjects

Alexandre Kirilov

Publications and source records attributed to Alexandre Kirilov.

At least 19 recordsLinked to original sources

arXiv2026

Top-Degree Global Solvability for Tube Complexes in Gevrey Ultradistributions

Let $s>1$, let $M$ be a connected, non-compact, oriented real-analytic manifold, and let $\omega_1,\ldots,\omega_m$ be real-valued closed $1$-forms of Gevrey order $s$ on $M$. We study the differential complex naturally associated with this family on $M\times\mathbb{T}^m$. We prove that its top-degree operator is globally solvable in Roumieu Gevrey ultradistributions, or equivalently that the corresponding top-degree cohomology vanishes. No global hypoellipticity assumption and no arithmetic condition on the periods of the defining forms are required. The proof is carried out in the physical variables and combines fiber translations, a local normal form, and a transport formula along paths in the base manifold. These tools yield propagation of Gevrey regularity, non-confinement of Gevrey singularities, and the support control needed to apply an abstract solvability criterion. The result highlights a sharp contrast with the compact setting, where compatibility conditions are unavoidable and solvability for compatible data may depend on exponential small-denominator conditions.

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arXiv2026

Vekua-Type Operators on Compact Lie Groups: Hypoellipticity, Solvability, and Self-Duality

We study global hypoellipticity and global solvability for Vekua-type operators associated with diagonal left-invariant operators on compact Lie groups. The conjugation term produces, on the Fourier side, a family of coupled systems whose determinants govern both regularity and solvability. Under a natural non-self-duality assumption, we obtain complete Diophantine-type characterizations of global hypoellipticity and global solvability, the latter on the natural space of admissible data. We then show how the theory must be modified in the self-dual setting by carrying out a complete analysis of the model case $\mathbb S^3\simeq SU(2)$, where conjugation acts inside each representation block. The resulting criteria exhibit two Fourier-side mechanisms for Vekua-type operators on compact Lie groups: coupling between distinct conjugate representations and coupling inside self-dual representation blocks. We also present examples on product groups illustrating how these mechanisms may coexist and how global solvability may hold even when global hypoellipticity fails.

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arXiv2026

Global Gevrey Hypoellipticity of Involutive Systems on Non-Compact Manifolds

We investigate the global Gevrey hypoellipticity of a class of first-order differential operators associated with tube-type involutive structures on $M\times\mathbb{T}^m$, where $M$ is a non-compact manifold diffeomorphic to the interior of a compact manifold with boundary and $\mathbb{T}^m$ is the $m$-dimensional torus. For $s>1$, we work in Gevrey classes of Roumieu and Beurling type. A key step is the construction, on $M$, of a scattering metric whose coefficients are Gevrey of order $s$ in every analytic chart; this allows us to use Hodge theory and obtain Gevrey regularity for the harmonic forms. Under a natural condition on the defining closed $1$-forms, we obtain a sharp criterion for global Gevrey hypoellipticity in terms of rationality and (Roumieu/Beurling) exponential Liouville behavior.

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arXiv2026

Solvability of a class of evolution operators on compact Lie groups

This paper provides sufficient conditions for the solvability of a class of first-order evolution operators of Vekua-type on the product of a one-dimensional torus and a compact Lie group. The conditions are expressed in terms of the time-dependent coefficients and the spectral behavior of a normalized left-invariant vector field on the group. The three-sphere case is discussed in detail, leading to more explicit criteria, and the main results are further extended to operators defined on finite products of compact Lie groups.

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arXiv2026

Differential Complexes in Time-Periodic Gelfand-Shilov Spaces

We study the global solvability of a class of differential complexes on the product manifold $\mathbb{T}^m \times \mathbb{R}^n$ associated with systems of evolution operators of the form $L_r = \partial_{t_r} + ia_r(t)P(x,D_x), r=1,\ldots,m,$ where the coefficients $a_r$ are real-valued Gevrey functions on the torus and $P(x,D_x)$ is a globally elliptic normal differential operator on $\mathbb{R}^n$. Within the framework of time-periodic Gelfand--Shilov spaces, we introduce a natural differential complex generated by these operators and investigate its solvability in both functional and ultradistributional settings. We provide a complete characterization of global solvability in terms of a Diophantine condition involving the constant part of the associated $1$-form and the spectrum of $P$. We also analyze global hypoellipticity of the complex. These results extend previous works on scalar operators and constant coefficient systems to the setting of differential complexes with time-dependent real coefficients.

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arXiv2025

Global Solvability for Involutive Systems on Non-Compact Manifolds

We establish necessary and sufficient conditions for the closedness of the range of a class of first-order differential operators associated with an involutive structure on $M\times\mathbb{T}^m$, where $M$ is a non-compact manifold satisfying suitable geometric assumptions and $\mathbb{T}^m$ is the $m$-dimensional torus. In addition, we prove that a weaker notion of global hypoellipticity ensures the closedness of the range for differential operators on smooth paracompact manifolds, thereby extending to the non-compact setting a result previously obtained by G.~Ara\'ujo, I.~Ferra, and L.~Ragognette [J. Anal. Math. 148, No. 1, 85-118, 2022] for compact manifolds.

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arXiv2025

Global Hypoellipticity for Systems in Time-Periodic Gelfand-Shilov Spaces

We investigate the global hypoellipticity of a class of overdetermined systems with coefficients depending both on time and space variables in the setting of time-periodic Gelfand-Shilov spaces. Our main result provides necessary and sufficient conditions for the global hypoellipticity of this class of systems, stated in terms of Diophantine-type estimates and sign-changing behavior of the imaginary parts of the coefficients. Through a reduction to a normal form and detailed construction of singular solutions, we fully characterize when the system fails to be globally hypoelliptic.

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arXiv2025

Global hypoellipticity for involutive systems on non-compact manifolds

We study the global hypoellipticity of the operator $\mathbb{L} = \mathrm{d}_t + \sum_{k=1}^m \omega_k \wedge \partial_{x_k}$, defined on differential forms over product manifolds of the form $M \times \mathbb{T}^m$, where $M$ is a non-compact manifold homeomorphic to the interior of a compact manifold with boundary, equipped with a scattering metric, and $\omega_1,\dots,\omega_m$ are smooth closed 1-forms on $M$. Extending previous results obtained in the compact setting, we characterize global hypoellipticity of $\mathbb{L}$ in terms of arithmetic properties of the forms $\omega_k$. The analysis relies on microlocal techniques adapted to the scattering setting and a version of the Hodge Theorem for scattering manifolds.

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arXiv2025

Global Hypoellipticity and Solvability with Loss of Derivatives on the Torus

This paper provides a complete characterization of global hypoellipticity and solvability with loss of derivatives for Fourier multiplier operators on the $n$-dimensional torus. We establish necessary and sufficient conditions for these properties and examine their connections with classical notions of global hypoellipticity and solvability, particularly in relation to the closedness of the operator's range. As an application, we explore the interplay between these properties and number theory in the context of differential operators on the two-torus. Specifically, we prove that the loss of derivatives in the solvability of the vector field $\partial_{x_1} - \alpha \partial_{x_2}$ is precisely determined by the well-known irrationality measure $\mu(\alpha)$ of its coefficient $\alpha$. Furthermore, we analyze the wave operator $\partial_{x_1}^2 - \eta^2 \Delta_{\mathbb{T}^n}$ and show how the loss of derivatives depends explicitly on the parameter $\eta > 0$.

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arXiv2025

Global solutions for systems of strongly invariant operators on closed manifolds

We study the global hypoellipticity and solvability of strongly invariant operators and systems of strongly invariant operators on closed manifolds. Our approach is based on the Fourier analysis induced by an elliptic pseudo-differential operator, which provides a spectral decomposition of $L^2(M)$ into finite-dimensional eigenspaces. This framework allows us to characterize these global properties through asymptotic estimates on the matrix symbols of the operators. Additionally, for systems of normal strongly invariant operators, we derive an explicit solution formula and establish sufficient conditions for global hypoellipticity and solvability in terms of their eigenvalues.

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arXiv2024

Diagonal systems of differential operators on compact Lie groups

We investigate the global hypoellipticity and global solvability of systems of left-invariant differential operators on compact Lie groups. Focusing on diagonal systems, we establish necessary and sufficient conditions for these global properties. Specifically, we show that global solvability is characterized by a Diophantine condition on the symbol of the system, while global hypoellipticity further requires that a set depending on the symbol to be finite. As an application, we provide a complete characterization of these properties for systems of vector fields defined on products of tori and spheres. Additionally, we present illustrative examples, including systems involving higher-order differential operators. Finally, we extend our analysis to triangular systems on compact Lie groups, introducing an additional condition related to the boundedness of the dimensions of the representations in these Lie groups.

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arXiv2024

Denjoy-Carleman solvability of Vekua-type periodic operators

This paper explores the solvability and global hypoellipticity of Vekua-type differential operators on the n-dimensional torus, within the framework of Denjoy-Carleman ultradifferentiability. We provide the necessary and sufficient conditions for achieving these global properties in the case of constant-coefficient operators, along with applications to classical operators. Additionally, we investigate a class of variable coefficients and establish conditions for its solvability.

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arXiv2024

On the Sobolev boundedness of vector fields on compact Riemannian manifolds

We analyze the sharpness of the Sobolev order for left-invariant vector fields on compact Riemannian manifolds. Utilizing techniques from pseudo-differential operator theory and microlocal analysis, we investigate the asymptotic behavior of eigenvalues associated with these vector fields. As an application, we demonstrate the ill-posedness of a class of Cauchy problems involving left-invariant vector fields on compact Lie groups.

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arXiv2024

Systems of differential operators in time-periodic Gelfand-Shilov spaces

This paper explores the global properties of time-independent systems of operators in the framework of Gelfand-Shilov spaces. Our main results provide both necessary and sufficient conditions for global solvability and global hypoellipticity, based on analysis of the symbols of operators. We also present a class of time-dependent operators whose solvability and hypoellipticity are linked to the same properties of an associated time-independent system, albeit with a loss of regularity for temporal variables.

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arXiv2024

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

In this note, we investigate Vekua-type periodic operators of the form $Pu=Lu-Au-B\bar u$, 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.

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arXiv2023

Global solvability and hypoellipticity for evolution operators on tori and spheres

In this paper, we study the global properties of a class of evolution-like differential operator with a 0-order perturbation defined on the product of $r+1$ tori and $s$ spheres $\mathbb{T}^{r+1}\times(\mathbb{S}^{3})^s$, with $r$ and $s$ non-negative integers. By varying the values of $r$ and $s$, we show that it is possible to recover results already known in the literature and present new results. The main tool used in this study is Fourier analysis, taken partially with respect to each copy of the torus and sphere. We obtain necessary and sufficient conditions related to Diophantine inequalities, change of sign and connectivity of level sets associated the operator's coefficients.

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arXiv2020

Global analytic hypoellipticity for a class of evolution operators on $\mathbb{T}^1\times\mathbb{S}^3$

In this paper, we present necessary and sufficient conditions to have global analytic hypoellipticity for a class of first-order operators defined on $\mathbb{T}^1 \times \mathbb{S}^3$. In the case of real-valued coefficients, we prove that an operator in this class is conjugated to a constant-coefficient operator satisfying a Diophantine condition, and that such conjugation preserves the global analytic hypoellipticity. In the case where the imaginary part of the coefficients is non-zero, we show that the operator is globally analytic hypoelliptic if the Nirenberg-Treves condition ($\mathcal{P}$) holds, in addition to a Diophantine condition.

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arXiv2020

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. We also investigate relations between the global hypoellipticity of $P$ and global subelliptic estimates.

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