We present necessary and sufficient conditions to have global hypoellipticity for a class of complex-valued coefficient first order evolution equations defined on $\mathbb{T}^1 \times G$, where $G$ is a compact Lie group. First, we show that the global hypoellipticity of the constant coefficient operator related to this operator is a necessary condition, but not a sufficient condition. Under certain hypothesis, we show that the global hypoellipticity of this class of operator is completely characterized by Nirenberg-Treves' condition $(\mathcal{P})$.

We analyze the sharpness of the Sobolev order for smooth 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.

We study the global hypoellipticity of the operator 𝕃 = d_t + ∑ _k=1^m ω _k ∧∂ _x_k , defined on differential forms over product manifolds of the form M ×𝕋^m , where M is a non-compact manifold, given by the interior of a scattering manifold, and ω _1,… ,ω _m are smooth closed 1-forms on M. Extending previous results obtained in the compact setting, we characterize the global hypoellipticity of 𝕃 in terms of arithmetic properties of the forms ω _1,… ,ω _m . The analysis relies on microlocal techniques, adapted to the scattering setting, and a version of the Hodge Theorem for scattering manifolds.

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.

We prove existence and uniqueness and give the analytical solution of heat and wave type equations on a compact Lie group $G$ by using a nonlocal (in time) differential operator and a positive left invariant operator (maybe unbounded) acting on the group. For heat type equations, solutions are given in $L^q(G)$ for data in $L^p(G)$ with $1<p\leqslant 2\leqslant q<+\infty $. We also provide some asymptotic estimates (large-time behavior) for the solutions. Some examples are given. Also, for wave-type equations, we give the solution on some suitable Sobolev spaces over $L^2(G)$. We complement our results, by studying a multi-term heat-type equation as well.

In this paper, we investigate global properties of a class of evolution differential operators defined on a product of tori and spheres. We present a comprehensive characterization of global solvability and hypoellipticity, providing necessary and sufficient conditions that involve Diophantine conditions and the connectedness of sublevel sets associated with the coefficients of the operator. Furthermore, we recover well-known results from existing literature and introduce novel contributions.

In this paper, we investigate the global properties of Fourier multipliers in the setting of nonharmonic analysis of boundary value problems. We give necessary and sufficient conditions for a Fourier multiplier to be globally hypoelliptic and also to be globally solvable. As an application, we consider operators on $[0,1]^2$ with non-periodic boundary conditions and we obtain results that extend what is already known in the periodic case.

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.

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.

Let \begin{document}$ G_1 $\end{document} and \begin{document}$ G_2 $\end{document} be compact Lie groups, \begin{document}$ X_1 \in \mathfrak{g}_1 $\end{document}, \begin{document}$ X_2 \in \mathfrak{g}_2 $\end{document} and consider the operator \begin{document}$ L_{aq} = X_1 + a(x_1)X_2 + q(x_1,x_2), $\end{document} where \begin{document}$ a $\end{document} and \begin{document}$ q $\end{document} are ultradifferentiable functions in the sense of Komatsu, and \begin{document}$ a $\end{document} is real-valued. Assuming certain condition on \begin{document}$ a $\end{document} and \begin{document}$ q $\end{document} we characterize completely the global hypoellipticity and the global solvability of \begin{document}$ L_{aq} $\end{document} in the sense of Komatsu. For this, we present a conjugation between \begin{document}$ L_{aq} $\end{document} and a constant-coefficient operator that preserves these global properties in Komatsu classes. We also present examples of globally hypoelliptic and globally solvable operators on \begin{document}$ \mathbb{T}^1\times \mathbb{S}^3 $\end{document} and \begin{document}$ \mathbb{S}^3\times \mathbb{S}^3 $\end{document} in the sense of Komatsu. In particular, we give examples of differential operators which are not globally \begin{document}$ C^\infty $\end{document}–solvable, but are globally solvable in Gevrey spaces.

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.

In this paper, we present necessary and sufficient conditions to have global analytic hypoellipticity for a class of first-order operators defined on T-1 x 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 (P) holds, in addition to an analytic Diophantine condition. (C) 2021 Elsevier Inc. All rights reserved.

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.

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

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.

In this note we investigate the partial Fourier series on a product of two compact Lie groups. We give necessary and sufficient conditions for a sequence of partial Fourier coefficients to define a smooth function or a distribution. As applications, we will study conditions for the global solvability of an evolution equation defined on $\mathbb{T}^1\times\mathbb{S}^3$ and we will show that some properties of this evolution equation can be obtained from a constant coefficient equation.

In this paper, we present necessary and sufficient conditions to have global analytic hypoellipticity for a class of first-order operators defined on 𝕋^1 ×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 (𝒫) holds, in addition to a Diophantine condition.

Let G1 and G2 be compact Lie groups, X1∈g1, X2∈g2 and consider the operator Laq=X1+a(x1)X2+q(x1,x2), where a and q are ultradifferentiable functions in the sense of Komatsu, and a is real-valued. We characterize completely the global hypoellipticity and the global solvability of Laq in the sense of Komatsu. For this, we present a conjugation between Laq and a constant-coefficient operator that preserves these global properties in Komatsu classes. We also present examples of globally hypoelliptic and globally solvable operators on T1×S3 and S3×S3 in the sense of Komatsu. In particular, we give examples of differential operators which are not globally C∞-solvable, but are globally solvable in Gevrey spaces.