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.Comment: 20 page

Global Solvability in Functional Spaces for Smooth Nonsingular Vector Fields in the Plane

We address some global solvability issues for classes of smooth nonsingular vector fields $L$ in the plane related to cohomological equations $Lu=f$ in geometry and dynamical systems. The first main result is that $L$ is not surjective in $C^\infty(\R^2)$ iff the geometrical condition -- the existence of separatrix strips -- holds. Next, for nonsurjective vector fields, we demonstrate that if the RHS $f$ has at most infra-exponential growth in the separatrix strips we can find a global weak solution $L^1_{loc}$ near the boundaries of the separatrix strips. Finally we investigate the global solvability for perturbations with zero order p.d.o. We provide examples showing that our estimates are sharp.Comment: 22 pages, 2 figures, submitted to the PDE volume of the proceedings of the ISAAC2009 conferenc

Global properties of vector fields on compact Lie groups in Komatsu classes. II. Normal forms

Let $G_1$ and $G_2$ be compact Lie groups, $X_1 \in \mathfrak{g}_1$, $X_2 \in \mathfrak{g}_2$ and consider the operator \begin{equation*} L_{aq} = X_1 + a(x_1)X_2 + q(x_1,x_2), \end{equation*} 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 $L_{aq}$ in the sense of Komatsu. For this, we present a conjugation between $L_{aq}$ 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 $\mathbb{T}^1\times \mathbb{S}^3$ and $\mathbb{S}^3\times \mathbb{S}^3$ in the sense of Komatsu. In particular, we give examples of differential operators which are not globally $C^\infty$-solvable, but are globally solvable in Gevrey spaces.Comment: 23 pages. arXiv admin note: text overlap with arXiv:1910.01922, arXiv:1910.0005

Partial Fourier series on compact Lie groups

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.Comment: 21 page

Global Properties for first order differential operators on Tr+1×(S3)s\mathbb{T}^{r+1}\times(\mathbb{S}^{3})^s

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

Global analytic hypoellipticity for a class of evolution operators on T1×S3\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.Comment: 24 page

Global properties of vector fields on compact Lie groups in Komatsu classes : II : normal forms

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