Global Hypoellipticity Research Articles

Global solvability of real analytic involutive systems on compact manifolds. Part 2

This work continues a previous study by Hounie and Zugliani on the global solvability of a locally integrable structure of tube type and a corank one, considering a linear partial differential operator L \mathbb L associated with a real analytic closed 1 1 -form defined on a real analytic closed n n -manifold. We deal now with a general complex form and complete the characterization of the global solvability of L . \mathbb L. In particular, we state a general theorem, encompassing the main result of Hounie and Zugliani. As in Hounie and Zugliani’s work, we are also able to characterize the global hypoellipticity of L \mathbb L and the global solvability of L n − 1 \mathbb L^{n-1} —the last nontrivial operator of the complex when M M is orientable—which were previously considered by Bergamasco, Cordaro, Malagutti, and Petronilho in two separate papers, under an additional regularity assumption on the set of the characteristic points of L . \mathbb L.

Perturbations of globally hypoelliptic operators on closed manifolds

Analyzing the behavior at infinity of the sequence of eigenvalues given by matrix symbol of a invariant operator with respect to a fixed elliptic operator, we obtain necessary and sufficient conditions to ensure that perturbations of globally hypoelliptic operators continue to have this property. As an application, we recover classical results about perturbations of constant vector fields on the torus and extend them for more general classes of perturbations. Additionally, we construct examples of low order perturbations that destroy the global hypoellipticity, in the presence of diophantine phenomena

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 positively homogeneous. Additionally, we explore the influence of irrational approximations of a real number in the global hypoellipticity.

Global hypoellipticity for first-order operators on closed smooth manifolds

The main goal of this paper is to address global hypoellipticity issues for the following class of operators: $L = D_t + C(t,x,D_x)$, where $(t,x) \in \mathbb{T} \times M$, $\mathbb{T}$ is the one-dimensional torus, $M$ is a closed manifold and $C(t,x,D_x)$ is a first order pseudo-differential operator on $M$, smoothly depending on the periodic variable $t$. In the case of separation of variables, namely, $C(t,x,D_x) = a(t)p(x,D_x)+ib(t)q(x,D_x)$, we give necessary and sufficient conditions for the global hypoellipticity of $L$. In particular, we show that, under suitable conditions, the famous (P) condition of Niremberg-Treves is neither necessary nor sufficient to guarantee the global hypoellipticity of $L$.

Global Gevrey hypoellipticity for the twisted Laplacian on forms

We study in this paper the global hypoellipticity property in the Gevrey category for the generalized twisted Laplacian on forms. Different from the 0-form case, where the twisted Laplacian is a scalar operator, this is a system of differential operators when acting on forms, each component operator being elliptic locally and degenerate globally. We obtain here the global hypoellipticity in anisotropic Gevrey space.

Global solvability and global hypoellipticity in Gevrey classes for vector fields on the torus

Let L=∂/∂t+∑j=1N(aj+ibj)(t)∂/∂xj be a vector field defined on the torus TN+1≃RN+1/2πZN+1, where aj, bj are real-valued functions and belonging to the Gevrey class Gs(T1), s>1, for j=1,…,N. We present a complete characterization for the s-global solvability and s-global hypoellipticity of L. Our results are linked to Diophantine properties of the coefficients and, also, connectedness of certain sublevel sets.

Born-Jordan pseudodifferential operators with symbols in the Shubin classes

We apply Shubin’s theory of global symbol classes Γ ρ m \Gamma _{\rho }^{m} to the Born-Jordan pseudodifferential calculus we have previously developed. This approach has many conceptual advantages and makes the relationship between the conflicting Born-Jordan and Weyl quantization methods much more limpid. We give, in particular, precise asymptotic expansions of symbols allowing us to pass from Born-Jordan quantization to Weyl quantization and vice versa. In addition we state and prove some regularity and global hypoellipticity results.

Global hypoellipticity and simultaneous approximability in ultradifferentiable classes

We start this work by recalling a class of globally hypoelliptic sublaplacians defined on the N-dimensional torus introduced by Himonas and Petronilho and we consider a new class of sublaplacians that generalizes this one and prove that it is globally {ω}-hypoelliptic if and only if the coefficients satisfy a diophantine condition involving a new concept of simultaneous approximability with exponent {ω}. Furthermore we prove that this new class is globally {ω}-hypoelliptic if and only if certain perturbations of its vector fields, by adding more derivatives with respect to other variables, are globally {ω}-hypoelliptic. We also recall the Petronilho's conjecture for the smooth hypoellipticity and present a new class of sublaplacians for which the Petronilho's conjecture holds true in the ultradifferentiable functions setup.

The generalized twisted bi-Laplacian and its spectral properties

Let L and be the twisted Laplacian and its transpose, respectively, and let and , be the renormalization of these operators. Now, we define a new partial differential operator , , which will be called the generalized twisted bi-Laplacian of type (m, n) or simply, the twisted bi-Laplacian of type (m, n). The aim of this paper is to study some spectral properties of the generalized twisted bi-Laplacian . The essential self-adjointness and the global hypoellipticity in the Schwartz space, in the Gelfand–Shilov spaces and in terms of a new and suitable family of Sobolev spaces are also studied. Our approach is based on the paper “Spectral theory of the twisted bi-Laplacian” by Gramchev et al. [Arch Math. 2009;93:565–575].

Global solvability of real analytic involutive systems on compact manifolds

The focus of this work is the smooth global solvability of a linear partial differential operator $${\mathbb {L}}$$ associated to a real analytic closed non-exact 1-form b—defined on a real analytic closed n-manifold—that may be naturally regarded as the first operator of the complex induced by a locally integrable structure of tube type and co-rank one. We define an appropriate covering projection $$\widetilde{M}\rightarrow M$$ such that the pullback of b has a primitive $$\widetilde{B}$$ and prove that the operator is globally solvable if and only if the superlevel and sublevel sets of $$\widetilde{B}$$ are connected. As a byproduct we obtain a new geometric characterization for the global hypoellipticity of the operator. When M is orientable we prove a connection between the global solvability of $${\mathbb {L}}$$ and that of $${\mathbb {L}}^{n-1}$$ which is the last non-trivial operator of the complex, in particular, we prove that $${\mathbb {L}}$$ is globally solvable if and only if $${\mathbb {L}}^{n-1}$$ is globally solvable. In the smooth category, we are able to provide analogous geometric characterizations of the global solvability and the global hypoellipticity when b is a Morse 1-form, i.e., when the structure is of Mizohata type.

Global Real Analyticity of the Kohn–Laplacian on Pseudoconvex CR Manifolds with Comparable Levi Form

In this paper, we study the global analytic hypoellipticity of the \(\Box _b\) operator on a general CR manifold of real dimension \((2n-1)\), with \(n\ge 3\). In particular, if M is a compact, real analytic, pseudoconvex CR manifold satisfying the \(D^{\epsilon }(q)\) and \((CR-P_q)\) conditions, and a special property for vector field T, we consider the following nonelliptic partial differential equation $$\begin{aligned} \Box _bu=f. \end{aligned}$$ If f is globally real analytic, we conclude that u is globally real analytic as well. The methods applied in this work are inspired from Tartakoff (1(4):283–311, 1976) and Tartakoff (89–90:85–116 1981).

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.

Global solvability and global hypoellipticity for a class of complex vector fields on the 3-torus

This work deals with global solvability and global hypoellipticity of complex vector fields of the form \(L=\partial /\partial t+ib_1(t)\partial /\partial x_1+ib_2(t)\partial /\partial x_2\), defined on \(\mathbb {T}^3\simeq \mathbb {R}^{3}/2\pi \mathbb {Z}^{3}\), where both \(b_1\) and \(b_2\) belong to \(\mathcal {C}^{\infty }(\mathbb {T}^1;\mathbb {R}).\) The solvability and hypoellipticity depend on condition (\(\mathcal P\)) and also on Diophantine properties of the coefficients.

Hörmander Class of Pseudo-Differential Operators on Compact Lie Groups and Global Hypoellipticity

Abstract In this paper we give several global characterisations of the Hörmander class $$\Psi ^m(G)$$ Ψ 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.

A symplectic extension map and a new Shubin class of pseudo-differential operators

For an arbitrary pseudo-differential operator A:S(Rn)→S′(Rn) with Weyl symbol a∈S′(R2n), we consider the pseudo-differential operators A˜:S(Rn+k)→S′(Rn+k) associated with the Weyl symbols a˜=(a⊗12k)∘s, where 12k(x)=1 for all x∈R2k and s is a linear symplectomorphism of R2(n+k). We call the operators A˜ symplectic dimensional extensions of A. In this paper we study the relation between A and A˜ in detail, in particular their regularity, invertibility and spectral properties. We obtain an explicit formula allowing to express the eigenfunctions of A˜ in terms of those of A. We use this formalism to construct new classes of pseudo-differential operators, which are extensions of the Shubin classes HGρm1,m0 of globally hypoelliptic operators. We show that the operators in the new classes share the invertibility and spectral properties of the operators in HGρm1,m0 but not the global hypoellipticity property. Finally, we study a few examples of operators that belong to the new classes and which are important in mathematical physics.

On Global Hypoellipticity

We consider a first order linear partial differential operator of principal type on a closed connected orientable two-dimensional manifold sending sections of one complex line bundle to sections of another. We prove that the assumption of global hypoellipticity of the operator implies a relation between the degrees of the line bundles and the Euler characteristic of the manifold.

Global hypoellipticity, global solvability and normal form for a class of real vector fields on a torus and application

Global hypoellipticity, global solvability and normal form for a class of real vector fields on a torus and application

The inverse, the heat semigroup, Liouville’s theorems and the spectrum for the Grushin operator

By decomposing the Grushin operator on \({\mathbb{R}2}\) into a family of parameterized Hermite operators, we give estimates for the inverses and the heat semigroups of these Hermite operators, which are then used to obtain Sobolev estimates for the inverse and the heat semigroup of the Grushin operator. Using the global hypoellipticity of the parametrized Hermite operators, Liouville’s theorems for harmonic functions of the Grushin operator on \({\mathbb{R}2}\) are obtained. The spectrum of the Grushin operator is computed.

The heat kernel of the τ-twisted Laplacian

For a family of τ-twisted Laplacians that includes the usual twisted Laplacian when τ = 1/2, we compute the heat kernel for each τ-twisted Laplacian, \({\tau\in [0,1]}\) . Applications of the heat kernel to the global hypoellipticity via an explicit formula of the Green function are described.

Spectral and regularity properties of a pseudo-differential calculus related to Landau quantization

The theme of this work is that the theory of charged particles in a uniform magnetic field can be generalized to a large class of operators if one uses an extended a class of Weyl operators which we call “Landau–Weyl pseudodifferential operators”. The link between standard Weyl calculus and Landau–Weyl calculus is made explicit by the use of an infinite family of intertwining “windowed wavepacket transforms” this makes possible the use of the theory of modulation spaces to study various regularity properties. Our techniques allow us not only to recover easily the eigenvalues and eigenfunctions of the Hamiltonian operator of a charged particle in a uniform magnetic field, but also to prove global hypoellipticity results and to study the regularity of the solutions to Schrodinger equations.