Global Hypoellipticity Research Articles

Global Hypoellipticity of a Mathieu Operator

We give a necessary and sufficient condition for the global hypoellipticity of a Mathieu operator on the torus ${\mathbb {T}^d}$ in terms of continued fractions. It is not hypoelliptic, nor does it satisfy a controllability condition, a Hörmander condition, or a Siegel condition. But it is still globally hypoelliptic (cf. [1, 3]).

Global hypoellipticity of a Mathieu operator

We give a necessary and sufficient condition for the global hypoellipticity of a Mathieu operator on the torus ${\mathbb {T}^d}$ in terms of continued fractions. It is not hypoelliptic, nor does it satisfy a controllability condition, a Hörmander condition, or a Siegel condition. But it is still globally hypoelliptic (cf. [1, 3]).

Global hypoellipticity property of a differential operator

Global hypoellipticity property of a differential operator

D-harmonic distributions and global hypoellipticity on nilmanifolds

<i>D</i>-harmonic distributions and global hypoellipticity on nilmanifolds

Globally hypoelliptic systems of vector fields on nilmanifolds

We show that a system L of real vector fields on a general compact nilmanifold Γ⧹; N induced by the Lie algebra N of N is globally hypoelliptic (GH) iff (1∘) The symbols of the vector fields of L projected onto the associated torus T = Γ[ N, N]⧹ N as functions on the integral lattice T ̂ collectively decrease at infinity not faster than a reciprocal of a polynomial and (2∘) the Lie subalgebra of N that L generates is not annihilated by any non-zero integral linear functional on any N j N j + 1 , j=0, 1,…, (N j + 1 = [ N , N j], N o = N) . It follows that (GH) is equivalent to injectivity of the system L on the dual on the space of C ∞-vectors of all the non-trivial representations in the spectrum of Γ⧹ N (a “Rockland type” condition) plus a number-theoretic condition on L on the associated torus (to avoid “small divisors”).

The global hypoellipticity of degenerate elliptic-parabolic operators

The global hypoellipticity of degenerate elliptic-parabolic operators

Global analytic hypoellipticity of the $$\bar \partial$$ -Neumann problem on circular domains

In this paper we show that if $$D \subseteq \mathbb{C}^n ,n \geqq 2$$ , is a smooth bounded pseudoconvex circular domain with real analytic defining functionr(z) such that $$\sum\limits_{k = 1}^n {z_k \frac{{\partial r}}{{\partial z_k }}} \ne 0$$ for allz near the boundary, then the solutionu to the $$\bar \partial$$ -Neumann problem, $$square u = (\bar \partial \bar \partial * + \bar \partial *\bar \partial )u = f,$$ is real analytic up to the boundary, if the given formf is real analytic up to the boundary. In particular, if $$D \subseteq \mathbb{C}^n ,n \geqq 2$$ , is a smooth bounded complete Reinhardt pseudoconvex domain with real analytic boundary. Then ▭ is analytic hypoelliptic.

The theory of infinite-dimensional lie groups and its applications

The purpose of this paper is to survey the theory of regular Frechet-Lie groups developed in [1–10]. Such groups appear and are useful in symplectic geometry and the theory of primitive infinite groups of Lie and Cartan [11]. From the group theoretical standpoint, general relativistic mechanics is a more closed system than Newtonian mechanics. Quantized objects of these classical groups are closely related to the group of Fourier integral operators [12]. These can also be managed as regular Frechet-Lie groups. However, there are many Frechet-Lie algebras which are not the Lie algebras of regular Frechet-Lie groups [13]. Thus, the enlargeability of the Poisson algebra is discussed in detail in this paper. Enlargeability is relevant to the global hypoellipticity [14, 15] of second-order differential operators.

The global hypoellipticity of a class of degenerate elliptic-parabolic operators

On donne un critere d'hypoellipticite globale plus fin que le resultat de Oleinik et Radkevich

On global hypoellipticity of vector fields

On global hypoellipticity of vector fields

Globally hypoelliptic and globally solvable first-order evolution equations

We consider global hypoellipticity and global solvability of abstract first order evolution equations defined either on an interval or in the unit circle, and prove that it is equivalent to certain conditions bearing on the total symbol. We relate this to known results about hypoelliptic vector fields on the 2-torus.

Remarks on Global Hypoellipticity

We study differential operators D which commute with a fixed normal elliptic operator E on a compact manifold M. We use eigenfunction expansions relative to E to obtain simple conditions giving global hypoellipticity. These conditions are equivalent to D having parametrices in certain spaces of functions or distributions. An example is given by M = compact Lie group and and E = Casimir operator, with D any invariant differential operator. The connections with global subelliptic estimates are investigated.

Remarks on global hypoellipticity

We study differential operators D which commute with a fixed normal elliptic operator E on a compact manifold M. We use eigenfunction expansions relative to E to obtain simple conditions giving global hypoellipticity. These conditions are equivalent to D having parametrices in certain spaces of functions or distributions. An example is given by M = compact Lie group and and E = Casimir operator, with D any invariant differential operator. The connections with global subelliptic estimates are investigated.

An example of a second-order nonhypoelliptic operator with the property of global hypoellipticity

It is proved that the operator $$P \equiv - \frac{{\partial ^2 }}{{\partial x_1^2 }} - \sum\nolimits_{k = 2}^n {\frac{\partial }{{\partial x_k }}\varphi ^2 (x)} \frac{\partial }{{\partial x_k }},$$ where ϕ e C∞(Ω) (Ω is a domain in Rn), {x: ϕ(x) = 0} is a compactun in Ω which is the closure of its internal points, has the property of global hypoellipticity in Ω, i.e., $$\begin{array}{*{20}c} {v \in D'(\Omega ),} & {Pv \in C^\infty } & {(\Omega ) \Rightarrow \upsilon \in C^\infty (\Omega ).} \\ \end{array} $$ . This operator is not hypoelliptic.

Hypoelliptic vector fields and continued fractions

We consider global analytic hypoellipticity of differential operators on the 2-torus, and prove that it is equivalent to a growth condition on the symbol. An example of a vector field which is globally analytic hypoelliptic but not globally hypoelliptic is constructed. Similar results are true on compact homogeneous spaces. 1. We discuss global analytic hypoellipticity for invariant differential operators on T2, and contrast it with global hypoellipticity (the topic of [G]). In particular, we construct a constant coefficient vector field on T2 which is globally analytic hypoelliptic but not globally hypoelliptic. 2. Let T2 = {(exp i01, exp iO2); Oj E R}. If L E D'(T2) (a distribution on T2) define L(n, m) = L(exp (-in01 im02)). Suppose P is an invariant differential operator: P = IN1=0 a7. Dk D , where ak E C and D' = ((1/i)a/a03)k. Define P(n, m) = ENI=k aklnlml, for n, m integers. We say P is globally hypoelliptic when: (GH) If g E C'(T2), and Pf = g, f ED 9'(T2), thenf E C'(T2). Let A(T2) denote the real analytic functions on T2. We say P is globally analytic hypoelliptic when: (GAH) If g E A(T2), and Pf = g, f E9'(T2), thenf E A(T2). The following theorem was proved in [G]. THEOREM 2.1. P is (GH) if and only if there are positive real numbers L and M so that: (LM) IP(n, m)l > L/(n2 + m2)M, for lnl, Iml sufficiently large. An analogous characterization of (GAH) is given below. THEOREM 2.2. P is (GAH) if and only iffor any positive real number K there is a positive integer NK so that: (KN) Jfi(n, m)l > exp (-K(n2 + m2)1/2),for lnl, Iml larger than NK. Received by the editors March 10, 1971. AMS 1970 subject classifications. Primary 35H05, 42A92, 43A75.

Global hypoellipticity and Liouville numbers

We consider global hypoellipticity of constant coefficient differential operators on the 2-torus, and prove that it is equivalent to an algebraic growth condition on the symbol. This is applied to give necessary and sufficient conditions that a constant coefficient vector field be globally hypoelliptic. Similar results are true on compact homogeneous spaces. Let T2 = {(exp iOl, exp i02); O0 e R}. If L eI'(T2) (a distribution on T2) define L(n, m) = L(exp (-inOlimO2)) (see the normalization below for functions). 1 is a function on Z x Z, the Fourier transform of L. We write L ' {L(n, m)} to indicate the correspondence. A function T: Z x Z -C is the Fourier transform of a distribution L iff there is K > 0 so that IT(n, m)I < K(n2 + m2 + 1)K. (T is of polynomial growth ([S, Chapter 7]).) L can be reconstructed from T by: L = 2 T(n, m) exp (inO1 + imO2). n,m (This implies the normalization f (g) = (2)J f (x, y)g(x, y) dx dy forf a function.) Suppose P is an invariant differential operator on T2:

Global Hypoellipticity and Liouville Numbers

Global Hypoellipticity and Liouville Numbers