Global Analytic Hypoellipticity Research Articles

(Semi-)global analytic hypoellipticity for a class of “sums of squares” which fail to be locally analytic hypoelliptic

The global and semi-global analytic hypoellipticity on the torus is proved for two classes of sums of squares operators, introduced by P. Albano, A. Bove, and M. Mughetti, satisfying the Hörmander condition and which fail to be either locally or microlocally analytic hypoelliptic.

Global analytic hypoellipticity and solvability of certain operators subject to group actions

On T × G T \times G , where T T is a compact real-analytic manifold and G G is a compact Lie group, we consider differential operators P P which are invariant by left translations on G G and are elliptic in T T . Under a mild technical condition, we prove that global hypoellipticity of P P implies its global analytic-hypoellipticity (actually Gevrey of any order s ≥ 1 s \geq 1 ). We also study the connection between the latter property and the notion of global analytic (resp. Gevrey) solvability, but in a much more general setup.

Global analytic hypoellipticity of involutive systems on compact manifolds

Global analytic hypoellipticity of involutive systems on compact manifolds

Global analytic hypoellipticity for a class of evolution operators on T1×S3

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.

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).

Global Analytic Regularity for Structures of Co-Rank One

We consider real analytic involutive structures 𝒱, of co-rank one, defined on a real analytic paracompact orientable manifold M. To each such structure we associate certain connected subsets of M which we call the level sets of 𝒱. We prove that analytic regularity propagates along them. With a further assumption on the level sets of 𝒱 we characterize the global analytic hypoellipticity of a differential operator naturally associated to 𝒱. As an application we study a case of tube structures.

Globally analytic hypoelliptic vector fields on compact surfaces

We present a characterization of the global analytic hypoellipticity of a complex, non-singular, real analytic vector field defined on a compact, connected, orientable, two-dimensional, real analytic manifold. In particular, we show that such vector fields exist only on the torus.

Global analytic hypoellipticity and pseudoperiodic functions

Global analytic hypoellipticity and pseudoperiodic functions

Remarks about global analytic hypoellipticity

We present a characterization of the operators L = 8/Ot + (a(t) + ib(t))&/&x which are globally analytic hypoelliptic on the torus. We give information about the global analytic hypoellipticity of certain overdetermined systems and of sums of squares.

Global analytic regularity for sums of squares of vector fields

We consider a class of operators in the form of a sum of squares of vector fields with real analytic coefficients on the torus and we show that the zero order term may influence their global analytic hypoellipticity. Also we extend a result of Cordaro-Himonas.

Global (and local) analyticity for second order operators constructed from rigid vector fields on products of tori

We prove global analytic hypoellipticity on a product of tori for partial differential operators which are constructed as rigid (variable coefficient) quadratic polynomials in real vector fields satisfying the Hörmander condition and where P P satisfies a “maximal” estimate. We also prove an analyticity result that is local in some variables and global in others for operators whose prototype is \[ P = ( ∂ ∂ x 1 ) 2 + ( ∂ ∂ x 2 ) 2 + ( a ( x 1 , x 2 ) ∂ ∂ t ) 2 P=\left ( \frac \partial {\partial x_1}\right ) ^2+\left ( \frac \partial { \partial x_2}\right ) ^2+\left ( a(x_1,x_2)\frac \partial {\partial t}\right )^2 \] (with analytic a ( x ) , a ( 0 ) = 0 a(x),a(0)=0 , naturally, but not identically zero). The results, because of the flexibility of the methods, generalize recent work of Cordaro and Himonas in [Global analytic hypoellipticity of a class of degenerate elliptic operators on the torus, Math. Res. Lett. 1 (1994), 501–510] and Himonas in [On degenerate elliptic operators of infinite type, Math. Z. (to appear)] which showed that certain operators known not to be locally analytic hypoelliptic (those of Baouendi and Goulaouic [Analyticity for degenerate elliptic equations and applications, Proc. Sympos. Pure Math., vol. 23, Amer. Math. Soc., Providence, RI, 1971, pp. 79–84], Hanges and Himonas [Singular solutions for sums of squares of vector fields, Comm. Partial Differential Equations 16 (1991), 1503–1511], and Christ [Certain sums of squares of vector fields fail to be analytic hypoelliptic, Comm. Partial Differential Equations 10 (1991), 1695–1707]) were globally analytic hypoelliptic on products of tori.

Global Analytic Hypoellipticity of a Class of Degenerate Elliptic Operators on the Torus

In this paper we prove global analytic hypoellipticity on the torus for certain classes of operators in the form of a sum of squares of vector fields with real-valued and real analytic coefficients. The main conditions are that every point of the torus is of finite type for the vector fields and that the coefficients are independent of some of the variables. We show, for instance, that the well known not locally analytic hypoelliptic operator of Baouendi and Goulaouic [1] is globally analytic hypoelliptic on the torus. Also, we show that the operators that were proved in Hanges-Himonas [11] and in Christ [6], [7] to be not locally analytic hypoelliptic, are globally analytic hypoelliptic. Global analytic hypoellipticity of sum of squares operators on a compact analytic manifold M is an open problem except for certain cases. From the complex analysis point of view, the most important situation is when M is the boundary of a domain in C n and the operator is Kohn’s Laplacian, b, for M. In the symplectic case the global analytic hypoellipticity of b follows from the fundamental result of Tartakoff [24], and Treves [26]. In

Global analytic hypoellipticity in the presence of symmetry

Global analytic hypoellipticity in the presence of symmetry

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.

Extendability of proper holomorphic mappings and global analytic hypoellipticity of the ∂̄-Neumann problem

IN THIS COMMUNICATION, THE PROOF OF THE FOLLOWING THEOREM IS SKETCHED: If D(1) is a bounded domain in C(n) with real analytic boundary whose partial differential-Neumann problem is globally real analytic hypoelliptic and f is a proper holomorphic mapping of D(1) onto a second bounded domain D(2) in C(n) with real analytic boundary, then the mapping f extends to be holomorphic in a neighborhood of D(1). The proof relies on a transformation formula for the Bergman projection under proper holomorphic mappings.

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.

Hypoelliptic Vector Fields and Continued Fractions

We consider global analytic hypoellipticity of constant coefficient differential operators on the $2$-torus, and prove that it is equivalent to a growth condition on the symbol. An example of a constant coefficient vector field which is globally analytic hypoelliptic but not globally hypoelliptic is constructed. Similar results are true on compact homogeneous spaces.