Remarks about global analytic hypoellipticity

We present a characterization of the operators

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.

     

Hyperfunctions and (analytic) hypoellipticity

In this work we discuss the problem of smooth and analytic regularity for hyperfunction solutions to linear partial differential equations with analytic coefficients. In particular we show that some well known “sum of squares” operators, which satisfy Hörmander’s condition and consequently are hypoelliptic, admit hyperfunction solutions that are not smooth (in particular they are not distributions).     

Global analytic and Gevrey hypoellipticity of sublaplacians under Diophantine conditions

In this paper we consider the problem of global Gevrey and analytic regularity for a class of partial differential operators on a torus in the form of a sum of squares of vector fields, which may not satisfy the bracket condition. We show that these operators are globally Gevrey or analytic hypoelliptic on the torus if and only if the coefficients satisfy certain Diophantine approximation properties.

     

On analytic microlocal hypoellipticity of linear partial differential operators of principal type

It is shown that if P is a linear partial differential uperator with analytic coefficients defined near a point xo in Rⁿ and if P in Rⁿ - 0 is such that: the principal symbol p_m,(x, ξ) vanishes at (x₀. ξ⁰). the differential of p_m, with respect to ξ is different from zero at (x₀, ξ⁰). the Poisson bracket {P_m, P_m} is zero at (x₀. ξ⁰) and the Poisson bracket {p_m, {p_m.p_m }} is different from zero at (x₀, ξ⁰), then P is analytic hypoelliptic at (x₀, ξ⁰). It is also proved that P is analytic hypoelliptic under the assumption that the first non-vanishing repeated Poisson bracket of p_m, and p_m, is of odd length and under some additional hypothesis on the commutators of the Hamilton fields of Re p_m, and Im p_m,     

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.     

An analytic characterization of the eigenvalues of self-adjoint extensions

Let be a self-adjoint extension in of a fixed symmetric operator A in . An analytic characterization of the eigenvalues of is given in terms of the Q-function and the parameter function in the Krein-Naimark formula. Here K and are Krein spaces and it is assumed that locally has the same spectral properties as a self-adjoint operator in a Pontryagin space. The general results are applied to a class of boundary value problems with λ-dependent boundary conditions.     

Leading coefficients of the eigenvalues of perturbed analytic matrix functions

This note contains some supplements to our earlier notes [LN II], [LN III], where the Newton diagram was used in order to obtain in a straightforward way information about the perturbed eigenvalues of an analytic and analytically perturbed matrix function.     

Nonlinear eigenvalues for anisotropic quasilinear degenerate elliptic equations

We consider the Dirichlet problem for a class of anisotropic degenerate elliptic equations.     

Comparison of polynomials and almost hypoellipticity

The paper investigates the hypoellipticity of polynomials in terms of comparisons.     

Nonlinear Cauchy problems with small analytic data

We study the lifespan of solutions to fully nonlinear Cauchy problems with small real- or complex-analytic data. Our proofs are based on the method of majorants and the fixed point theorem for a contraction mapping.

     

Graded hypoellipticity of BGG sequences

Annals of Global Analysis and Geometry - We study covariant Sobolev spaces and $$nabla $$ -differential operators with coefficients in general Hermitian vector bundles on Riemannian manifolds,...     

Gevrey hypoellipticity for linear and non-linear Fokker-Planck equations

This paper studies the Gevrey regularity of weak solutions of a class of linear and semi-linear Fokker-Planck equations.     

Compactness in kinetic transport equations and hypoellipticity

We establish improved hypoelliptic estimates on the solutions of kinetic transport equations, using a suitable decomposition of the phase space. Our main result shows that the relative compactness in all variables of a bounded family of nonnegative functions fλ(x,v)∈L¹ satisfying some appropriate transport relation

     

On global hypoellipticity of degenerate elliptic operators

We consider a class of degenerate elliptic operators on a torus and prove that global hypoellipticity is equivalent to an algebraic condition involving Liouville vectors and simultaneous approximability. For another class of operators we show that the zero order term may influence global hypoellipticity. Received August 13, 1997