×

A progress report on analytic hypoellipticity. (English) Zbl 0924.47029

Noguchi, J. (ed.) et al., Geometric complex analysis. Proceedings of the conference held at the 3rd International Research Institute of the Mathematical Society of Japan, Hayama, March 19–29, 1995. Singapore: World Scientific. 123-146 (1996).
The concept of hypoellipticity for partial differential equations and its properties is as old as the existence of the class of functions. But in the recent years, important progress has been made in the theory of analytic hypoellipticity, due to the development of partial differential equations having multiple characteristics and on three-dimensional Cauchy-Riemann manifolds. The author defines a linear partial differential operator \(L\), to be analytically hypoelliptic, having \(c^W\) coefficients of the class of real analytic functions, in the open set \(V\). It is established that the analytic hypoellipticity is of two types, “global” and “local”, under certain necessary and sufficient conditions. Having analyzed the characteristic variety of the operator, necessary and sufficient conditions are reported for analytic hypoelliptic neighbourhoods.
If the operator is having the property of pseudo-convexity and symplecticity, it is said to be micro elliptic and degenerates regularly at \(0\), even if it is not elliptic. The author also explains, through remarkable illustrations that, if the operator is non-elliptic then it is analytically hypoelliptic, if it is not analytically hypoelliptic then it is a stronger operator, elliptic everywhere at an isolated point and globally analytically hypoelliptic. This article also defines that, if local analytic hypoellipticity fails in two-dimensional space, a Gevrey class may then be introduced to provide the operator to be \(G^9\).
As local and global analytic hypoellipticity rely on nonlinear eigenvalue problems, using nonlinear eigenvalues, the family of ordinary differential operators, linear partial differential operators \(D\) and constant coefficient partial differential operators \(T\) satisfy, for holomorphic functions, the analytic hypoellipticity. The importance of the present study is that it provides with an account of the recent progress and a fairly good account of an alternative method for the subject. The Szegö projection is also reported, which incidentally does not preserve global analyticity.
For the entire collection see [Zbl 0903.00037].

MSC:

47F05 General theory of partial differential operators
65H10 Numerical computation of solutions to systems of equations
35J99 Elliptic equations and elliptic systems
35D10 Regularity of generalized solutions of PDE (MSC2000)
PDFBibTeX XMLCite