Remarks on Analytic Hypoellipticity

We will compare the foIlowing ideas: analytic hypoeIlipticity on open subsets of Euclidean space; global analytic hypoeIlipticity; analytic hypoeIlipticity in the sense of germs. We present a new operator which posseses Treves curves, yet is analytic hypoelliptic in the sense of germs. That is, the analog of the Treves conjecture, in the sense of germs, is false

Hypoellipticity in spaces of ultradistributions-Study of a model case

In this work we study C (a)-hypoellipticity in spaces of ultradistributions for analytic linear partial differential operators. Our main tool is a new a-priori inequality, which is stated in terms of the behaviour of holomorphic functions on appropriate wedges. In particular, for sum of squares operators satisfying Hormander's condition, we thus obtain a new method for studying analytic hypoellipticity for such a class. We also show how this method can be explicitly applied by studying a model operator, which is constructed as a perturbation of the so-called Baouendi-Goulaouic operator.NSF Grant [INT 0227100]CNPqFAPES

Explicit formulas for the Szegö kernel for some domains in C2

AbstractWe study the Szegö kernel for a class of strictly pseudoconvex domains in C2. An explicit algorithm is given to compute the complete asymptotic expansion for the symbol of the Szegö kernel for these domains. It is then easy to compute the first three terms explicitly in terms of the defining function and its derivatives. We give an example where the first three terms (including the logarithmic term) are all non zero. Finally, we show that if the second term vanishes identically, then the boundary is locally biholomorphic to the surface Im w = ¦z¦2

PARAMETRICES AND LOCAL SOLVABILITY FOR A CLASS OF SINGULAR HYPERBOLIC OPERATORS.

Abstract not availabl

A NEW PROOF OF OKAJI'S THEOREM FOR A CLASS OF SUM OF SQUARES OPERATORS

Let P be a linear partial differential operator with analytic coefficients. We assume that P is of the form ""sum of squares"", satisfying Hormander's bracket condition. Let q be a characteristic point; for P. We assume that q lies on a symplectic Poisson stratum of codimension two. General results of Okaji Show that P is analytic hypoelliptic at q. Hence Okaji has established the validity of Treves' conjecture in the codimension two case. Our goal here is to give a simple, self-contained proof of this fact

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 Hormander`s condition and consequently are hypoelliptic, admit hyperfunction solutions that are not smooth (in particular they are not distributions).NSF[INT 0227100]NSFConselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)CNPqFapespFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP