Rüdiger W. Braun

Ultradifferentiable functions and Fourier analysis

Classes of non-quasianalytic functions are classically defined by imposing growth conditions on the derivatives of the functions. It was Beuding [1] (see Bjorck [2]) who pointed out that decay properties of the Fourier transform of a compactly supported function can be used for this purpose equally well. In the present article we modify Beudings approach. More precisely, we call w: [0,00[--+ [0, oo[ a weight function if w is continuous and satisfies

The geometry of analytic varieties satisfying the local Phragmén-Lindelöf condition and a geometric characterization of the partial differential operators that are surjective on {}({ℝ}⁴)

The local Phragmen-Lindelof condition for analytic subvarieties of C n at real points plays a crucial role in complex analysis and in the theory of constant coefficient partial differential operators, as Hormander has shown. Here, necessary geometric conditions for this Phragmen-Lindelof condition are derived. They are shown to be sufficient in the case of curves in arbitrary dimension and of surfaces in C 3 . The latter result leads to a geometric characterization of those constant coefficient partial differential operators which are surjective on the space of all real analytic functions on R 4 .

Applications of the Projective Limit Functor to Convolution and Partial Differential Equations

Let e {ω{(ℝ N ) denote the non-quasianalytic class of all {ω{-ultradifferentiable functions on ℝ N . This notion is an extension of the classical Gevrey classes Γ{d{(ℝ N ), d > 1. Reporting on our work [5] and [6], we explain how the projective limit functor introduced by Palamodov [21] can be used to characterize the surjectivity of (1) convolution operators T μ on e {ω{(ℝ) and (2) linear partial differential operators P(D) on e {ω{(ℝ N ).

Higher Order Tangents to Analytic Varieties along Curves. II

Let V be an analytic variety in some open set in Cn. For a real analytic curve with (0) = 0 and d ≥ 1, define Vt = t d (V − (t)). It was shown in a previous paper that the currents of integration over Vt converge to a limit current whose support T,dV is an algebraic variety as t tends to zero. Here, it is shown that the canonical defining function of the limit current is the suitably normalized limit of the canonical defining functions of the Vt. As a corollary, it is shown that T,dV is either inhomogeneous or coincides with T,�V for allin some neighborhood of d. As another application it is shown that for surfaces only a finite number of curves lead to limit varieties that are interesting for the investigation of Phragm´¨ of conditions. Corresponding results for limit varieties T�,�W of algebraic varieties W along real analytic curves tending to infinity are derived by a reduction to the local case.

Algebraic varieties on which the classical Phragmén-Lindelöf estimates hold for plurisubharmonic functions

Abstract. Algebraic varieties V are investigated on which the natural analogue of the classical Phragmén-Lindelöf principle for plurisubharmonic functions holds. For a homogeneous polynomial P in three variables it is shown that its graph has this property if and only if P has real coefficients, no elliptic factors, is locally hyperbolic in all real characteristics, and the localizations in these characteristics are square-free. The last condition is shown to be necessary in any dimension.

Local radial Phragmén-Lindelöf estimates for plurisubharmonic functions on analytic varieties

We give a sufficient condition for a local radial Phragmen-Lindelof principle on analytic varieties. This condition is expressed in terms of existence of hyperbolic directions.

Nearly Hyperbolic Varieties and Phragmén-Lindelöf Conditions

By a classical estimate in function theory, each subharmonic function u(z) on the unit disk that is bounded above by 1 and bounded by 0 on the real axis must satisfy a bound of the form u(z) ≤ A|Imz| on smaller subdisks. When an analogous estimate holds for the plurisubharmonic functions in a neighborhood of a real point ξ in an analytic variety, the variety is said to satisfy the local Phragmen-Lindelof condition at ξ. Interest in such conditions originated from a theorem of Hormander who showed that the surjective constant coefficient linear partial differential operators on the space of real analytic functions on ℝ n are characterized in terms of these conditions. We give a new geometric condition on a local variety that is necessary in order that the local Phragmen-Lindelof condition holds, and is sufficient in the case of varieties of dimension 1 or 2.

An example concerning the local radial Phragmén-Lindelöf condition

Abstract An example of an algebraic surface in ℂ 3 is given which satisfies the local radial Phragmen- Lindelof condition RPL loc (0) but which fails a certain hyperholicity condition. This provides a counterexample to the converse of a result by the present authors .