Bogdan Ziemian

On G-invariant distributions

Abstract Let G be a connected Lie group. In this paper we give a characterization of G -invariant distributions on certain regular subsets of a G -manifold in terms of distributions on the orbit space. In order to obtain this characterization we introduce and exploit distributions on manifolds which are not assumed to be Hausdorff spaces.

Local existence and regularity of solutions of singular elliptic operators on manifolds with corner singularities

Abstract The local existence and regularity of solutions of singular elliptic operators on manifolds with corner singularities are studied by means of a perturbation technique based on the theory of multidimensional Mellin transformation developed in this paper. Relations with 2-microlocal regularity are established.

Special solutions of the equations Pu = 0, Pu = δ for invariant linear differential operators with polynomial coefficients

Explicit formulas are given for certain solutions of the equations Pu = δ and Pu = 0 with P being a differential operator with polynomial coefficients preserving the form ∑i = 1n xip − ∑i = 1m yiq for arbitrary even integers p, q. These formulas are a direct consequence of the invariance of P and depend only up to a constant factor upon the operator P.

A remark on Nilsson type integrals

We investigate ramification properties with respect to parameters of integrals (distributions) of a class of singular functions over an unbounded cycle which may intersect the singularities of the integrand. We generalize the classical result of Nilsson dealing with the case where the cycle is bounded and contained in the set of holomorphy of the integrand. Such problems arise naturally in the study of exponential representation at infinity of solutions to certain PDE’s (see [Z]).

Fuchsian Type Singular Operators

Before passing to the study of Fuchsian type differential operators with smooth coefficients we must acquaint ourselves with the notion of an asymptotic expansion of a function into polyhomogeneous terms (cf. [J]). It can be regarded as a generalization of the notion of smoothness.

Elliptic corner operators in spaces with continuous radial asymptotics II

Asymptotic expansions at the origin with respect to the radial variable are established for solutions to equations with smooth 2-dimensional singular Fuchsian type operators. Introduction. This paper completes the results of paper [11] in which regularity of solutions to equations with elliptic corner operators (i.e. n-dimensional Fuchsian operators) is studied in the spaces M(Ω; %) of distributions with continuous radial asymptotics. Here we introduce subspaces Z(Ω; %) and Zd(Ω; %) of M(Ω; %) and prove a generalized Taylor formula for elements of those subspaces. This is preceded by preliminaries on the modified Cauchy transformation needed to establish a Mittag-Leffler type decomposition for holomorphic functions with a growth control along the imaginary axis. Then we study solutions to homogeneous equations R(x1, x2, x1∂/∂x1, x2∂/∂x2)u = 0 with R(x1, x2, ζ1, ζ2) an elliptic symbol, on proper cones in the positive quadrant in R. The solutions u are shown to belong to Zd(Ω; %). In contrast to solutions to Fuchsian equations in the sense of Baouendi– Goulaouic, the solutions u to Ru = 0 do not expand in discrete powers of the radial variable. Instead, for n = 2, we have “continuous” asymptotic expansions whose densities are distributions supported by several half lines parallel to the real axis. The densities are equal to the boundary values of the Mellin transforms of u times the factor (2πi)−1. Moreover, they extend to holomorphic functions with logarithmic singularities situated in a discrete lattice in C. This is resemblant of the resurgence phenomenon of Jean Ecalle and is further investigated in a forthcoming paper [12]. The paper ends with an explicit example covering the case of the operator ∆̃ = (x1∂/∂x1) + (x2∂/∂x2). Some results of this paper appeared in [13]. The reader interested in a system-