Quasihomogeneous Distributions, Volume 165

(Almost) Quasihomogeneous Functions. Definitions and Basic Properties. (Almost) Quasihomogeneous Distributions. Definitions and Basic Properties. Constructing (Almost) Quasihomogeneous Functions by Taking Quasihomogeneous Averages of Functions with M-Bounded Support. Constructing (Almost) Quasihomogeneous Distributions by Taking Quasihomogeneous Averages. The Case: X is Locally M-Bounded. Constructing (Almost) Quasihomogeneous Functions by Taking Quasihomogeneous Averages of Functions Not Necessarily Having M-Bounded Support. Constructing (Almost) Quasihomogeneous Distributions by Taking Quasihomogeneous Averages. The Case: (1.14) Holds. Solvability of Quasihomogeneous Multiplication Equations and Partial Differential Equations. Extending (Almost) Quasihomogeneous Distributions on X+ to the Whole of X. Quasihomogeneous Wave Front Sets. References. Index.

This is a systematic exposition of the basics of the theory of quasihomogeneous (in particular, homogeneous) functions and distributions (generalized functions). A major theme is the method of taking quasihomogeneous averages. It serves as the central tool for the study of the solvability of quasihomogeneous multiplication equations and of quasihomogeneous partial differential equations with constant coefficients. Necessary and sufficient conditions for solvability are given. Several examples are treated in detail, among them the heat and the Schrödinger equation. The final chapter is devoted to quasihomogeneous wave front sets and their application to the description of singularities of quasihomogeneous distributions, in particular to quasihomogeneous fundamental solutions of the heat and of the Schrödinger equation.