A - The theory of associated and quasi associated homogeneous real distributions

Summary

Introduction

In this appendix we construct and study associated homogeneous distributions (AHDs) and quasi associated homogeneous distributions (QAHDs) for the real case. These results are based on the paper [223]. The results of this appendix are used in Chapter 6 to develop the theory of p-adic associated and quasi associated homogeneous distributions.

The concept of AHD was first introduced and studied in the book [95, Ch.I, §4.1.] (see Definitions A.2 and A.3 by analogy with the notion of the associated eigenvector (A.2.2)). Later the concept of an AHD was introduced in the paper [232, Ch.X, 8.] by Definition A.4, and in the books [87, (2.6.19)], [88, (2.110)] by Definition A.5. In the book [95, Ch.I, §4] and in the paper [232, Ch.X, 8.] a theorem was given (without proof), in which all AHDs were described (see Proposition A.2.1). In Section A.2.2 we discuss and analyse Definitions A.3, A.4, A.5, (A.2.9) of an AHD and show that they are selfcontradictory for k ≥ 2. Moreover, these definitions come into conflict with Proposition A.2.1. According to Section A.2.2, there exist only AHDs of order k = 0, i.e., homogeneous distributions (HDs) (given by Definition A.1) and of order k = 1 (given by Definition (A.2.3) or Definition A.2). Thus one can see that the concept of an AHD requires a special study.