Riesz Kernels and Pseudodifferential Operators Attached to Quadratic Forms Over p-adic Fields

TL;DR

This paper investigates Riesz kernels and pseudodifferential equations linked to quadratic forms over p-adic fields, establishing fundamental solutions and algebraic structures for these distributions.

Contribution

It introduces a family of Riesz kernels associated with elliptic quadratic forms over p-adic fields and proves their group structure under convolution.

Findings

Riesz kernels form an Abelian group under convolution

Existence of fundamental solutions for certain pseudodifferential equations

Extension of classical pseudodifferential theory to p-adic quadratic forms

Abstract

We study pseudodifferential equations and Riesz kernels attached to certain quadratic forms over p-adic fields. We attach to an elliptic quadratic form of dimension two or four a family of distributions depending on a complex parameter, the Riesz kernels, and show that these distributions form an Abelian group under convolution. This result implies the existence of fundamental solutions for certain pseudodifferential equations like in the classical case.