Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A structure theorem for fundamental solutions of analytic multipliers in $\mathbb{R}^n$

David Scott Winterrose

Abstract

Using a version of Hironaka's resolution of singularities for real-analytic functions, any elliptic multiplier $\mathrm{Op}(p)$ of order $d>0$, real-analytic near $p^{-1}(0)$, has a fundamental solution $μ_0$. We give an integral representation of $μ_0$ in terms of the resolutions supplied by Hironaka's theorem. This $μ_0$ is weakly approximated in $H^t_{\mathrm{loc}}(\mathbb{R}^n)$ for $t<d-\frac{n}{2}$ by a sequence from a Paley-Wiener space. In special cases of global symmetry, the obtained integral representation can be made fully explicit, and we use this to compute fundamental solutions for two non-polynomial symbols.

A structure theorem for fundamental solutions of analytic multipliers in $\mathbb{R}^n$

Abstract

Using a version of Hironaka's resolution of singularities for real-analytic functions, any elliptic multiplier of order , real-analytic near , has a fundamental solution . We give an integral representation of in terms of the resolutions supplied by Hironaka's theorem. This is weakly approximated in for by a sequence from a Paley-Wiener space. In special cases of global symmetry, the obtained integral representation can be made fully explicit, and we use this to compute fundamental solutions for two non-polynomial symbols.

Paper Structure

This paper contains 6 sections, 9 theorems, 43 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Solution Operator
  4. Sums of powers of the Laplacian
  5. A perturbation of the Laplacian
  6. Acknowledgements

Key Result

Theorem 3.1

Let $U\subset \mathbb{R}^n$ be an open neighbourhood of $0$, and let $f$ be a function $0 \not\equiv f \in C^\omega(U)$. Then there is an open $0 \in V\subset U$, a real-analytic manifold $M$, and a map It has the following properties:

Figures (2)

  • Figure 1: Deformation of a star-convex zero-set onto a circle.
  • Figure 2: Covering the zero-set of $p$ except for a point. Here $a=\frac{1}{4}$ and $n=12$.

Theorems & Definitions (18)

  • Definition 2.1: The Paley-Wiener spaces
  • Theorem 3.1: Local embedded version of Hironaka's theorem. From Atiyah atiyah1970
  • Theorem 3.2
  • Lemma 3.3
  • Proof 1
  • Lemma 3.4
  • Proof 2
  • Lemma 3.5
  • Proof 3
  • Lemma 3.6
  • ...and 8 more