# Differences between fundamental solutions of general higher order elliptic operators and of products of second order operators

## TL;DR

This paper investigates the sign behavior of fundamental solutions of high-order elliptic operators with constant coefficients, revealing how their positivity or negativity near singularities depends on the dimension and operator structure.

## Contribution

It provides a systematic analysis of sign changes in fundamental solutions across dimensions and explicit formulas for specific cases, extending understanding beyond the polyharmonic operator.

## Key findings

- Fundamental solutions can change sign depending on dimension.
- Sign positivity is guaranteed in certain low dimensions.
- Explicit formulas illustrate the dependence of sign on dimension.

## Abstract

We study fundamental solutions of elliptic operators of order $2m\geq4$ with constant coefficients in large dimensions $n\ge 2m$, where their singularities become unbounded. For compositions of second order operators these can be chosen as convolution products of positive singular functions, which are positive themselves. As soon as $n\geq3$, the polyharmonic operator $(-\Delta)^m$ may no longer serve as a prototype for the general elliptic operator. It is known from examples of [V.G. Maz'ya, S. A. Nazarov, Math. Notes 39 (1986); Transl. of Mat. Zametki 39 (1986)] and [E.B. Davies, Journal Differ. Equations 135 (1997)] that in dimensions $n\ge 2m+3$ fundamental solutions of specific operators of order $2m\geq4$ may change sign near their singularities: there are ``positive'' as well as ``negative'' directions along which the fundamental solution tends to $+\infty$ and $-\infty$ respectively, when approaching its pole. In order to understand this phenomenon systematically we first show that existence of a ``positive'' direction directly follows from the ellipticity of the operator. We establish an inductive argument by space dimension which shows that sign change in some dimension implies sign change in any larger dimension for suitably constructed operators. Moreover, we deduce for $n=2m$, $n=2m+2$ and for all odd dimensions an explicit closed expression for the fundamental solution in terms of its symbol. From such formulae it becomes clear that the sign of the fundamental solution for such operators depends on the dimension. Indeed, we show that we have even sign change for a suitable operator of order $2m$ in dimension $n=2m+2$. On the other hand we show that in the dimensions $n=2m$ and $n=2m+1$ the fundamental solution of any such elliptic operator is always positive around its singularity.