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

Abstract

We study fundamental solutions of elliptic operators of order 2m≥4 with constant coefficients in large dimensions n≥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≥3, the polyharmonic operator (−Δ)m may no longer serve as a prototype for the general elliptic operator. It is known from examples of Maz’ya and Nazarov (Math. Notes 39:14–16, 1986; Transl. of Mat. Zametki 39, 24–28, 1986) and Davies (J Differ Equ 135:83–102, 1997) that in dimensions n≥2m+3 fundamental solutions of specific operators of order 2m≥4 may change sign near their singularities: there are “positive” as well as “negative” directions along which the fundamental solution tends to +∞ and −∞ 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.