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We study fundamental solutions of elliptic operators of order $2m\geq4$ withconstant coefficients in large dimensions $n\ge 2m$, where their singularitiesbecome unbounded. For compositions of second order operators these can bechosen as convolution products of positive singular functions, which arepositive themselves. As soon as $n\geq3$, the polyharmonic operator$(-\Delta)^m$ may no longer serve as a prototype for the general ellipticoperator. It is known from examples of [V.G. Maz'ya, S. A. Nazarov, Math. Notes39 (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 ofspecific operators of order $2m\geq4$ may change sign near their singularities:there are ``positive'' as well as ``negative'' directions along which thefundamental solution tends to $+\infty$ and $-\infty$ respectively, whenapproaching its pole. In order to understand this phenomenon systematically wefirst show that existence of a ``positive'' direction directly follows from theellipticity of the operator. We establish an inductive argument by spacedimension which shows that sign change in some dimension implies sign change inany larger dimension for suitably constructed operators. Moreover, we deducefor $n=2m$, $n=2m+2$ and for all odd dimensions an explicit closed expressionfor the fundamental solution in terms of its symbol. From such formulae itbecomes clear that the sign of the fundamental solution for such operatorsdepends on the dimension. Indeed, we show that we have even sign change for asuitable operator of order $2m$ in dimension $n=2m+2$. On the other hand weshow that in the dimensions $n=2m$ and $n=2m+1$ the fundamental solution of anysuch elliptic operator is always positive around its singularity.

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