We have a "nonnegative" distribution $\mu$ with compact support in $\mathbb{R}^2$ which is not a measure, as we can produce a linear function $f(x,y)=x-1$ such that the integral of $f^{2k}$ w.r.t. $\mu$ is zero for $k\geq 2$, but positive for $k=1$. This prompts us to suspect that $\mu$ can be expressed as the product $\zeta(x)\rho(y)$ of a linear combination $\zeta(x)$ of derivatives of delta-function (which all only depend upon $x$), and a 1-dimensional distribution $\rho(y)$.

We would like to show that this is indeed the case, by checking that the Fantappie transform $$F(u,v)=\int \frac{d\mu(x,y)}{(1-ux-vy)^3}$$ of $\mu$, which we know, e.g. $$F(u,v)=\frac{1}{(1-u-v)(1-u-2v)(1-u+v)(1-u+2v)},$$ equals the Fantappie transform of $\zeta(x)\rho(y)$. And indeed, if we naively integrate first w.r.t. $x$, and then w.r.t. $y$, we have equality. However, we cannot find anything like Fubini theorem for the product of measures for such more general case, and in fact something seems to be not quite right here, as different $\zeta(x)$ seem to give the same answer. Could someone point out the way out here? Is there perhaps some helpful formalism for such a setting?

The Analysis of Linear Partial Differential Operators I. – Jochen Wengenroth Dec 13 '13 at 8:00