In the 6-volume "Generalized functions" a treatment of Fantappie transformations is promised in Vol. 1 (bottom of p.461 of the Russian edition) to come in Vol. 5. However, there is no Fantappie anywhere in Vol. 5.

A Fantappie transformation is an integral transform $$ \mu\mapsto\int_{\mathbb{R}^n} \frac{d\mu(X)}{(1-\langle T,X\rangle)^{m+1}} $$

Does anyone have an idea why the plan was changed?

My interest in this topic stems from not so well-known fact that certain Fantappie transforms of uniform measures supported on polyhedra are rational functions of $T$.

After a "conification" Fantappie transform becomes a more familiar Laplace transform (I guess it might be the real reason it was not treated after all). Embed $\mathbb{R}^n$ as an affine hyperplane at $X_0=1$, and extend $\mu$ with exponentially decaying density at direction $X_0$. E.g. the simplest case is of $\mu$ being uniform on $\Omega\subset\mathbb{R}^n$, and $m=n$, that is we have, after a convenient rescaling by $n!=\Gamma(n+1)=\int_0^\infty t^{n} e^{-t}dt$, that $$ F_{n,\Omega}(T):=\Gamma(n+1)\int_{\Omega} \frac{dX}{(1-\langle T,X\rangle)^{n+1}}=\int_{\Omega}\int_0^\infty\exp(-t+t\langle T,X\rangle)t^ndtdX. $$ The latter after the variables change $Y_0=t$, $Y_1=tX_1$,... ,$Y_n=tX_n$ becomes the Laplace transform $$ \int_{\Omega}\int_0^\infty\exp\langle \tilde{T},Y\rangle dY=F_{n,\Omega}(T), $$ where $\tilde{T}:=(1,T_1,\dots,T_n)$.