I know that for a general tempered distribution (see here) $f$ the convolution $f\star P_t$ is not meaningful. Where $P_t$ is the Poisson kernel (see here) which is given by $P(x)=\dfrac{c_n}{(1+|x|^2)^{(n+1)/2}}$ and $P_t(x)=t^{-n}P(t/x)$.
Could we construct some examples for $f$, tempered distributions, such that $f\star P_t$ is not well-defined?
For example, $f(x)=|x|^{n+1}$. Trying to convolve $f$ with $P_t$ produces an integral of a positive function that has positive limit at infinity; such an integral diverges. This happens at every point.
The problem is that the decay of $P$ (and subsequently, of $P_t$) at infinity is offset by the growth of $f$.