  Let $K$ be a field of characteristic zero, $X$ and $Y$ be smooth
$K$-varieties, and let $G$ be a algebraic $K$-group. Given two algebraic
morphisms $\varphi:X\rightarrow G$ and $\psi:Y\rightarrow G$, we define their
convolution $\varphi*\psi:X\times Y\to G$ by
$\varphi*\psi(x,y)=\varphi(x)\cdot\psi(y)$. We then show that this operation
yields morphisms with improved smoothness properties. More precisely, we show
that for any morphism $\varphi:X\rightarrow G$ which is dominant when
restricted to each absolutely irreducible component of $X$, by convolving it
with itself finitely many times, one can obtain a flat morphism with reduced
fibers of rational singularities, generalizing the main result of our previous
paper. Uniform bounds on families of morphisms are given as well. Moreover, as
a key analytic step, we also prove the following result in motivic integration;
if
$\{f_{\mathbb{Q}_{p}}:\mathbb{Q}_{p}^{n}\rightarrow\mathbb{C}\}_{p\in\mathrm{primes}}$
is a collection of functions which is motivic in the sense of Denef-Pas, and
$f_{\mathbb{Q}_{p}}$ is $L^{1}$ for any $p$ large enough, then in fact there
exists $\epsilon>0$ such that $f_{\mathbb{Q}_{p}}$ is $L^{1+\epsilon}$ for any
$p$ large enough.
