Let $K$ be a field of characteristic zero, $X$ and $Y$ be smooth $K$-varieties, and let $G$ be an algebraic $K$-group. Given two algebraic morphisms $\phi :X\to G$ and $\psi :Y\to G$, we define their convolution $\phi \ast \psi :X\times Y\to G$ by $\phi \ast \psi (x,y)=\phi (x)·\psi (y)$. We then show that this operation yields morphisms with improved smoothness properties. More precisely, we show that for any morphism $\phi :X\to G$ which is dominant when restricted to each geometrically irreducible component of $X$, by convolving it with itself finitely many times, one obtains a flat morphism with reduced fibers of rational singularities. 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 ${\{f_{{\mathbb{Q}}_p}:{\mathbb{Q}}_p^n\to \mathbb{C}\}}_{p\in \mathrm{primes}}$ is a collection of motivic functions, 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.

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关于代数态射卷积的奇异性——一般情况

让 $钾$ 是特征为零的域， $X$ 和 $是$ 圆滑 $钾$- 品种，并让 $G$ 成为代数 $钾$-团体。给定两个代数态射$\phi ：X\to G$ 和 $\psi ：是\to G$，我们定义它们的卷积 $\phi ＊\psi ：X\times 是\to G$ 经过 $\phi ＊\psi (X,是)=\phi (X)·\psi (是)$. 然后我们证明这个操作产生了具有改进平滑特性的态射。更准确地说，我们证明对于任何态射$\phi ：X\to G$ 当限制到每个几何上不可约的分量时，它占主导地位 $X$，通过将它与自身有限多次卷积，可以得到一个具有减少的有理奇点纤维的平面态射。还给出了态射族的统一边界。此外，作为关键的分析步骤，我们还证明了动机整合的以下结果：如果${\{F_{{\mathbb{问}}_{磷}}：{\mathbb{问}}_{磷}^n\to \mathbb{C}\}}_{磷\in \mathrm{素数}}$ 是一个动机函数的集合，并且 $F_{{\mathbb{问}}_{磷}}$ 是 ${升}^1$ 对于任何 $磷$ 足够大，那么实际上存在 $\epsilon >0$ 以至于 $F_{{\mathbb{问}}_{磷}}$ 是 ${升}^{1+\epsilon }$ 对于任何 $磷$ 足够大。