Uniform in "p" bounds for orbital integrals

Let $G$ be a reductive p-adic group (such as the group $GL_{n}(\mathbb{Q}_{p})$). Harish-Chandra introduced the notion of an orbital integral on $G$, and proved that the orbital integrals, normalized by the discriminant, are bounded, in a certain sense. However, it is not easy to see how this bound behaves if we let the p-adic field vary (for example, if the group $G$ is defined over a number field $F$, and we consider the family $G_{v}=G(F_{v})$, as $v$ runs over the set of finite places of $F$).

Using motivic integration and the recent transfer principles described by I. Halupczok in his talk, we prove that the bound on orbital integrals can be taken to be a fixed power (depending on $G$) of the cardinality of the residue field. This statement has an application in number theory. I will also talk about other applications of motivic integration and transfer principles in harmonic analysis on p-adic groups - namely, about the local integrability of Harish-Chandra characters in large positive characteristic. This project is joint work with R. Cluckers and I. Halupczok.