Bounds for Fourier transforms of regular orbital integrals on $p$-adic Lie algebras

Let $G$ be a connected reductive $p$-adic group and let $\mathfrak g$ be its Lie algebra. Let $\mathcal O$ be a $G$-orbit in $\mathfrak g$. Then the orbital integral $\mu _{\mathcal O}$ corresponding to $\mathcal O$ is an invariant distribution on $\mathfrak g$, and Harish-Chandra proved that its Fourier transform $\hat \mu _{\mathcal O }$ is a locally constant function on the set $\mathfrak g'$ of regular semisimple elements of $\mathfrak g$. Furthermore, he showed that a normalized version of the Fourier transform is locally bounded on $\mathfrak g$. Suppose that $\mathcal O$ is a regular semisimple orbit. Let $\gamma$ be any semisimple element of $\mathfrak g$, and let $\mathfrak m$ be the centralizer of $\gamma$. We give a formula for $\hat \mu _{\mathcal O }(tH)$ (in terms of Fourier transforms of orbital integrals on $\mathfrak m$), for regular semisimple elements $H$ in a small neighborhood of $\gamma$ in $\mathfrak m$ and $t\in F^{\times}$ sufficiently large. We use this result to prove that Harish-Chandra's normalized Fourier transform is globally bounded on $\mathfrak g$ in the case that $\mathcal O$ is a regular semisimple orbit.