Abstract. Let G be a connected reductive p-adic group and let g be its Lie algebra. Let O be a G-orbit in g. Then the orbital integral µO corresponding to O is an invariant distribution on g, and Harish-Chandra proved that its Fourier transform ˆµO is a locally constant function on the set g ′ of regular semisimple elements of g. Furthermore, he showed that a normalized version of the Fourier transform is locally bounded on g. Suppose that O is a regular semisimple orbit. Let γ be any semisimple element of g, andletmbethe centralizer of γ. We give a formula for ˆµO(tH) (in terms of Fourier transforms of orbital integrals on m), for regular semisimple elements H in a small neighborhood of γ in m and t ∈ F × sufficiently large. We use this result to prove that Harish-Chandra’s normalized Fourier transform is globally bounded on g in the case that O is a regular semisimple orbit. 1
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