Local integrability results in harmonic analysis on reductive groups in large positive characteristic

Let G be a connected reductive algebraic group over a non-Archimedean local field K, and let g be its Lie algebra. By a theorem of Harish-Chandra, if K has characteristic zero, the Fourier transforms of orbital integrals are represented on the set of regular elements in g(K) by locally constant functions, which, extended by zero to all of g(K), are locally integrable. In this paper, we prove that these functions are in fact specializations of constructible motivic exponential functions. Combining this with the Transfer Principle for integrability of [8], we obtain that Harish-Chandra's theorem holds also when K is a non-Archimedean local field of sufficiently large positive characteristic. Under the hypothesis that mock exponential map exists, this also implies local integrability of Harish-Chandra characters of admissible representations of G(K), where K is an equicharacteristic field of sufficiently large (depending on the root datum of G) characteristic.