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Let $G$ be a connected reductive algebraic group over a non-Archimedean localfield $K$, and let $g$ be its Lie algebra. By a theorem of Harish-Chandra, if$K$ has characteristic zero, the Fourier transforms of orbital integrals arerepresented on the set of regular elements in $g(K)$ by locally constantfunctions, which, extended by zero to all of $g(K)$, are locally integrable. Inthis paper, we prove that these functions are in fact specializations ofconstructible motivic exponential functions. Combining this with the TransferPrinciple for integrability [R. Cluckers, J. Gordon, I. Halupczok, "Transferprinciples for integrability and boundedness conditions for motivic exponentialfunctions", preprint arXiv:1111.4405], we obtain that Harish-Chandra's theoremholds also when $K$ is a non-Archimedean local field of sufficiently largepositive characteristic. Under the hypothesis on the existence of the mockexponential map, this also implies local integrability of Harish-Chandracharacters of admissible representations of $G(K)$, where $K$ is anequicharacteristic field of sufficiently large (depending on the root datum of$G$) characteristic.

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