Stable Harmonic Analysis and Stable Transfer

Langlands posed the question of whether a local functorial transfer map of stable tempered characters can be interpolated by the transpose of a linear operator between spaces of stable orbital integrals of test functions. These so-called stable transfer operators are intended to serve as the main local ingredient in Beyond Endoscopy, his proposed strategy for proving the Principle of Functoriality. Working over a local field of characteristic zero and assuming a hypothesis on the local Langlands correspondence for p-adic groups, we prove the existence of continuous stable transfer operators between spaces of stable orbital integrals of Harish-Chandra Schwartz functions, test functions, and K-finite test functions. This is achieved via stable Paley--Wiener theorems for each of the three types of function spaces. The stable Paley--Wiener theorem for Harish-Chandra Schwartz functions is new and includes the result that stable tempered characters span a weak-* dense subspace of the space of stable tempered distributions, a result previously unknown for p-adic groups.