Title: Lions' formula for RKHSs of real harmonic functions on Lipschitz domains,
Abstract: Let $ \Omega$ be a bounded Lipschitz domain of $ \mathbb{R}^{d}.$ The purpose
of this paper is to establish Lions' formula for reproducing kernel Hilbert
spaces $\mathcal H^s(\Omega)$ of real harmonic functions elements of the usual
Sobolev space $H^s(\Omega)$ for $s\geq 0.$ To this end, we provide a functional
characterization of $\mathcal H^s(\Omega)$ via some new families of positive
self-adjoint operators, describe their trace data and discuss the values of $s$
for which they are RKHSs. Also a construction of an orthonormal basis of
$\mathcal H^s(\Omega)$ is established.