On the formula of Jacques-Louis Lions for reproducing kernels of harmonic and other functions

We give a simpler proof of the formula, due to J.-L. Lions, for the reproducing kernel of the space of harmonic functions on a domain Omegasubset ofR(n) whose boundary values belong to the Sobolev space H-s(partial derivativeOmega), and also obtain generalizations of this formula when instead of harmonic functions one considers functions annihilated by a given elliptic partial differential operator. Further, we compute the reproducing kernels explicitly in several examples, which leads to an occurrence of new special functions. Some spaces of caloric functions are also briefly considered.