Putinar: The exponential transform: a renormalized Riesz potential at critical exponent

ABSTRACT. For an arbitrary domain in Rn we consider the ex-ponential of a suitably normalized Riesz potential of first nonin-tegrable index. This gives a positive function, with certain mono-tonicity properties, defined in the complement of the domain, and vanishing on its boundary. In two dimensions this func-tion, polarized into two complex variables, has previously been studied and proved to be useful within operator theory, moment problems and other problems of domain identification, and for proving regularity of free boundaries. Even in the absence of a natural polarization there are note-worthy properties of the higher dimensional exponential trans-form. For instance we show that this function is superharmonic and that it tends to zero at smooth points of the boundary, with slope bounded from above and below by constants depending solely on the local curvature. The analytic continuation configu-ration of the exponential transform of a convex polyhedron shows that it is indeed a natural defining function of its boundary. In addition, computations of this transform are carried out in full detail for some domains bounded by lower degree algebraic sur-faces. There is also an inner exponential transform, defined inside the domain. If the domain is convex then the ordinary Newto-nian potential of it can, for points inside the domain, be inter-preted is an arithmetic meanvalue with respect to the solid angle of the squared distance to the boundary. The interior exponential transform is then the corresponding geometric meanvalue