A renormalized Riesz potential and applications

The convolution in R with |x|−n is a very singular operator. Endowed with a proper normalization, and regarded as a limit of Riesz potentials, it is equal to Dirac’s distribution δ. However, a different normalization turns the non-linear operator: Ef = exp( −2 |Sn−1| |x|−n ∗ f), into a remarkable transformation. Its long history (in one dimension) and some of its recent applications in higher dimensions make the subject of this exposition. A classical extremal problem studied by A. A. Markov is related to the operation E in one real variable. Later, the theory of the spectral shift of self-adjoint perturbations was also based on E. In the case of two real variables, the transform E has appeared in operator theory, as a determinantal-characteristic function of certain close to normal operators. The interpretation of E in complex coordinates reveals a rich structure, specific only to the plane setting. By exploiting an inverse spectral function problem for hyponormal operators, applications of this exponential transform to image reconstruction, potential theory and fluid mechanics have recently been discovered. In any number of dimensions, the transformation E, applied to characteristic functions of domains Ω, can be regarded as the geometric mean of the distance function to the boundary. This interpretation has a series of unexpected geometric and analytic consequences. For instance, for a convex algebraic Ω, it turns out that the operation E is instrumental in converting finite external data (such as field measurements or tomographic pictures) into an equation of the boundary. §