Bruno Pini and the Parabolic Harnack Inequality: The Dawning of Parabolic Potential Theory

Abstract

In this article, we describe the pioneering work of Bruno Pini toward the modern Potential Analysis of second order parabolic Partial Differential Equations. We mainly focus on the parabolic Harnack inequality, discovered by Pini in 1954, jointly, and independently, with Jacques Hadamard. Pini made of this inequality one the crucial tools in his construction of a generalized Wiener-type solution to the Dirichlet problem for the Heat equation. To this end, he also used an average operator on the level sets of the Heat kernel, characterizing caloric and sub-caloric functions, in analogy with the classical Gauss–Koebe, Blaschke–Privaloff and Saks Theorems for harmonic and sub-harmonic functions. He also proved a Riesz-type representation Theorem for subcaloric functions. To complete his research design, Pini established the notion of Heat capacity, and proved a Wiener-type criterion for regularity of the boundary points in the generalized Dirichlet problem for the Heat equation.