Caloric Harnack Inequality, Mean Value Theorem and Capacity: the Bruno Pini Work Towards Modern Parabolic Potential Theory

We describe the pioneering work of Bruno Pini towards the modern Potential Analysis of linear second order parabolic Partial Differential Equations. We mainly focus on the caloric Harnack Inequality discovered by Bruno Pini in 1954, jointly, and independently, with Jacques Hadamard. Pini made of this inequality the crucial tools in his construction of a Wiener-type solution to the ''Dirichlet problem'' for the Heat equation. To this end he also introduced an average operator on the level set 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. Pini also established, and used, the notion of caloric capacity to study the boundary behavior of his Wiener-type solution to the first boundary value problem for the Heat equation.