The central formula is a tautological rearrangement of the Poisson kernel definition, and the claimed higher-dimensional Harnack generalization is missing.
Pini , Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico, Rend
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Geometric aspects of the Harnack Inequality for a nonlocal heat equation
The central formula is a tautological rearrangement of the Poisson kernel definition, and the claimed higher-dimensional Harnack generalization is missing.