Harnack Inequalities: An Introduction

The aim of this article is to give an introduction to certain inequalities named after Carl Gustav Axel von Harnack. These inequalities were originally defined for harmonic functions in the plane and much later became an important tool in the general theory of harmonic functions and partial differential equations. We restrict ourselves mainly to the analytic perspective but comment on the geometric and probabilistic significance of Harnack inequalities. Our focus is on classical results rather than latest developments. We give many references to this topic but emphasize that neither the mathematical story of Harnack inequalities nor the list of references given here is complete.