Potential Theory Results for a Class of PDOs Admitting a Global Fundamental Solution

  • Andrea BonfiglioliEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 275)


We outline several results of Potential Theory for a class of linear partial differential operators \(\mathcal {L}\) of the second order in divergence form. Under essentially the sole assumption of hypoellipticity, we present a non-invariant homogeneous Harnack inequality for \(\mathcal {L}\); under different geometrical assumptions on \(\mathcal {L}\) (mainly, under global doubling/Poincaré assumptions), it is described how to obtain an invariant, non-homogeneous Harnack inequality. When \(\mathcal {L}\) is equipped with a global fundamental solution \(\varGamma \), further Potential Theory results are available (such as the Strong Maximum Principle). We present some assumptions on \(\mathcal {L}\) ensuring that such a \(\varGamma \) exists.


Fundamental solutions Hypoelliptic operators Harnack inequalities Potential theory 



The results of this paper were presented by the author at the Conference “Noncommutative Analysis and Partial Differential Equations”, 11–15 April, 2016, Imperial College, London; the author wishes to express his gratitude to the Organizing Committee of the Conference for the hospitality.


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Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di BolognaBolognaItaly

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