Joint BCAM-UPV/EHU Analysis and PDE seminar: Hardy inequalities in the Heisenberg group

In this talk, we revisit Hardy inequalities for the Heisenberg sub-Laplacian, first proved by Garofalo and Lanconelli in the 90s. 

We discuss what happens when the Hardy weight is not of Garofalo-Lanconelli-type, and is instead associated with the sub-Riemannian distance, i.e., the control distance induced by the Heisenberg distribution. Surprising phenomena concerning the optimal constant emerge, raising several unanswered questions. 

Time permitting, we delve into Hardy inequalities for Folland-Stein operators, which generalize the Heisenberg sub-Laplacian, and their connection with a newly introduced concept of magnetic sub-Laplacian.

Based on joint works with Dario Prandi and Biagio Cassano, David Krejčiřík and Dario Prandi.