**abstract:**
In his seminal 1934 paper on Brownian motion and the theory of gases Kolmogorov introduced a second order hypoellipitc evolution equation which displays many challenging features. Despite the large amount of work done by many people over the past thirty years, some basic questions presently remain unsettled such as Hardy-Littlewood-Sobolev and Isoperimetric inequalities, a Calder\’on-Zygmund theory, and the study of local and nonlocal minimal surfaces. In this lecture I will present a fractional calculus adapted to a class of equations modelled on Kolmogorov’s, and using such calculus I will discuss some interesting developments in the above program. The natural semigroups attached to such equations need not be symmetric or doubling, thus the existing theories do not apply.

Wed 17 Apr, 8:40 - 9:30, Aula Dini

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