This project studies two fundamental notions in analysis on metric spaces: modulus and capacity. While these notions are ancient, there are new results one can obtain regarding them. One specific type of question asks when the modulus of curves is in some sense dual to a modulus of a family of separating surfaces. It is remarkable, that in Euclidean space this duality holds, with a constant one. In metric spaces such results also hold, but with a relatively strong assumption of a doubling and a Poincare inquality (see e.g. work of Rajala, Lohvansuu, Shanmugalingam, Lahti and Jones). Here, contrary to the Euclidean setting, constants emerge that depend on the auxiliary structures. Indications exist that some of the constants should not depend on any such data, and be universal.
Remarkably, in Graphs such a duality also holds with constant unity. One might expect, that via discretization and a limiting process the result on graphs would yield a general metric space result. However, there are several fundamental obstacles to this simple approach. Together with my coauthors, we are studying which sharp results hold in a metric space setting, and whether the conditions can be relaxed to the space being simply locally complete and separable. The initial work gave a sharp lower bound for duality, and we will push this result further to give other insights.
On the Sharp Lower Bound for Duality of Modulus
Sylvester Eriksson-Bique, Pietro Poggi-Corradini