Mathematics

The independence on p of p-weak gradients is a classical problem in the Sobolev space metric theory. However, if no regularity assumption is enforced to the underlying space, the dependence of weak gradients on the integrable exponent is typically expected. In this seminar, we propose a new strategy based on optimal transportation techniques to achieve a strong kind of independence of p-weak gradients on spaces satisfying Wasserstein interpolation L^{\infty}-estimates. This improves previously available results in settings relying on Doubling & Poincarè assumptions. We push then this analysis to deduce fundamental information concerning the Sobolev and BV calculus on metric measure spaces. This is based on joint works with N. Gigli and E. Pasqualetto, T. Schultz.