Dmitriy Stolyarov


St. Petersburg State University


Thursday, November 4, 2021 - 11:00am


zoom ID: 949 5980 5461. Password: the last four digits of the zoom ID in the reverse order

I will speak about an unusual way to correct the (invalid) endpoint case of the Hardy—Littlewood—Sobolev inequality. Usually the correction is done by imposing additional linear constraints on the function we apply the Riesz potential to. Being the gradient of another function is an example of such a constraint. The inequalities obtained this way are often called Bourgain—Brezis inequalities. In 2010, Maz’ya suggested another approach: instead of constraining the right hand side we should replace the L_p norm on the left with an expression \Phi, which alongside with having the same homogeneity properties as the L_p norm, possesses additional cancellations. He conjectured that if \Phi satisfies a natural necessary condition, then the modified Hardy—Littlewood—Sobolev inequality holds true. I will try to survey the proof of Maz’ya’s conjecture. Based on https://arxiv.org/abs/2109.08014