EVENT DETAILS AND ABSTRACT

Title: Hardy-Littlewood-Sobolev/Burkholder-Gundy, Doob
Speaker: Rodrigo Banuelos
Speaker Info: Purdue
Brief Description:
Special Note:
Abstract:

The Hardy-Littlewood-Sobolev (HLS) inequality (containing the classical Sobolev inequality) has been refined and extended in many directions, including to the setting of symmetric Markovian semigroups. The latter is a 1985 celebrated result of N. Varopoulos which has had many applications. In this talk we present a proof of the HLS inequality given by L. Hedberg in 1972 which extends, with almost no change, to Markovian semigroups. We give a stochastic integral formulation of the HLS inequality and adapt Hedberg’s argument to prove it based on the Burkholder-Gundy and Doob inequalities. The motivation for this approach comes from efforts to employ probabilistic techniques which have been extremely successful in proving sharp inequalities for various singular integrals, to study (and extend to other geometric settings) the sharp form of the HLS inequality proved by E. Lieb in 1983 and which has been of great interest to many in recent years.