LisMath Seminar 

13/07/2023, 10:00 — 10:30 — Online
Rodrigo Duarte, Instituto Superior Técnico, Universidade de Lisboa

Applications of Dyadic Harmonic Analysis to Weighted Inequalities

In this talk, we introduce some ideas from Dyadic Harmonic Analysis and we apply these methods to obtain weighted norm inequalities. We focus mainly on Gagliardo-Nirenberg inequalities with the Lebesgue measure modified by a weight. The classical Gagliardo-Nirenberg inequality generalizes the Sobolev inequality and is an absolutely central tool in the study of PDEs. The first results using weighted norms were given by Caffarelli, Kohn and Nirenberg, and they were subsequently improved by Chang-Shou Lin, who settled the inequality with homogeneous radial weights and integer derivatives. Here we will explain how to use the modern methods of weighted inequalities in Harmonic Analysis to extend Lin's inequality to the fractional derivatives setting. Moreover, these methods are flexible enough to give rise to similar inequalities with other weights, including non-homogeneous weights.