Sharp weighted estimates and further improvements via Reverse Holder Inequalities

EZEQUIEL RELA

Barranquilla

Congreso; V Congreso Latinoamericano de Matemáticos; 2016

UMALCA - Universidad del Norte

In the theory of weighted inequalities for Ap weights, there are a lot of inequalities proved for a large class of classical operators, including maximal operator and singular integrals. To test the sharpness ofthe exponent , each situation demands a specific example. Usually, a 1-parameter family of functions and weights asymptotically verifying that thesame inequality cannot hold for a smaller . In this talk I will present arecent discovery of a close connection between the best possible for theabove inequality and the asymptotic behaviour of the unweighted Lp normkTkLp as p goes to 1 and +1. Using this method, we verify the sharpness ofknow bounds and provide a lower bound for the exponent for those caseswhen it is still not know if this best possible value is attained.Moreover, we will show that this sharp bound can be further improved interms of mixed constants Ap - A-infty. This improvement is achieved by usinga precise quantitative formulation of the open condition satised by the Apclasses. The key here is the recently obtained Reverse Holder Inequalityinvolving A-infty constants.