On Lerner's inequality and maximal functions for the Schrödinger operator

BONGIOANNI, BRUNO; CABRAL, ENRIQUE ADRIÁN; HARBOURE, ELEONOR

Córdoba

Congreso; IV Congreso Latinoamericano de Matemáticos; 2012

Unión Matemática de América Latina y el Caribe - Facultad de Matemática, Astronomía y Física - UNC - Centro de Investigación y Estudios de Matemática - CIEM

We consider the Schrödinger operator L= - Laplaciano + V where the potential V is non-negative and satisfies a reverse Hölder inequality. We obtain some weighted inequalities of the form begin{equation} int_{R^n}|Tf|^p w leq C int_{R^n} |Sf|^p w, end{equation} where T is an operator of the Harmonic Analysis that generates L, controlled by an appropriate maximal function S. The weight w belongs to a class that is larger than Muckenhoupt class A_infty, conditioning the weights to behave only locally as A_infty weights. To this end we use an extrapolation argument and point-wise inequalities that we prove for each case of  T. Another important tool that we develop in this work is an adapted version of Lerner´s inequality for the maximal and sharp operators which appear in this context.