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

BONGIOANNI, CABRAL, HARBOURE

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 Schr"odinger operator $mathcal{L}=-Delta + V$ where the potential $V$ is non-negative and satisfies a reverse H"older inequality. We obtain some weighted inequalities of the form egin{equation}label{ineq} int_{mathbb{R}^n}|Tf|^p w leq Cint_{mathbb{R}^n} |Sf|^p w, end{equation} where $T$ is an operator of the Harmonic Analysis that generates $mathcal{L}$, controlled by an appropriate maximal function $S$ (see cite{BHS2}). 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 (see cite{BHS1}). 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 (see cite{lerner}) for the maximal and sharp operators which appear in this context. As a particular instance of eqref{ineq} we obtain a weighted Fefferman-Stein type inequality.