Lerner’s inequality associated to a critical radius function and applications

This work deals with weighted inequalities of the type ∫ R d | T f ( x ) | p w ( x ) d x ≤ C ∫ R d | S f ( x ) | p w ( x ) d x , where S is some maximal operator and T is an operator that comes from the harmonic analysis associated to a critical radius function. The weight w belongs to an appropriate family and 0 < p < ∞ . The proofs are based on an adapted Lerner’s inequality and some point-wise estimates. The results can be applied to obtain inequalities for several operators associated to the Schrödinger semigroup.