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

This work deals with weighted inequalities of the type ∫Rd|Tf(x)|pw(x)dx≤C∫Rd|Sf(x)|pw(x)dx,∫Rd|Tf(x)|pw(x)dx≤C∫Rd|Sf(x)|pw(x)dx, where SS is some maximal operator and TT is an operator that comes from the harmonic analysis associated to a critical radius function. The weight ww belongs to an appropriate family and 0