Vol.:(0123456789) Collectanea Mathematica https://doi.org/10.1007/s13348-020-00287-1 1 3 On the optimal numerical parameters related with two weighted estimates for commutators of classical operators and extrapolation results Gladis Pradolini 1  · Wilfredo Ramos 2  · Jorgelina Recchi 3 Received: 6 December 2019 / Accepted: 27 April 2020 © Universitat de Barcelona 2020 Abstract We give two-weighted norm estimates for higher order commutator of classical operators such as singular integral and fractional type operators, between weighted L p and certain spaces that include Lipschitz, BMO and Morrey spaces. We also give the optimal param- eters involved with these results, where the optimality is understood in the sense that the parameters defining the corresponding spaces belong to certain region out of which the classes of weights are satisfied by trivial weights. We also exhibit pairs of non-triv- ial weights in the optimal region satisfying the conditions required. Finally, we exhibit an extrapolation result that allows us to obtain boundedness results of the type described above in the variable setting and for a great variety of operators, by starting from analogous inequalities in the classical context. In order to get this result we prove a Calderón–Scott type inequality with weights that connects adequately the spaces involved. Keywords Fractional operators · Singular integral operators · Conmutators · Extrapolation · Weights Mathematics Subject Classification 42B20 · 42B25 · 42B35 * Jorgelina Recchi drecchi@uns.edu.ar; jrecchi@gmail.com Gladis Pradolini gladis.pradolini@gmail.com Wilfredo Ramos wilfredo.ramos@comunidad.unne.edu.ar 1 Departamento de Matemática, Facultad de Ingeniería Química, CONICET, UNL, 3000 Santa Fe, Argentina 2 Departamento de Matemática, Facultad de Ciencias Exactas y Naturales y Agrimensura, CONICET, UNNE, 3400 Corrientes, Argentina 3 Departamento de Matemáticas, Instituto de Matemáticas(INMABB), Universidad Nacional del Sur (UNS)-CONICET, 8000 Bahía Blanca, Argentina