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On the optimal numerical parameters related with two weighted estimates for commutators of classical operators and extrapolation results

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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 parameters 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-trivial 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.

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Acknowledgements

The authors are supported by CONICET and UNL, UNNE and UNS respectively.

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Authors and Affiliations

  1. Departamento de Matemática, Facultad de Ingeniería Química, CONICET, UNL, 3000, Santa Fe, Argentina

    Gladis Pradolini

  2. Departamento de Matemática, Facultad de Ciencias Exactas y Naturales y Agrimensura, CONICET, UNNE, 3400, Corrientes, Argentina

    Wilfredo Ramos

  3. Departamento de Matemáticas, Instituto de Matemáticas(INMABB), Universidad Nacional del Sur (UNS)-CONICET, 8000, Bahía Blanca, Argentina

    Jorgelina Recchi

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  1. Gladis Pradolini
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  3. Jorgelina Recchi
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Correspondence to Jorgelina Recchi.

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Pradolini, G., Ramos, W. & Recchi, J. On the optimal numerical parameters related with two weighted estimates for commutators of classical operators and extrapolation results. Collect. Math. 72, 229–259 (2021). https://doi.org/10.1007/s13348-020-00287-1

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