On two weighted problems for commutators of classical operators with optimal behaviour on the parameters involved and extrapolation results

TL;DR

This paper establishes weighted norm estimates for higher order commutators of classical operators, identifies optimal parameters for these estimates, and develops an extrapolation method to extend results to variable settings and diverse operators.

Contribution

It provides the first comprehensive analysis of weighted bounds for higher order commutators with optimal parameters and introduces an extrapolation technique for variable exponent spaces.

Findings

Derived weighted norm estimates for higher order commutators.

Identified optimal parameters for the classes of weights and spaces.

Developed an extrapolation result extending classical inequalities to variable settings.

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 Calder\'on-Scott type inequality with weights that connects adequately the spaces involved.