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Session S13 - Harmonic Analysis, Fractal Geometry, and Applications

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Boundedness and compactness for commutators of singular integrals related to a critical radius function

Pablo Quijano

IMAL (UNL - CONICET), Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We work in the general framework of a family of singular integrals with kernels controlled in terms of a critical radius function $\rho$. This family models the harmonic analysis derived from the Schrödinger operator $L= -\Delta+V$, where the non-negative potential $V$ satisfies an appropriate reverse H\"older condition. For their commutators, we find sufficient conditions on the symbols for boundedness and/or compactness when acting on weighted $L^p$ spaces. In all cases, the classes of symbols and weights are larger than their classical counterparts, $\textup{BMO}$, $\textup{CMO}$ and $A_p$. When these general results are applied to the Schrödinger context, we obtain boundedness and compactness for commutators of operators like $\nabla L^{-1/2}$, $\nabla^2 L^{-1}$, $V^{1/2} L^{-1/2}$, $V^{1/2} \nabla L^{-1}$, $VL^{-1}$ and $L^{i\alpha}$. As in Uchiyama's classical paper, we give a full description of the class for compactness, $\textup{CMO}^\infty_\rho$, assuming $\rho$ to be bounded. Finally, we provide examples showing that $\textup{CMO}$ is strictly contained in $\textup{CMO}^\infty_\rho$ for any $\rho$, bounded or not.

Joint work with Bruno Bongioanni IMAL (UNL - CONICET) and and Eleonor Harboure IMAL (UNL - CONICET).

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