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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. document.getElementById('cloakce9ced30ebab2a30f3537564f0c15f5f').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyce9ced30ebab2a30f3537564f0c15f5f = 'p&#97;bl&#111;q&#117;&#105;j&#97;n&#111;&#97;r' + '&#64;'; addyce9ced30ebab2a30f3537564f0c15f5f = addyce9ced30ebab2a30f3537564f0c15f5f + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m'; var addy_textce9ced30ebab2a30f3537564f0c15f5f = 'p&#97;bl&#111;q&#117;&#105;j&#97;n&#111;&#97;r' + '&#64;' + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m';document.getElementById('cloakce9ced30ebab2a30f3537564f0c15f5f').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyce9ced30ebab2a30f3537564f0c15f5f + '\'>'+addy_textce9ced30ebab2a30f3537564f0c15f5f+'<\/a>';

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).