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arXiv · 2309.15819

On the $T1$ theorem for compactness of Calder\'on-Zygmund operators

Mishko Mitkovski·Cody B. Stockdale
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Abstract

We give a new formulation of the $T1$ theorem for compactness of Calder\'on-Zygmund singular integral operators. In particular, we prove that a Calder\'on-Zygmund operator $T$ is compact on $L^2(\mathbb{R}^n)$ if and only if $T1,T^*1\in \text{CMO}(\mathbb{R}^n)$ and $T$ is weakly compact. Compared to existing compactness criteria, our characterization more closely resembles David and Journ\'e's classical $T1$ theorem for boundedness, avoids technical conditions involving the Calder\'on-Zygmund kernel, and follows from a simpler argument.

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Mishko Mitkovski, Cody B. Stockdale. 2023-09-27. On the $T1$ theorem for compactness of Calder\'on-Zygmund operators. https://arxiv.org/abs/2309.15819

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