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arXiv · 2303.10965

A compact extension of Journ\'{e}'s $T1$ theorem on product spaces

Mingming Cao·Kôzô Yabuta·Dachun Yang
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Abstract

We prove a compact version of the $T1$ theorem for bi-parameter singular integrals. That is, if a bi-parameter singular integral operator $T$ admits the compact full and partial kernel representations, and satisfies the weak compactness property, the diagonal $\mathrm{CMO}$ condition, and the product $\mathrm{CMO}$ condition, then $T$ can be extended to a compact operator on $L^p(w)$ for all $1<p<\infty$ and $w \in A_p(\mathbb{R}^{n_1} \times \mathbb{R}^{n_2})$. Even in the unweighted setting, it is the first time to give a compact extension of Journ\'{e}'s $T1$ theorem on product spaces.

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Mingming Cao, Kôzô Yabuta, Dachun Yang. 2023-03-20. A compact extension of Journ\'{e}'s $T1$ theorem on product spaces. https://doi.org/10.1090/tran%2F9206

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