A compact extension of Journé’s 𝑇1 theorem on product spaces

We prove a compact version of the T 1 T1 theorem for bi-parameter singular integrals. That is, if a bi-parameter singular integral operator T T admits the compact full and partial kernel representations, and satisfies the weak compactness property, the diagonal C M O CMO condition, and the product C M O CMO condition, then T T can be extended to a compact operator on L p ( w ) L^p(w) for all 1 > p > ∞ 1>p>\infty and w ∈ A p ( R n 1 × R n 2 ) 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 Journé’s T 1 T1 theorem on product spaces.