A compact extension of Journé's \(T1\) theorem on product spaces (Q6597534)

The authors provide a general theory of compactness for a class of bi-parameter singular integrals. For them, they impose appropriate hypotheses on operators \(T\) in the bi-parameter case since, in general, bi-parameter singular integrals are not compact on \(L^{p}(\mathbb{R}^{n_1} \times \mathbb{R}^{n_2})\). These hypotheses are\N\N-- (H1) \(T\) admits the compact full kernel representation,\N\N-- (H2) \(T\) admits the compact partial kernel representation,\N\N-- (H3) \(T\) satisfies the weak compactness property,\N\N-- (H4) \(T\) satisfies the diagonal CMO condition,\N\N-- (H5) \(T\) satisfies the product CMO condition.\N\NThe main result of this work is the following compact version of the \(T1\) theorem for bi-parameter singular integrals: If \(T\) is a bi-parameter singular integral operator satisfying (H1)--(H5), then \(T\) is compact on \(L^p(\omega)\) for all \(1 < p < \infty\) and all \(\omega \in \mathcal{A}_p (\mathbb{R}^{n_1} \times \mathbb{R}^{n_2})\). In particular, for \(\omega \equiv 1\), this result extends Theorem 2.21 in [\textit{P. Villarroya}, J. Math. Pures Appl. (9) 104, No. 3, 485--532 (2015; Zbl 1343.42024)] to the case of bi-parameter singular integrals and improves Journé's \(T1\) theorem on product spaces established in [\textit{J.-L. Journé}, Rev. Mat. Iberoam. 1, No. 3, 55--91 (1985; Zbl 0634.42015)] to the corresponding compact version.\N\NThe main tools used in this paper are the weighted interpolation for compact operators, BMO theory, wavelet decomposition, and dyadic analysis.