We develop the compactness theory of multilinear singular integrals on product spaces using a modern point of view. The first main result is a compact $T1$ theorem for multilinear Calder\'{ o}n--Zygmund operators on product spaces. More specifically, we prove that a multilinear singular integral operator $T$ on product spaces can be extended to a compact multilinear operator from $L^{ p_1}(w_1^{ p_1}) \times \cdots \times L^{ p_m}(w_m^{ p_m})$ to $L^p(w^p)$ for all exponents $\frac1p = \sum_{ j=1}^m \frac{ 1}{ p_j}>0$ with $p_1, \ldots, p_m \in (1, \infty]$ and for all weights $\vec{ w} \in A_{ \vec{ p}}(\mathbb{ R}^{ n_1} \times \mathbb{ R}^{ n_2})$ if the following hypotheses are satisfied: (H1) $T$ admits a compact full kernel representation, (H2) $T$ admits a compact partial kernel representation, (H3) $T$ satisfies the weak compactness property, (H4) $T$ satisfies the diagonal $\mathrm{ CMO}$ condition, and (H5) $T$ satisfies the product $\mathrm{ CMO}$ condition. This is a multilinear compact extension of Journ\'{ e}'s $T1$ theorem on product spaces. The second main result establishes the mean continuity of commutators $[\boldsymbol{ b}, T]_{ \boldsymbol{ \alpha}}$ on weighted Lebesgue spaces as above, which can be viewed as a substitution of compactness because the compactness of $[\boldsymbol{ b}, T]_{ \boldsymbol{ \alpha}}$ is equivalent to $\boldsymbol{ b} \equiv \text{ constant}$ when $T$ is a non-degenerate bi-parameter singular integral. Our main tools include multilinear bi-parameter dyadic representation, multilinear extrapolation, multilinear interpolation, and Kolmogorov--Riesz compactness criterion.
Categories: math.CA