Dyadic analysis of compactness on product spaces

Abstract

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 Calderón--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 $\frac{1}{p}={\sum }_{j=1}^m\frac{1}{p_j}>0$ with $p_1,\dots ,p_m\in (1,\infty ]$ 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 Journé's $T1$ theorem on product spaces. The second main result establishes the mean continuity of commutators $[\mathbf{b},T{]}_{\hat{\mathbf{I}}\pm }$ on weighted Lebesgue spaces as above, which can be viewed as a substitution of compactness because the compactness of $[\mathbf{b},T{]}_{\hat{\mathbf{I}}\pm }$ is equivalent to $\mathbf{b}\equiv \text{constant}$ 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.