Dyadic analysis of compactness on product spaces

TL;DR

This work develops a comprehensive compactness theory for multilinear bi-parameter Calderón–Zygmund operators on product spaces. It introduces a compact dyadic representation that decomposes operators into shifts, partial paraproducts, and full paraproducts across two parameters, and proves weighted compactness under the full set of kernel-cancellation hypotheses. A Rubio de Francia extrapolation framework is established for both compactness and mean continuity of commutators on weighted spaces, and mean continuity is shown to be weaker than compactness in the bi-parameter setting. The results extend Journé’s $T1$ theory to the multilinear, bi-parameter, weighted context and provide robust tools for further study of compactness in multi-parameter harmonic analysis and applications to PDEs.

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 $\frac1p = \sum_{j=1}^m \frac{1}{p_j}>0$ 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 Journé's $T1$ theorem on product spaces. The second main result establishes the mean continuity of commutators $[\boldsymbol{b}, T]_{\boldsymbolα}$ on weighted Lebesgue spaces as above, which can be viewed as a substitution of compactness because the compactness of $[\boldsymbol{b}, T]_{\boldsymbolα}$ 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.

Theorems & Definitions (72)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1
• Definition 2.12