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Multilinear Wavelet Compact T(1) Theorem

Anastasios Fragkos, A. Walton Green, Brett D. Wick

TL;DR

This work addresses the problem of characterizing compactness for multilinear Calderón-Zygmund operators via a $T(1)$-type testing framework. The authors develop a two-pronged approach: (i) a multilinear $T(1)$ characterization of compact CZOs using wavelet-based testing and a weak compactness property, and (ii) a sharp representation theorem showing every compact CZO is a finite linear combination of compact wavelet forms and compact paraproduct forms with $\mathrm{CMO}$-symbols. The key contributions include extending compactness testing to the multilinear setting, proving a representation theorem that decomposes compact CZOs into two building blocks, and establishing essential-norm estimates for paraproducts to control compactness at endpoints. The results unify and extend prior linear and multilinear compactness theories, providing a concrete framework to construct all compact CZOs and analyze their adjoints, with potential implications for endpoint estimates and weighted theories.

Abstract

We prove a wavelet $T(1)$ theorem for compactness of multilinear Calderón-Zygmund (CZ) operators. Our approach characterizes compactness in terms of testing conditions and yields a representation theorem for compact CZ forms in terms of wavelet and paraproduct forms that reflect the compact nature of the operator.

Multilinear Wavelet Compact T(1) Theorem

TL;DR

This work addresses the problem of characterizing compactness for multilinear Calderón-Zygmund operators via a -type testing framework. The authors develop a two-pronged approach: (i) a multilinear characterization of compact CZOs using wavelet-based testing and a weak compactness property, and (ii) a sharp representation theorem showing every compact CZO is a finite linear combination of compact wavelet forms and compact paraproduct forms with -symbols. The key contributions include extending compactness testing to the multilinear setting, proving a representation theorem that decomposes compact CZOs into two building blocks, and establishing essential-norm estimates for paraproducts to control compactness at endpoints. The results unify and extend prior linear and multilinear compactness theories, providing a concrete framework to construct all compact CZOs and analyze their adjoints, with potential implications for endpoint estimates and weighted theories.

Abstract

We prove a wavelet theorem for compactness of multilinear Calderón-Zygmund (CZ) operators. Our approach characterizes compactness in terms of testing conditions and yields a representation theorem for compact CZ forms in terms of wavelet and paraproduct forms that reflect the compact nature of the operator.
Paper Structure (8 sections, 14 theorems, 98 equations)

This paper contains 8 sections, 14 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Wavelets
  3. Analysis of the parameter space
  4. Wavelet classes and forms
  5. Wavelet representation of compact Calderón-Zygmund forms
  6. Compact wavelet forms
  7. Compactness of paraproducts
  8. Main result and proof

Key Result

Theorem 1

Let $T$ be a bilinear CZO, $\Lambda$ the associated trilinear form, and $\mathfrak b_j \in \mathrm{BMO}(\mathbb R^d)$ defined by e:T1. Then, the following are equivalent.

Theorems & Definitions (32)

  • Theorem 1
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • ...and 22 more