On the sense of convergence in the dyadic representation theorem
Tuomas Hytönen
TL;DR
The paper sharpens the dyadic representation for Calderón-Zygmund operators by proving a convergence sense that applies to all pairs $(f,g)\in L^p(\mathbb{R}^d)\times L^{p'}(\mathbb{R}^d)$ under essentially minimal kernel regularity and standard $T(1)$-type hypotheses. The authors develop a finite, explicit Beylkin-Coifman-Rochberg style decomposition with a controllable error term, then average over random dyadic grids to convert off-diagonal interactions into a single-series of generalized dyadic shifts and paraproducts. The main result is a representation formula expressing $\langle Tf,g\rangle$ as an average of Haar multipliers, paraproducts, and a convergent sum of dyadic shifts, valid for all $f\in L^p$ and $g\in L^{p'}$, which in turn yields $L^p$-boundedness and a robust structural decomposition. The framework extends to UMD spaces, indicating a broad vector-valued applicability and connecting to vector-valued paraproduct theory and $R$-boundedness considerations, with potential implications for multiparameter and non-convolution settings.
Abstract
The dyadic representation of any singular integral operator, as an average of dyadic model operators, has found many applications. While for many purposes it is enough to have such a representation for a "suitable class" of test functions, we show that, under quite general assumptions (essentially minimal ones to make sense of the formula), the representation is actually valid for all pairs $(f,g)\in L^p(\mathbb R^d)\times L^{p'}(\mathbb R^d)$, not just test functions.