Equivalence of sparse and Carleson coefficients for general sets

TL;DR

This paper demonstrates the equivalence of sparse and Carleson coefficients for general sets, extending known results from dyadic rectangles to broader collections of Borel sets, with implications for bi-parameter singular integrals.

Contribution

It shows that sparse and Carleson coefficients are equivalent for all countable collections of Borel sets, generalizing previous results to more complex set collections.

Findings

Equivalence of sparse and Carleson coefficients for general Borel sets

Extension of dual reformulation to arbitrary sets

Simplified proof of the reformulation

Abstract

We remark that sparse and Carleson coefficients are equivalent for every countable collection of Borel sets and hence, in particular, for dyadic rectangles, the case relevant to the theory of bi-parameter singular integrals. The key observation is that a dual refomulation by I. E. Verbitsky for Carleson coefficients over dyadic cubes holds also for Carleson coefficients over general sets. We give a simple proof for this reformulation.