Equivalence of sparse and Carleson coefficients for general sets

We show 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.

Keywords

Carleson coefficients, sparse coefficients

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The author is supported by the Academy of Finland through funding of his postdoctoral researcher post (Funding Decision No 297929). He is a member of the Finnish Centre of Excellence in Analysis and Dynamics Research.

Received 9 October 2017

Accepted 19 February 2018

Published 24 May 2022