New York Journal of Mathematics
Volume 26 (2020), 1232-1272


Gianmarco Brocchi

A sparse quadratic T(1) theorem

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Published: October 28, 2020.
Keywords: Sparse domination, $T(1)$ theorem, Littlewood--Paley square functions, Carleson condition.
Subject: 42B20, 42B25.

We show that any Littlewood--Paley square function S satisfying a minimal Carleson condition is dominated by a sparse form. This implies strong weighted Lp estimates for all Ap weights with sharp dependence on the Ap characteristic. In particular, the Carleson condition and the sparse domination are equivalent. The proof uses random dyadic grids, decomposition in the Haar basis, and a stopping time argument.


The author was supported by the UK Engineering and Physical Sciences Research Council (EPSRC).

Author information

Gianmarco Brocchi:
School of Mathematics
University of Birmingham
B15 2TT, Birmingham, UK