Year: 2026
Type: preprint
Abstract: Lacey used sparse domination to study the sharp weighted norm estimate of the maximal function of predictable multipliers in discrete time filtration spaces. Domelevo, Petermichl, and Škreb developed the self similarity argument known as sparse domination in an abstract martingale setting with a continuous time parameter. In our investigation, we establish sparse domination for discrete-time martingale transforms, introducing the novel concept of conditional sparsity as a core property of our approach. The conditional sparsity framework enables derivation of sharp weighted estimates and a mixed-norm estimate \( A_p^αA_r^β\) that improves upon known sharp \( L^p \) bounds. Moreover, we develop dedicated sparse domination specifically for Doob's maximal operator, recovering the sharp bound as a direct application. Finally, we focus on the application of sparse theory to quantitative two-weight estimates.
Source: arXiv (Cornell University)
Authors: Chen, Wei, Zhang, Chaoyue, Zhang, Gege
Institution: Yangzhou University
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G6740207049
OpenAlex ID: W7204483248
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