Borderline Weak--Type Estimates for Sparse Bilinear Forms Involving A_∞ Maximal Functions

2020/03/02 by Rob Rahm, Rahm, Rob
Computer Science · Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2003.01058

openalex publication_date 2020/03/02 · openalex created_date 2020/03/06 · openalex updated_date 2026/07/28

Abstract

For any operator T whose bilinear form can be dominated by a sparse bilinear form, we prove that T is bounded as a map from L¹(\widetildeMw) into weak--L¹(w). Our main innovation is that \widetildeM is a maximal function defined by directly using the local A_∞ characteristic of the weight (rather than Orlicz norms). Prior results are due to Coifman&Fefferman, Pérez, Hytönen&Pérez, and Domingo-Salazar&Lacey&Rey. As we discuss, but do not prove, the maximal functions we use seem to be on the order of M_L(log log L) (log log log L) (log log log log L)^1+ε.

Citations

• Weighted inequalities for the dyadic square function without dyadic A∞
• Some Maximal Inequalities

Related

• The Sparse T1 Theorem
• Sparse Domination of Singular Bilinear Forms on Non-Homogeneous spaces
• An elementary approach to certain bilinear estimates
• A sparse estimate for multisublinear forms involving vector-valued maximal functions
• Endpoint weak-type bounds beyond Calderón-Zygmund theory