The Linear Bound for the Natural Weighted Resolution of the Haar Shift

2013/08/24 by Sandra Pott, Maria Carmen Reguera, Pott, Sandra +5
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Analysis and Transform Methods #math.CA #math.CV

paper · pdf · doi:10.48550/arxiv.1308.5349

v1: 21 pages; v2: 22 pages, typos corrected, main results slightly modified

openalex publication_date 2013/08/24 · arxiv created 2014/01/12 · arxiv updated 2014/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Hilbert transform has a linear bound in the A₂ characteristic on weighted L², \Vert H\Vert _{L²(w)→ L²(w)}\lesssim [ w ] _A₂, and we extend this linear bound to the nine constituent operators in the natural weighted resolution of the conjugation M_w^(1)/(2)S M_w^-(1)/(2) induced by the canonical decomposition of a multiplier into paraproducts:% M_f=P_f^-+P_f⁰+P_f⁺. The main tools used are composition of paraproducts, a product formula for Haar coefficients, the Carleson Embedding Theorem, and the linear bound for the square function.

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