2023/03/25 by Daewon Chung, Chung, Daewon, Weiyan Huang +7
Computer Science · Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.48550/arxiv.2303.14556
openalex publication_date 2023/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we provide necessary and sufficient conditions on a triple of weights (u,v,w) so that the t-Haar multipliers Ttw,σ, t∈ \R, %defined in \citeP when σ=1, are uniformly (on the choice of signs σ) bounded from L2(u) into L2(v). These dyadic operators have symbols s(x,I)=σI (w(x)/⟨ w⟩I)t which are functions of the space variable x∈\R and the frequency variable I∈ D, making them dyadic analogues of pseudo-differential operators. Here D denotes the dyadic intervals, σI=±1, and ⟨ w⟩I denotes the integral average of w on I. When w≡ 1 we have the martingale transform and our conditions recover the known two-weight necessary and sufficient conditions of Nazarov, Treil and Volberg. %We will discuss some relations between the three weights inequality for these operators given the inequality for other dyadic operators. We also show how these conditions are simplified when u=v. In particular, the martingale one-weight and the t-Haar multiplier unsigned and unweighted (corresponding to σI≡ 1 and u=v≡ 1) known results are recovered or improved. We also obtain necessary and sufficient testing conditions of Sawyer type for the two-weight boundedness of a single variable Haar multiplier similar to those known for the martingale transform.