On two-weight norm inequalities for positive dyadic operators

2018/09/27 by Hänninen, Timo S., Verbitsky, Igor E.
#47G40 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Primary 42B25 #Secondary 42B35

paper · doi:10.48550/arxiv.1809.10800

Abstract

Let σ and ω be locally finite Borel measures on ℝ^d, and let p∈(1,∞) and q∈(0,∞). We study the two-weight norm inequality ‖ T(fσ) ‖_L^q(ω)≤ C ‖ f ‖_L^p(σ), for all f ∈ L^p(σ), for both the positive summation operators T=T_λ(⋅ σ) and positive maximal operators T=M_λ(⋅ σ). Here, for a family \λ_Q\ of non-negative reals indexed by the dyadic cubes Q, these operators are defined by T_λ(fσ):=∑_Q λ_Q ⟨ f⟩^σ_Q 1_Q and M_λ(fσ):=sup_Q λ_Q ⟨ f⟩^σ_Q 1_Q, where ⟨ f⟩^σ_Q:=(1)/(σ(Q)) ∫_Q |f| d σ. We obtain new characterizations of the two-weight norm inequalities in the following cases: 1. For T=T_λ(⋅σ) in the subrange q

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