2023/08/30 by Michel Alexis, Alexis, Michel, José Luis Luna-Garcia +5
Mathematics · #42B20 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2308.15739
openalex publication_date 2023/08/30 · openalex created_date 2023/09/02 · openalex updated_date 2026/07/28
We show that for an individual Riesz transform in the setting of doubling measures, the scalar T1 theorem fails when p ≠ 2: for each p ∈ (1, ∞) ∖ \2\, we construct a pair of doubling measures (σ, ω) on ℝ2 with doubling constant close to that of Lebesgue measure that also satisfy the scalar Ap condition and the full scalar Lp-testing conditions for an individual Riesz transform Rj, and yet ( Rj )σ : Lp (σ) \not → Lp (ω). On the other hand, we improve upon the quadratic, or vector-valued, T1 theorem of Sawyer-Wick when p ≠ 2 on pairs of doubling measures: we dispense with their vector-valued weak boundedness property to show that for pairs of doubling measures, the two-weight Lp norm inequality for the vector Riesz transform is characterized by a quadratic Muckenhoupt condition Ap ℓ2, local, and a quadratic testing condition. Finally, in the appendix, we use constructions of Kakaroumpas-Treil to show that the two-weight norm inequality for the maximal function cannot be characterized solely by the Ap condition when the measures are doubling, contrary to reports in the literature.