The scalar $T1$ theorem for pairs of doubling measures fails for Riesz transforms when p not 2

TL;DR

The paper shows that for $p \neq 2$ the scalar two-weight $T1$ theorem fails for an individual Riesz transform even on pairs of doubling measures with nearly Lebesgue doubling constants. It then proves that the vector (quadratic) two-weight inequality for the Riesz transform is characterized by a quadratic Muckenhoupt condition $A_p^{\ell^2,local}$ together with a quadratic testing condition, removing the need for a quadratic weak-boundedness property. The authors also demonstrate that the full testing condition cannot be reduced to a scalar $A_p$ in the doubling setting, via a remodeling/construction based on Kakaroumpas–Treil that preserves scalar $A_p$ but inflates the quadratic $A_p$. Finally, they establish that in the plane the quadratic Muckenhoupt condition controls the norm inequality for the vector Riesz transform $\mathbf{R}$ and provide a detailed counterexample showing the necessity of quadratic phenomena for the scalar case, clarifying the boundary between scalar and vector two-weight theories.

Abstract

We show that for an individual Riesz transform in the setting of doubling measures, the scalar $T1$ theorem fails when $p \neq 2$: for each $ p \in (1, \infty) \setminus \{2\}$, we construct a pair of doubling measures $(σ, ω)$ on $\mathbb{R}^2$ with doubling constant close to that of Lebesgue measure that also satisfy the scalar $\mathcal{A}_p$ condition and the full scalar $L^p$-testing conditions for an individual Riesz transform $R_j$, and yet $\left ( R_j \right )_σ : L^p (σ) \not \to L^p (ω)$. On the other hand, we improve upon the quadratic, or vector-valued, $T1$ theorem of Sawyer-Wick when $p \neq 2$ 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 $L^p$ norm inequality for the vector Riesz transform is characterized by a quadratic Muckenhoupt condition $A_{p} ^{\ell^2, \operatorname{local}}$, 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 $A_p$ condition when the measures are doubling, contrary to reports in the literature.

Figures (6)

• Figure 1: Picture of $\overline{\mathbf{I}}$ and $\mathbf{K}_{\mathbf{I}}$
• Figure 2: Graphs of the truncated random walks $\Sigma_{I,1}$, $\Sigma_{I,2}$ and $\Sigma_{I,3}$
• Figure 3: Two generations of stopping intervals when $d=2$
• Figure 4: Remodeling of a dyadic step function with step size $\frac{1}{4}$
• Figure 5: Graph of $\operatorname{Osc}^{\left [\frac{1}{2},1 \right )}_{k_2}$ relative to stopping and transition intervals.

Theorems & Definitions (67)

• Theorem 1
• Remark 2
• Theorem 3: NTV2Neu
• proof : Proof sketch
• Theorem 4
• Remark 5
• Remark 6
• Remark 7
• Lemma 8: Fefferman-Stein type inequalities
• Remark 9