The scalar $T1$ theorem for pairs of doubling measures fails for Riesz transforms when p not 2
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| Format: | Preprint |
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2023
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| _version_ | 1866911258630946816 |
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| author | Alexis, Michel Luna-Garcia, José Luis Sawyer, Eric Uriarte-Tuero, Ignacio |
| author_facet | Alexis, Michel Luna-Garcia, José Luis Sawyer, Eric Uriarte-Tuero, Ignacio |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_15739 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The scalar $T1$ theorem for pairs of doubling measures fails for Riesz transforms when p not 2 Alexis, Michel Luna-Garcia, José Luis Sawyer, Eric Uriarte-Tuero, Ignacio Classical Analysis and ODEs Functional Analysis 42B20 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. |
| title | The scalar $T1$ theorem for pairs of doubling measures fails for Riesz transforms when p not 2 |
| topic | Classical Analysis and ODEs Functional Analysis 42B20 |
| url | https://arxiv.org/abs/2308.15739 |