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The two weight T 1 theorem for fractional Riesz transforms when one measure is supported on a curve
Journal d'Analyse Mathématique ( IF 0.8 ) Pub Date : 2021-01-26 , DOI: 10.1007/s11854-020-0141-4
Eric T. Sawyer , Chun-Yen Shen , Ignacio Uriarte-Tuero

Let σ and ω be locally finite positive Borel measures on ℝn. We assume that at least one of the two measures σ and ω is supported on a regular C1,δ curve in ℝn. Let Rα,n be the α-fractional Riesz transform vector on ℝn. We prove the T1 theorem for Rα,n: namely that Rα,n is bounded from L2(σ) to L2(σ) if and only if the \({\cal A}_2^\alpha \) conditions with holes hold, the punctured \(A_2^\alpha \) conditions hold, and the cube testing condition for Rα,n and its dual both hold. The special case of the Cauchy transform, n = 2 and α = 1, when the curve is a line or circle, was established by Lacey, Sawyer, Shen, Uriarte-Tuero and Wick in [LaSaShUrWi]. This T1 theorem represents essentially the most general T1 theorem obtainable by methods of energy reversal. More precisely, for the pushforwards of the measures σ and ω, under a change of variable to straighten out the curve to a line, we use reversal of energy to prove that the quasienergy conditions in [SaShUr7] are implied by the \({\cal A}_2^\alpha \) with holes, punctured \(A_2^\alpha \), and quasicube testing conditions for Rα,n. Then we apply the main theorem in [SaShUr7] to deduce the T1 theorem above.



中文翻译:

当曲线上支持一种度量时,分数Riesz变换的两个权重T 1定理

σω上ℝ局部有限的积极博雷尔措施ñ。我们假设measures n中的规则C 1,δ曲线上至少支持两个度量σω之一。让- [R α中,nα -fractional中Riesz变换矢量上ℝ Ñ。我们证明了Ť 1定理[R α中,n:即[R α中,n是从有界大号2σ)到大号2σ)当且仅当所述\({\ CAL A} _2 ^ \阿尔法\)具有孔的条件成立,则刺破\(A_2 ^ \阿尔法\)的条件成立,而对于立方体试验条件- [R α中,n和它的对偶两者都成立。Lauchy,Sawyer,Shen,Uriarte-Tuero和Wick在[LaSaShUrWi]中建立了Cauchy变换的特殊情况,当曲线为直线或圆时,n = 2和α = 1。该T 1定理实质上表示通过能量逆转方法可获得的最通用的T 1定理。更确切地说,对于度量σω的前推,在改变变量以将曲线拉直至一条直线的情况下,我们使用能量逆转来证明[SaShUr7]中的准能量条件由带孔的\({\ cal A} _2 ^ \ alpha \)隐含,打孔\(A_2 ^ \阿尔法\) ,和用于quasicube测试条件- [R α,N。然后,我们应用[SaShUr7]中的主定理来推导上面的T 1定理。

更新日期:2021-01-28
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