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 C^1,δ curve in ℝⁿ. Let R^α,n be the α-fractional Riesz transform vector on ℝⁿ. We prove the T1 theorem for R^α,n: namely that R^α,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 R^α,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 R^α,n. Then we apply the main theorem in [SaShUr7] to deduce the T1 theorem above.

让σ和ω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σ和ω之一。让- [R ^α中，n是α -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²（σ）到大号²（σ）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σ和ω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。