This article was written in 1999, and was posted as a preprint in CRM(Barcelona) preprint series $n^0\, 519$ in 2000. However, recently CRM erasedall preprints dated before 2006 from its site, and this paper becameinacessible. It has certain importance though, as the reader shall see. Formally this paper is a proof of the (qualitative version of the) Vitushkinconjecture. The last section is concerned with the quantitative version. Thisquantitative version turns out to be very important. It allowed Xavier Tolsa toclose the subject concerning Vtushkin's conjectures: namely, using thequantitative nonhomogeneous $Tb$ theorem proved in the present paper, he provedthe semiadditivity of analytic capacity. Another "theorem", which is implicitlycontained in this paper, is the statement that any non-vanishing $L^2$-functionis accretive in the sense that if one has a finite measure $\mu$ on the complexplane ${\mathbb C}$ that is Ahlfors at almost every point (i.e. for$\mu$-almost every $x\in {\mathbb C}$ there exists a constant $M>0$ such that$\mu(B(x,r))\le Mr$ for every $r>0$) then any one-dimensional antisymmetricCalder\'on-Zygmund operator $K$ (e.g. a Cauchy integral type operator)satisfies the following "all-or-nothing" princple: if there exists at least onefunction $\phi\in L^2(\mu)$ such that $\phi(x)\ne 0$ for $\mu$-almost every$x\in {\mathbb C}$ and such that {\it the maximal singular operator}$K^*\phi\in L^2(\mu)$, then there exists an everywhere positive weight $w(x)$,such that $K$ acts from $L^2(\mu)$ to $L^2(wd\mu)$.
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机译：这篇文章写于1999年，并作为预印本张贴在2000年巴塞罗那的CRM（巴塞罗那）预印本系列$ n ^ 0 \，519 $中。但是，最近，CRM从其网站上删除了2006年之前的所有预印本，因此这篇论文变得不必要了。正如读者所看到的，它具有一定的重要性。形式上，本文正式证明了维图什金猜想（定性版本）。最后一部分与定量版本有关。定量版本非常重要。它允许泽维尔·托尔萨（Xavier Tolsa）结束有关伏图什金猜想的话题：即，使用本文证明的定量非齐次$ Tb $定理，他证明了分析能力的半可加性。另一个隐含在本文中的“定理”是这样的陈述：任何不消失的$ L ^ 2 $函数都是有增值作用的，即如果在复平面$ {\ mathbb C上有一个有限量度$ \ mu $ } $在几乎所有点上都是Ahlfors（即，对于$ \ mu $-几乎每个$ x \在{\ mathbb C} $中，存在一个常数$ M> 0 $使得$ \ mu（B（x，r） ）\ r Mr $（每$ r> 0 $），那么任何一维反对称Calder \'on-Zygmund运算符$ K $（例如Cauchy整型运算符）都满足以下“全有或全无”原则：在L ^ 2（\ mu）$中至少存在一个函数$ \ phi \，使得$ \ mu $的$ \ phi（x）\ ne 0 $-{\ mathbb C} $中的几乎每个$ x \ {\它是最大奇异算子} $ K ^ * \ phi \ in L ^ 2（\ mu）$，那么到处都存在正权重$ w（x）$，使得$ K $从$ L ^ 2起作用（\ mu）$到$ L ^ 2（wd \ mu）$。
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