  This article was written in 2005 and subsequently lost (at least by the third
author). Recently it resurfaced due to one of the colleagues to whom a hard
copy has been sent in 2005. We consider here a problem of finding necessary and
sufficient conditions for the boundedness of two weight Calder\'on-Zygmund
operators. We give such necessary and sufficient conditions in very natural
terms, if the operator is the Hilbert transform, and the weights satisfy some
very natural condition. The condition on weights was lifted in a recent paper
of Michael Lacey, Eric Sawyer and Ignacio Uriarte-Tuero: "A characterization of
the two weight norm inequality for the Hilbert transform", arXiv:1001.4043
[math.CA] 31 January 2010. The paper of Lacey--Sawyer-Uriarte-Tuero alliviated
the "pivotal" condition used in a present article and replaced it by the very
interesting and correct energy condition, which, unlike the "pivotal" condition
turned out to be also necessary. The paper of Lacey-Sawyer-Uriarte-Tuero used
the present article in its main aspect. The thrust of the present article is to
use the methods of nonhomogeneous Harmonoc Analysis together with a several
paraproducts arising from a certain stopping time argument. In view of the
importance of the present article for Lacey--Sawyer-Uriarte-Tuero's paper
arXiv:1001.4043 [math.CA] 31 January 2010, we present it to the attention of
the reader. Drawing no parallels, "Darwin spent 1838-1859 getting ready to
publish "On the Origin of Species" without actually publishing it, only
brooding over beaks of finches".
