  We give a new proof of the sharp one weight $L^p$ inequality for any operator
$T$ that can be approximated by Haar shift operators such as the Hilbert
transform, any Riesz transform, the Beurling-Ahlfors operator. Our proof avoids
the Bellman function technique and two weight norm inequalities. We use instead
a recent result due to A. Lerner to estimate the oscillation of dyadic
operators. Our method is flexible enough to prove the corresponding sharp
one-weight norm inequalities for some operators of harmonic analysis: the
maximal singular integrals associated to $T$, Dyadic square functions and
paraproducts, and the vector-valued maximal operator of C. Fefferman-Stein.
Also we can derive a very sharp two-weight bump type condition for $T$.

Classical Analysis and ODEs;Functional Analysis