We establish borderline regularity for solutions of the Beltrami
equation $f\sb z-\mu f\sb {\overline {z}}=0$ on the plane, where $\mu$
is a bounded measurable function, $\parallel\mu\parallel\sb
\infty=k<1$. What is the minimal requirement of the type $f\in W
\sp {1,q}\sb {{\rm loc}}$ which guarantees that any solution of the
Beltrami equation with any $\parallel\mu\parallel\sb \infty=k<1$ is
a continuous function? A deep result of K. Astala says that $f\in W
\sp {1,1+k+\varepsilon}\sb {{\rm loc}}$ suffices if
$\varepsilon>0$. On the other hand, O. Lehto and T. Iwaniec showed
that $q<1+k$ is not sufficient. In [2], the following question was
asked: What happens for the borderline case $q=1+k$? We show that the
solution is still always continuous and thus is a quasiregular
map. Our method of proof is based on a sharp weighted estimate of the
Ahlfors-Beurling operator. This estimate is based on a sharp weighted
estimate of a certain dyadic singular integral operator and on using
the heat extension of the Bellman function for the problem. The sharp
weighted estimate of the dyadic operator is obtained by combining
J. Garcia-Cuerva and J. Rubio de Francia's extrapolation technique and
two-weight estimates for the [26].