  In this article we present a new strategy of addressing the (variable
coefficient) thin obstacle problem. Our approach is based on a (variable
coefficient) Carleman estimate. This yields semi-continuity of the vanishing
order, lower and uniform upper growth bounds of solutions and sufficient
compactness properties in order to carry out a blow-up procedure. Moreover, the
Carleman estimate implies the existence of homogeneous blow-up limits along
certain sequences and ultimately leads to an almost optimal regularity
statement. As it is a very robust tool, it allows us to consider the problem in
the setting of Sobolev metrics, i.e. the coefficients are only $W^{1,p}$
regular for some $p>n+1$. These results provide the basis for our further
analysis of the free boundary, the optimal ($C^{1,1/2}$-) regularity of
solutions and a first order asymptotic expansion of solutions at the regular
free boundary which is carried out in a follow-up article in the framework of
$W^{1,p}$, $p>2(n+1)$, regular coefficients.
