  We study the vertical and conical square functions defined via elliptic
operators in divergence form. In general, vertical and conical square functions
are equivalent operators just in $L^2$. But when this square functions are
defined through the heat or Poisson semigroup that arise from an elliptic
operator, we are able to find open intervals containing $2$ where the
equivalence holds. The intervals in question depend ultimately on the range
where the semigroup is uniformly bounded or has off-diagonal estimates. As a
consequence we obtain new boundedness results for some square functions.
Besides, we consider a non-tangential maximal function associated with the
Poisson semigroup and extend the known range where that operator is bounded.
Our methods are based on the use of extrapolation for Muckenhoupt weights and
change of angle estimates. All our results are obtained in the general setting
of a degenerate elliptic operator, where the degeneracy is given by an $A_2$
weight, in weighted Lebesgue spaces. Of course they are valid in the unweighted
and/or non-degenerate situations, which can be seen as special cases, and
provide new results even in those particular settings.
  We also consider the square root of a degenerate elliptic operator in
divergence form $L_w$ and improve the lower bound of the interval where this
operator is known to be bounded on $L^p(vdw)$. To finish we give unweighted
boundedness results for the degenerate operators under consideration.
