  Let $L$ be a linear operator in $L^2(\mathbb{R}^n)$ which generates a
semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy the Gaussian upper bound.
In this paper, we investigate several kinds of weighted norm inequalities for
the conical square function $S_{\alpha,L}$ associated with an abstract operator
$L$. We first establish two-weight inequalities including bump estimates, and
Fefferman-Stein inequalities with arbitrary weights. We also present the local
decay estimates using the extrapolation techniques, and the mixed weak type
estimates corresponding Sawyer's conjecture by means of a Coifman-Fefferman
inequality. Beyond that, we consider other weak type estimates including the
restricted weak-type $(p, p)$ for $S_{\alpha, L}$ and the endpoint estimate for
commutators of $S_{\alpha, L}$. Finally, all the conclusions aforementioned can
be applied to a number of square functions associated to $L$.
