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

Weak and strong types estimates for square functions associated with operators