Quadratic sparse domination and Weighted Estimates for non-integral Square Functions

We prove a quadratic sparse domination result for general non-integral square functions $S$. That is, we prove an estimate of the form \begin{equation*} \int_{M} (S f)^{2} g , $\mathrm{d}$\mu \le c \sum_{P \in $\mathcal{S}$} \left(\frac{1}{\lvert 5P \rvert}\int_{5 P} \lvert f\rvert^{p_{0}} , $\mathrm{d}$\mu\right)^{2/p_{0}} \left(\frac{1}{\lvert 5P \rvert} \int_{5 P} \lvert g\rvert^{q_{0}^},$\mathrm{d}$\mu\right)^{1/q_{0}^} \lvert P\rvert, \end{equation*} where $q_0^{\ast }$ is the H"{o}lder conjugate of $q_0\mathrm{/}2$, $M$ is the underlying doubling space and $\mathcal{S}$ is a sparse collection of cubes on $M$. Our result will cover both square functions associated with divergence form elliptic operators and those associated with the Laplace-Beltrami operator. This sparse domination allows us to derive optimal norm estimates in the weighted space $L^p(w)$.