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, for $p_0\in [1,2)$ and $q_0\in (2,\infty ]$ , we prove an estimate of the form where $q_0^{\ast }$ is the Hölder conjugate of $q_0/2$ , M is the underlying doubling space and $S$ 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 $L^p\left(w\right)$ .