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 p0 ∈ [1, 2) and q0 ∈ (2, ∞], we prove an estimate of the form

∫M(S f)2g dμ ≤ c ∑P∈S(−∫5P| f |p0 dμ)2/p0 (−∫5P|g|q∗0 dμ)1/q∗0|P|,

where q∗ 0 is the Hölder conjugate of q0/2, 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(w).