Quadratic Sparse Domination and Weighted Estimates for Non-integral Square Functions

Abstract

We prove a quadratic sparse domination result for general non-integral square functions S. That is, for $$p_0 \in [1,2)$$ p 0 ∈ [ 1 , 2 ) and $$q_0 \in (2,\infty ]$$ q 0 ∈ ( 2 , ∞ ] , we prove an estimate of the form "Equation missing"where $$q_{0}^{*}$$ q 0 ∗ is the Hölder conjugate of $$q_{0}/2$$ q 0 / 2 , M is the underlying doubling space and $${\mathcal {S}}$$ 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 $$L^{p}(w)$$ L p ( w ) .