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, we prove an estimate of the form \begin{equation*} \int_{M} (S f)^{2} g \, \mathrm{d}μ\le c \sum_{P \in \mathcal{S}} \left(\frac{1}{\lvert 5P \rvert}\int_{5 P} \lvert f\rvert^{p_{0}} \, \mathrm{d}μ\right)^{2/p_{0}} \left(\frac{1}{\lvert 5P \rvert} \int_{5 P} \lvert g\rvert^{q_{0}^*}\,\mathrm{d}μ\right)^{1/q_{0}^*} \lvert P\rvert, \end{equation*} where $q_0^{\ast }$ is the Hölder conjugate of $q_0/2$, $M$ 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 $L^p(w)$.

31 pages

References in corpus (3)

• Dyadic harmonic analysis and weighted inequalities: the sparse revolution
• Sparse domination implies vector-valued sparse domination
• A sparse quadratic $T1$ theorem

Cited by in corpus (3)

• Operator-free sparse domination
• Multi-scale sparse domination
• On pointwise ${\ell }^r$-sparse domination in a space of homogeneous type