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 $L^p(w)$.

arXiv comment31 pages

References in corpus (18)

• Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights
• An elementary proof of the $A_2$ Bound
• On pointwise estimates involving sparse operators
• A sparse domination principle for rough singular integrals
• Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, II
• Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, I
• Sharp weighted norm estimates beyond Calderón-Zygmund theory
• A pointwise estimate for positive dyadic shifts and some applications
• Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: Harmonic analysis of elliptic operators
• Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part II: Off-diagonal estimates on spaces of homogeneous type
• Domination of multilinear singular integrals by positive sparse forms
• On weighted norm inequalities for the Carleson and Walsh-Carleson operators
• Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part IV: Riesz transforms on manifolds and weights
• Dyadic harmonic analysis and weighted inequalities: the sparse revolution
• Sparse Bounds for Bochner-Riesz Multipliers
• Sparse domination implies vector-valued sparse domination
• Endpoint sparse bounds for Walsh-Fourier multipliers of Marcinkiewicz type
• 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