[Paper Review] Intrinsic square functions with arbitrary aperture

Pavel Zorin‐Kranich|arXiv (Cornell University)|May 10, 2016

Advanced Harmonic Analysis Research24 references3 citations

TL;DR

This paper establishes sharp weighted bounds for intrinsic square functions with arbitrary aperture $β \geq 1$ on spaces of homogeneous type, using log-Dini continuous test functions. It proves optimal $L^p$ and weak-type estimates with explicit dependence on the aperture $β$, the Dini/log-Dini modulus of continuity, and the doubling constant, achieving optimal dependence on $γ = \sup_x \mu(B(x,\beta t))/\mu(B(x,t))$, which scales as $\beta^D$ in Euclidean space.

ABSTRACT

We consider intrinsic square functions defined using (log-)Dini continuous test functions on spaces of homogeneous type. We prove weighted estimates with optimal (at least in the Euclidean case) dependence on the aperture of the cone used to define the square function and linear dependence on the (log-)Dini modulus of continuity.

Motivation & Objective

• To establish weighted $L^p$ and weak-type estimates for intrinsic square functions with arbitrary aperture $\beta \geq 1$ on spaces of homogeneous type.
• To quantify the dependence of these estimates on the aperture $\beta$, the modulus of continuity $\omega$, and the doubling constant of the measure.
• To prove that the dependence on $\beta$ is optimal in the Euclidean case, particularly for power weights.
• To extend previous results for $\beta = 1$ to general $\beta$ by avoiding reduction to the $\beta = 1$ case.
• To derive sparse domination and two-weight estimates for intrinsic square functions with arbitrary aperture.

Proposed method

• Define the intrinsic square function $G_{\omega,\beta}f(x)$ using a supremum over test functions with log-Dini regularity, supported in balls of radius $\kappa^k$, with $L^1$-normalization and Hölder control via $\omega$.
• Use dyadic decomposition and sparse domination techniques to control the square function via sparse operators, leveraging known sparse operator bounds.
• Establish weak-type $(1,1)$ estimates via sparse domination and apply known sparse operator estimates to derive $L^p$ and weak-type $L^p$ bounds.
• Use the doubling and reverse doubling properties of the measure to control the growth of the measure of balls under scaling by $\beta$, leading to the quantity $\gamma = \sup_x \mu(B(x,\beta t))/\mu(B(x,t))$.
• Prove optimality of the aperture dependence by constructing a counterexample on $\mathbb{R}^d$ with power weights, showing the exponent of $\gamma$ cannot be improved.
• Apply metrization theorems to extend results from metric to quasimetric spaces, noting only notational changes are required.

Experimental results

Research questions

• RQ1 What is the sharp dependence of weighted $L^p$ and weak-type estimates for intrinsic square functions on the aperture $\beta$?
• RQ2 Can the dependence on the aperture $\beta$ be improved beyond the $\gamma^{1/2}$ factor, where $\gamma = \sup_x \mu(B(x,\beta t))/\mu(B(x,t))$?
• RQ3 Is the log-Dini norm of the modulus of continuity $\omega$ necessary for the weak-type estimate, or can it be replaced by the Dini norm under additional geometric assumptions?
• RQ4 How does the intrinsic square function with arbitrary aperture compare to the classical case $\beta = 1$ in terms of weighted norm bounds?
• RQ5 Can the sparse domination approach be used to derive two-weight $L^p$ and weak-type estimates for intrinsic square functions with arbitrary aperture?

Key findings

• The weighted $L^2$ estimate satisfies $\int (G_{\omega,\beta}f)^2 v \lesssim \Phi \|\omega\|_{\mathrm{Dini}} \int |f|^2 Mv$, with optimal dependence on the Dini norm and $\Phi$, and no additional dependence on $\beta$ beyond $\Phi$.
• The weak-type estimate satisfies $\sup_\lambda \lambda v\{G_{\omega,\beta}f > \lambda\} \lesssim \gamma^{1/2} \|\omega\|_{\mathrm{log-Dini}} \int |f| Mv$, with $\gamma = \sup_x \mu(B(x,\beta t))/\mu(B(x,t))$, showing optimal dependence on the aperture.
• In the Euclidean case $\mathbb{R}^d$, $\gamma \sim \beta^d$, and the dependence on $\beta$ is optimal, as shown by a counterexample with power weights $w(x) = |x|^\alpha$.
• The two-weight $L^p$ and weak-type $L^p$ estimates in Corollary 1.7 show that the dependence on $\gamma^{1/2}$ and $\|\omega\|_{\mathrm{log-Dini}}$ is sharp, with explicit dependence on $[w,\sigma]_{A_p}$, $[w]_{A_\infty}$, and $[\sigma]_{A_\infty}$.
• Under the reverse doubling condition, the log-Dini norm in the weak-type estimate can be replaced by the Dini norm, improving the dependence in certain geometric settings.
• The dependence on $\beta$ in the $L^p$ operator norm is optimal for power weights on $\mathbb{R}^d$, as shown in Appendix B by constructing a function $f$ and weight $w$ such that $\|G_{\omega,\beta}f\|_{L^{p,\infty}(w)} \gtrsim \beta^{-dp/2 + d + \alpha/2}$, matching the predicted power of $\gamma = \beta^d$.