[Paper Review] Sharp Weighted $L^2$ inequalities for square functions

Rodrigo Bañuelos, Adam Osękowski|arXiv (Cornell University)|Mar 16, 2016

Advanced Harmonic Analysis Research28 references3 citations

TL;DR

This paper establishes sharp weighted $L^2$ inequalities for dyadic square functions using the Bellman function method, achieving optimal dependence on the $A_2$ characteristic of the weight. It provides explicit constants and extends results to classical Lusin and Littlewood-Paley square functions in both analytic and probabilistic settings.

ABSTRACT

Using Bellman function approach, we present new proofs of weighted $L^2$ inequalities for square functions, with the optimal dependence on the $A_2$ characteristics of the weight and further explicit constants. We study the estimates both in the analytic and probabilistic context, and, as application, obtain related estimates for the classical Lusin and Littlewood-Paley square functions.

Motivation & Objective

• To establish sharp weighted $L^2$ bounds for dyadic square functions with optimal dependence on the $A_2$ characteristic of the weight.
• To provide new proofs of known weighted inequalities using the Bellman function technique, ensuring explicit constants.
• To extend the results to classical Lusin and Littlewood-Paley square functions in both analytic and probabilistic frameworks.
• To analyze the behavior of square functions on Riemannian manifolds with non-negative Ricci curvature, linking heat kernel bounds to weighted inequalities.
• To unify and refine existing results on weighted square function estimates by achieving sharpness in the $A_2$ norm dependence.

Proposed method

• Employ the Bellman function approach to construct special functions that satisfy majorization and concavity properties required for sharp $L^2$ estimates.
• Use the dyadic square function defined via Haar functions on $[0,1]$, with projections onto Haar systems and $L^2$-norm comparisons.
• Derive weighted inequalities by analyzing the $A_2$ characteristic $[w]_{A_2}$ and proving that the optimal exponent is 1 for the upper bound and $1/2$ for the reverse inequality.
• Apply the Bellman function method to martingale square functions and extend the results to continuous-time settings via Skorokhod embeddings.
• Establish pointwise inequalities between square functions and their weighted variants using heat kernel bounds and Riemannian geometry.
• Use the semigroup property and the carré du champ identity to relate the square function to the gradient of the heat semigroup on manifolds.

Experimental results

Research questions

• RQ1 What is the optimal dependence of the weighted $L^2$ norm of the square function on the $A_2$ characteristic of the weight?
• RQ2 Can the Bellman function method be used to derive sharp constants in weighted square function inequalities?
• RQ3 How do the sharp estimates for dyadic square functions extend to classical square functions such as Lusin and Littlewood-Paley?
• RQ4 What is the role of geometric assumptions like non-negative Ricci curvature in extending weighted square function inequalities to Riemannian manifolds?
• RQ5 How do heat kernel bounds and volume growth conditions affect the weighted $A_p$-type conditions on manifolds?

Key findings

• The paper establishes the sharp weighted $L^2$ inequality $\|S(\varphi)\|_{L^2_w} \leq C[w]_{A_2} \|\varphi\|_{L^2_w}$ with the optimal exponent 1 on the $A_2$ characteristic.
• It proves the reverse inequality $\|\varphi\|_{L^2_w} \leq C[w]_{A_2}^{1/2} \|S(\varphi)\|_{L^2_w}$, showing that the exponent $1/2$ is also optimal.
• The Bellman function method yields explicit constants in the inequalities, improving upon previous results that only gave qualitative sharpness.
• The results are extended to the classical Lusin and Littlewood-Paley square functions, confirming their sharp weighted $L^2$ bounds.
• On complete Riemannian manifolds with non-negative Ricci curvature, the paper proves that the weighted square function inequalities hold under the heat kernel $A_2$ condition, with the same sharp dependence on $[w]_{A_2}$.
• The paper shows that the condition $\sup_x p_t(x,x) \to 0$ as $t \to \infty$ ensures the validity of the inequalities on manifolds, and this holds automatically if the volume growth is at least $r^n$.