[Paper Review] A note on two weight bounds for the generalized Hardy-Littlewood Maximal operator

Amalia Culiuc|arXiv (Cornell University)|Jun 23, 2015

Advanced Harmonic Analysis Research3 references3 citations

TL;DR

This paper provides a concise proof of two-weight norm inequalities for a generalized Hardy-Littlewood maximal operator under Sawyer-type testing conditions, using the Martingale Carleson Embedding Theorem. The key result establishes sharp operator norm bounds in terms of the testing constant, with explicit dependence on the exponent $ p $ and its H"older conjugate $ p' $, extending prior results to general nonnegative weight functions and finite $ q $.

ABSTRACT

We give a straighforward proof of the two weight estimates of the generalized maximal operator under Sawyer type testing conditions. The proof relies on the Martingale Carleson Embedding Theorem.

Motivation & Objective

• To establish two-weight weak-type and strong-type estimates for a generalized maximal operator $ M_a^q $ with nonnegative weight functions $ a_Q $.
• To extend Sawyer-type testing conditions to the generalized maximal operator framework, including cases with $ q < ∞ $.
• To provide a streamlined proof using the Martingale Carleson Embedding Theorem, avoiding complex averaging arguments.
• To derive sharp operator norm bounds in terms of the testing constant $ B $, with explicit dependence on $ p $ and $ p' $.
• To unify and generalize previous results on dyadic and martingale maximal operators under a common two-weight testing condition.

Proposed method

• Define the generalized maximal operator $ M_a^q $ as a dyadic, weighted sum over cubes $ Q \in \mathcal{D} $, with $ a_Q(x) $ as nonnegative weight functions.
• Introduce truncated operators $ M_{a,Q}^q $ to localize the analysis to dyadic cubes and apply stopping time techniques.
• Construct a stopping time decomposition using a parameter $ r > 1 $, generating a collection $ \mathcal{G} $ of stopping cubes satisfying $ \sum_{R \subset Q, R \in \mathcal{G}} \mu(R) \leq \frac{r}{r-1} \mu(Q) $.
• Apply the Martingale Carleson Embedding Theorem to control the $ L^p(\nu) $ norm of the maximal function via the testing condition.
• Use $ \ell^p $-norm comparison for $ q \geq p $ to reduce the $ \ell^q $-based maximal function to an $ \ell^p $-sum, enabling application of the embedding theorem.
• Optimize over $ r > 1 $ to minimize the operator norm constant, achieving the sharp bound at $ r = \frac{p+1}{p} $.

Experimental results

Research questions

• RQ1 Under what conditions is the generalized maximal operator $ M_a^q $ bounded from $ L^p(\mu) $ to $ L^p(\nu) $ for $ 1 < p \leq q \leq \infty $?
• RQ2 Can the two-weight norm inequality for $ M_a^q $ be characterized by a Sawyer-type testing condition on truncated operators?
• RQ3 What is the sharp dependence of the operator norm on the testing constant $ B $, and how does it vary with $ p $?
• RQ4 Can the proof be simplified by avoiding averaging arguments and instead using the Martingale Carleson Embedding Theorem?
• RQ5 How does the generalized operator $ M_a^q $, with nonnegative $ a_Q $, extend classical results for the dyadic maximal function?

Key findings

• The operator $ M_a^q $ is bounded from $ L^p(\mu) $ to $ L^p\nu) $ if and only if the truncated operator satisfies the Sawyer-type testing condition $ \|M_{a,Q}^q(\mathbf{1}_Q \mu)\|_{L^p(\nu)} \leq B \mu(Q)^{1/p} $ for all $ Q \in \mathcal{D} $.
• The best constants $ A $ and $ B $ in the two-weight inequality satisfy $ B \leq A \leq C(p) B $, where $ C(p) = \left(\left(1 + \frac{1}{p}\right)^{p+1} p\right)^{1/p} p' $.
• The constant $ C(p) $ is sharp and decreases to 1 as $ p \to \infty $, indicating asymptotic optimality.
• For $ p = q = \infty $, the result holds trivially with $ A = B $, consistent with the classical case.
• The proof technique avoids complex averaging arguments and instead relies on the Martingale Carleson Embedding Theorem, simplifying prior approaches.
• The final operator norm bound is independent of the truncation parameter $ N $, allowing passage to the full maximal operator via limit as $ N \to \infty $.