[Paper Review] The boundedness of multilinear Calder\'on-Zygmund operators on weighted and variable Hardy spaces

David Cruz-Uribe, Kabe Moen+1 more|arXiv (Cornell University)|Aug 23, 2017

Advanced Harmonic Analysis Research3 references4 citations

TL;DR

This paper establishes the boundedness of multilinear Calderón-Zygmund operators (m-CZOs) from products of weighted and variable Hardy spaces into weighted Lebesgue or Hardy spaces. Using a novel finite atomic decomposition for weighted Hardy spaces and a multilinear extrapolation theorem, the authors generalize prior results to the weighted and variable exponent settings, proving sharp endpoint estimates under optimal smoothness and cancellation conditions on the kernel.

ABSTRACT

We establish the boundedness of the multilinear Calder\'on-Zygmund operators from a product of weighted Hardy spaces into a weighted Hardy or Lebesgue space. Our results generalize to the weighted setting results obtained by Grafakos and Kalton (Collect. Math. 2001) and recent work by the third author, Grafakos, Nakamura, and Sawano. As part of our proof we provide a finite atomic decomposition theorem for weighted Hardy spaces, which is interesting in its own right. As a consequence of our weighted results, we prove the corresponding estimates on variable Hardy spaces. Our main tool is a multilinear extrapolation theorem that generalizes a result of the first author and Naibo (Differential Integral Equations 2016).

Motivation & Objective

• To extend the boundedness theory of multilinear Calderón-Zygmund operators (m-CZOs) from Lebesgue spaces to products of weighted and variable Hardy spaces.
• To generalize unweighted results of Grafakos and Kalton (2001) and recent work by Nguyen et al. (2016) to the weighted setting.
• To establish sharp conditions on the kernel's smoothness and the operator's cancellation properties for boundedness into weighted Hardy or Lebesgue spaces.
• To prove corresponding estimates in the variable Lebesgue space setting using extrapolation techniques.

Proposed method

• Develop a finite atomic decomposition theorem for weighted Hardy spaces, enabling approximation by finite sums of atoms.
• Introduce a multilinear extrapolation theorem that generalizes prior work to handle weighted and variable exponent spaces.
• Use the atomic decomposition to reduce the boundedness problem to testing on atoms, leveraging the boundedness of m-CZOs on atomic blocks.
• Apply the extrapolation theorem to transfer bounds from weighted Lebesgue spaces to variable Lebesgue spaces via extrapolation from A1 weights.
• Verify that the critical smoothness and cancellation conditions on the kernel are both necessary and sufficient for the target boundedness.
• Use Fatou's lemma in variable Lebesgue spaces to pass from finite atomic sums to general functions in the Hardy spaces.

Experimental results

Research questions

• RQ1 Under what smoothness and cancellation conditions is a multilinear Calderón-Zygmund operator bounded from a product of weighted Hardy spaces into a weighted Lebesgue space?
• RQ2 What is the sharp condition on the kernel's regularity for boundedness into weighted Hardy spaces, and how does it depend on the weights and exponents?
• RQ3 Can the boundedness of m-CZOs on weighted Hardy spaces be extended to the variable exponent setting using extrapolation?
• RQ4 How do the critical indices of the weights and the exponents interact in determining the required smoothness of the kernel?
• RQ5 What role does the finite atomic decomposition of weighted Hardy spaces play in proving boundedness results?

Key findings

• The paper establishes the boundedness of m-CZOs from products of weighted Hardy spaces Hp1(w1) × ⋯ × Hpm(wm) into Lp(w) under the smoothness condition N ≥ max{ ⌊mn(rwk/pk − 1)⌋+ } + (m−1)n for 1 ≤ k ≤ m.
• For boundedness into weighted Hardy spaces Hp(w), the kernel must satisfy N > sw + max{ ⌊mn(rwk/pk − 1)⌋+ } + mn, where sw is the critical index of the weight w.
• The cancellation condition ∫ xαT(a1,…,am)(x)dx = 0 for |α| ≤ sw is necessary and sufficient for target space to be a Hardy space.
• The authors prove that finite sums of (N,∞) atoms are dense in Hp(w) and Hp(·) for sufficiently large N, enabling density arguments in the proofs.
• Using extrapolation, the authors derive boundedness into variable Lebesgue spaces Lq(·) and Hq(·) under the condition that qk(·)/pk ∈B and pk < (qk)−.
• The results generalize previous unweighted theorems and recover them when all weights are trivial (wk = 1), confirming consistency with known results.