[Paper Review] On the Hardy-Littlewood maximal function for the cube

Jean Bourgain|arXiv (Cornell University)|Dec 11, 2012

Advanced Harmonic Analysis Research6 references4 citations

TL;DR

This paper establishes dimension-free $ L^p $ bounds for the Hardy-Littlewood maximal function associated with the cube in $ \mathbb{R}^n $, proving that the operator norm is uniformly bounded for all $ p > 1 $, independent of dimension. The proof relies on a Fourier-analytic decomposition of the cube's indicator function, localization in frequency space, and interpolation between $ L^2 $ and $ L^p $ estimates using tailored multipliers and Gaussian smoothing.

ABSTRACT

It is shown that the Hardy-Littlewood maximal function associated to the cube in $\mathbb R^n$ obeys dimensional free bounds in $L^p$ fir $p>1$. Earlier work only covered the range $p>\frac 32$.

Motivation & Objective

• To establish uniform $ L^p $ operator norm bounds for the Hardy-Littlewood maximal function associated with the $ n $-dimensional cube, independent of dimension $ n $, for all $ p > 1 $.
• To overcome the limitation of prior results that only achieved dimension-free bounds for $ p > \frac{3}{2} $, by exploiting the faster Fourier decay of the cube's indicator function in specific frequency regions.
• To analyze the maximal function via a decomposition of the cube's characteristic function into frequency-localized components using Gaussian smoothing and difference operators.
• To use interpolation between $ L^2 $ and $ L^p $ estimates, with careful control of multiplier norms and decay rates in different conical regions of frequency space.

Proposed method

• Decomposes the cube's indicator function $ 1_B $ as a sum of frequency-localized kernels $ \Omega^{(s)} = 1_B * H_{2^{-s}} - 1_B * H_{2^{-s+1}} $, where $ H $ is the Gaussian kernel.
• Applies a Fourier multiplier estimate (Lemma 1) to bound the $ L^2 $ operator norm of the maximal function associated with each $ \Omega^{(s)} $, yielding $ \| \sup_t |f * (\Omega^{(s)})_t| \|_2 \leq C 2^{-s/2} \|f\|_2 $.
• Uses interpolation between $ L^2 $ and $ L^p $ estimates to control the maximal function on $ L^p $, with the key step being the control of the multiplier $ \prod_{i=1}^n \hat{\eta}(t\xi_i)(1 - e^{-t_0^2|\xi|^2}) $, where $ \hat{\eta} $ is the Fourier transform of a smooth cutoff.
• Employs a dyadic decomposition in frequency space, isolating regions where $ \xi $ is aligned with coordinate axes (worst-case decay) and exploiting rapid decay elsewhere.
• Applies a randomization trick via $ \varepsilon \in \{1, -1\}^n $ to control the $ L^p $ norm of a square function involving localized operators $ \Gamma_i $, using Khintchine-type inequalities.
• Establishes $ L^2 $ bounds for the error terms via pointwise decay of Fourier multipliers, showing they are $ O(R^{-\varepsilon}) $, and interpolates with $ L^\infty $ to obtain $ L^p $ bounds with $ R^{-\varepsilon/p} $ decay.

Experimental results

Research questions

• RQ1 Can the Hardy-Littlewood maximal function for the cube in $ \mathbb{R}^n $ be bounded in $ L^p $ with a constant independent of dimension for all $ p > 1 $?
• RQ2 Why do previous methods based on $ L^2 $ estimates fail to extend beyond $ p > \frac{3}{2} $, and how can faster Fourier decay in conical regions be exploited to improve this range?
• RQ3 To what extent does the structure of the cube's Fourier transform $ \hat{1}_B(\xi) = \prod_{j=1}^n \frac{\sin \pi \xi_j}{\pi \xi_j} $ allow for better $ L^p $ bounds than general isotropic convex bodies?
• RQ4 Can interpolation techniques between $ L^2 $ and $ L^p $, combined with frequency localization, yield uniform bounds when the worst-case decay $ |\hat{1}_B(\xi)| \lesssim |\xi|^{-1} $ is only present in narrow regions?

Key findings

• The Hardy-Littlewood maximal function associated with the $ n $-dimensional cube satisfies $ \|M_B f\|_p \leq C_p \|f\|_p $ for all $ p > 1 $, with $ C_p $ independent of $ n $, resolving a long-standing question in dimension-free harmonic analysis.
• The proof shows that the maximal function's $ L^2 $ operator norm decays as $ 2^{-s/2} $ for each frequency-localized component $ \Omega^{(s)} $, enabling effective interpolation to $ L^p $.
• The key innovation lies in exploiting the fact that the worst-case Fourier decay $ |\hat{1}_B(\xi)| \lesssim |\xi|^{-1} $ only occurs in narrow conical regions along the coordinate axes, allowing for localized analysis.
• The error terms arising from Gaussian smoothing and frequency truncation are shown to be $ O(R^{-\varepsilon}) $ in $ L^2 $, leading to $ O(R^{-\varepsilon/p}) $ decay in $ L^p $, which is sufficient for interpolation.
• The final bound $ A_p(R) < C_p(\varepsilon) R^{8\varepsilon} $ is obtained, and since $ \varepsilon > 0 $ is arbitrary, the constant is independent of $ R $, implying uniform $ L^p $ bounds.
• This result contrasts sharply with the weak-type $ (1,1) $ case, where $ C_1(B_\infty^{(n)}) \gtrsim (\log n)^{1-\varepsilon} $, showing that $ L^p $ bounds for $ p > 1 $ are fundamentally more robust than weak-type estimates for the cube.